Displaying 1-10 of 32 results found.
1, 2, 4, 6, 12, 18, 24, 30, 48, 60, 78, 90, 120, 150, 180, 210, 330, 360, 390, 420, 630, 840, 1050, 1260, 1470, 1680, 1890, 2100, 2310, 3360, 3570, 3990, 4200, 4620, 5460, 6300, 6930, 9240, 10710, 10920, 11550, 13860, 16380, 17220, 17850, 18480, 20790, 27720, 30030, 39270
COMMENTS
Numbers m such that the smallest solution k to the equation phi(m+k) = phi(k) is larger than all the corresponding smallest solutions for all numbers below m.
The corresponding record values are 1, 4, 8, 24, 48, 52, 96, ... (see the link for more values).
Apparently, a(n) is even for n > 1, divisible by 6 for n > 3, by 30 for n > 9, and by 210 for n > 19. These observations are based on data up to n=100.
It seems that in general, for all k >= 1 there is a number n_k such that all the terms a(n) with n > n_k are divisible by the first k primes.
Furthermore, it seems that all the terms are of the form m*p^e, were m is a term of A055932, and p^e is a prime power ( A000961).
EXAMPLE
The first 6 terms of A007015 are 1, 4, 3, 8, 5 and 24. The record values, 1, 4, 8 and 24 occur at 1, 2, 4 and 6, the first 4 terms of this sequence.
MATHEMATICA
f[n_] := Module[{k = 1}, While[EulerPhi[n + k] != EulerPhi[k], k++]; k]; fm =0; s = {}; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]]; , {n, 1, 1000}]; s
1, 3, 4, 3, 8, 5, 24, 5, 13, 9, 20, 7, 48, 13, 16, 13, 26, 17, 52, 19, 37, 21, 44, 13, 96
Number of terms in the Zeckendorf representation of n (write n as a sum of non-consecutive distinct Fibonacci numbers).
+10
149
0, 1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 3, 1, 2, 2, 2, 3, 2, 3, 3, 1, 2, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 4, 1, 2, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 4, 2, 3, 3, 3, 4, 3, 4, 4, 1, 2, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 4, 2, 3, 3, 3, 4, 3, 4, 4, 2, 3, 3, 3, 4, 3, 4, 4, 3, 4, 4, 4, 5, 1, 2, 2, 2, 3, 2, 3, 3, 2, 3, 3, 3, 4, 2, 3, 3
COMMENTS
Also number of 0's (or B's) in the Wythoff representation of n -- see the Reble link. See also A135817 for references and links for the Wythoff representation for n >= 1. - Wolfdieter Lang, Jan 21 2008; N. J. A. Sloane, Jun 28 2008 Or, a(n) is the number of applications of Wythoff's B sequence A001950 needed in the unique Wythoff representation of n >= 1. E.g., 16 = A(B(A(A(B(1))))) = ABAAB = `10110`, hence a(16) = 2. - Wolfdieter Lang, Jan 21 2008 Let M(0) = 0, M(1) = 1 and for i > 0, M(i+1) = f(concatenation of M(j), j from 0 to i - 1) where f is the morphism f(k) = k + 1. Then the sequence is the concatenation of M(j) for j from 0 to infinity. - Claude Lenormand (claude.lenormand(AT)free.fr), Dec 16 2003
Let m be the number of parts in the listing of the compositions of n into odd parts as lists of parts in lexicographic order, a(k) = (n - length(composition(k)))/2 for all k < Fibonacci(n) and all n (see example).
Let m be the number of parts in the listing of the compositions of n into parts 1 and 2 as lists of parts in lexicographic order, a(k) = n - length(composition(k)) for all k < Fibonacci(n) and all n (see example).
A000120 gives the equivalent for (all) compositions. (End)
a(n) is also the minimum number of Fibonacci numbers which sum to n, regardless of adjacency or duplication. - Alan Worley, Apr 17 2015 This follows from the fact that the sequence is subadditive: a(n+m) <= a(n) + a(m) for nonnegative integers n,m. See Lemma 2.1 of the Stoll link. - Robert Israel, Apr 17 2015 This sequence is a morphic sequence on an infinite alphabet, i.e., (a(n)) is a letter-to-letter projection of a fixed point of a morphism tau.
The alphabet is {0,1,...,j,...}X{0,1}, and tau is given by
tau((j,0)) = (j,0) (j+1,1),
tau((j,1)) = (j,0).
The letter-to-letter map is given by the projection on the first coordinate: (j,i)->j for i=0,1.
To prove this, note first that the second coordinate of the letters generates the infinite Fibonacci word = A003849 = 0100101001001.... This implies that for all n and j one has
|tau^n(j,0)| = F(n+2),
where |w| denotes the length of a word w, and (F(n)) = A000045 are the Fibonacci numbers. Secondly, we need the following simple, but crucial observation. Let the Zeckendorf representation of n be Z(n) = A014417(n). For example, Z(0) = 0, Z(1) = 1, Z(2) = 10, Z(3) = 100, Z(4) = 101, Z(5) = 1000.
From the unicity of the Zeckendorf representation it follows that for the positions i = 0,1,...,F(n)-1 one has
Z(F(n+1)+i) = 10...0 Z(i),
where zeros are added to Z(i) to give the total representation length n-1.
This gives for i = 0,1,...,F(n)-1 that
a(F(n+1)+i) = a(i) + 1.
From the first observation follows that the first F(n+1) letters of tau^n(j,0) are equal to tau^{n-1}(j,0), and the last F(n) letters of tau^n(j,0) are equal to tau^{n-1}(j+1,1) = tau^{n-2}(j+1,0).
Combining this with the second observation shows that the first coordinate of the fixed point of tau, starting from (0,0), gives (a(n)).
It is of course possible to obtain a morphism tau' on the natural numbers by changing the alphabet: (j,0)-> 2j (j,1)-> 2j+1, which yields the morphism
tau'(2j) = 2j, 2j+3, tau'(2j+1) = 2j.
The fixed point of tau' starting with 0 is
u = 03225254254472544747625...
The corresponding letter-to-letter map lambda is given by lambda(2j)=j, lambda(2j+1)= j. Then lambda(u) = (a(n)).
(End)
REFERENCES
Cornelius Gerrit Lekkerkerker, Voorstelling van natuurlijke getallen door een som van getallen van Fibonacci, Simon Stevin 29 (1952), 190-195.
F. Weinstein, The Fibonacci Partitions, preprint, 1995.
Édouard Zeckendorf, Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liège 41, 179-182, 1972.
LINKS
Paul Baird-Smith, Alyssa Epstein, Kristen Flint, and Steven J. Miller, The Zeckendorf Game, arXiv:1809.04881 [math.NT], 2018.
EXAMPLE
a(46) = a(1 + 3 + 8 + 34) = 4.
Connection to the compositions of n into odd parts (see comment):
[ #]: a(n) composition into odd parts
[ 0] [ 0] 1 1 1 1 1 1 1 1
[ 1] [ 1] 1 1 1 1 1 3
[ 2] [ 1] 1 1 1 1 3 1
[ 3] [ 1] 1 1 1 3 1 1
[ 4] [ 2] 1 1 1 5
[ 5] [ 1] 1 1 3 1 1 1
[ 6] [ 2] 1 1 3 3
[ 7] [ 2] 1 1 5 1
[ 8] [ 1] 1 3 1 1 1 1
[ 9] [ 2] 1 3 1 3
[10] [ 2] 1 3 3 1
[11] [ 2] 1 5 1 1
[12] [ 3] 1 7
[13] [ 1] 3 1 1 1 1 1
[14] [ 2] 3 1 1 3
[15] [ 2] 3 1 3 1
[16] [ 2] 3 3 1 1
[17] [ 3] 3 5
[18] [ 2] 5 1 1 1
[19] [ 3] 5 3
[20] [ 3] 7 1
Connection to the compositions of n into parts 1 or 2 (see comment):
[ #]: a(n) composition into parts 1 and 2
[ 0] [0] 1 1 1 1 1 1 1
[ 1] [1] 1 1 1 1 1 2
[ 2] [1] 1 1 1 1 2 1
[ 3] [1] 1 1 1 2 1 1
[ 4] [2] 1 1 1 2 2
[ 5] [1] 1 1 2 1 1 1
[ 6] [2] 1 1 2 1 2
[ 7] [2] 1 1 2 2 1
[ 8] [1] 1 2 1 1 1 1
[ 9] [2] 1 2 1 1 2
[10] [2] 1 2 1 2 1
[11] [2] 1 2 2 1 1
[12] [3] 1 2 2 2
[13] [1] 2 1 1 1 1 1
[14] [2] 2 1 1 1 2
[15] [2] 2 1 1 2 1
[16] [2] 2 1 2 1 1
[17] [3] 2 1 2 2
[18] [2] 2 2 1 1 1
[19] [3] 2 2 1 2
[20] [3] 2 2 2 1
(End)
The third iterate of the morphism tau generating this sequence:
tau^3((0,0)) = (0,0)(1,1)(1,0)(1,0)(2,1)
= (a(0),0)(a(1),1)(a(2),0)(a(3),0)(a(4),1). (End)
MAPLE
# With the following Maple program (not the best one), B(n) (n >= 1) yields the number of terms in the Zeckendorf representation of n.
with(combinat): B := proc (n) local A, ct, m, j: A := proc (n) local i: for i while fibonacci(i) <= n do n-fibonacci(i) end do end proc: ct := 0; m := n: for j while 0 < A(m) do ct := ct+1: m := A(m) end do: ct+1 end proc: 0, seq(B(n), n = 1 .. 104);
N:= 1000: # to get a(n) for n <= N
m:= ceil(log[(1+sqrt(5))/2](sqrt(5)*N)):
Z:= Vector(m):
a[0]:= 0:
for n from 1 to N do
if Z[1] = 0 then Z[1]:= 1; q:= 1;
else Z[2]:= 1; Z[1]:= 0; q:= 2;
fi;
while Z[q+1] = 1 do
Z[q]:= 0;
Z[q+1]:= 0;
Z[q+2]:= 1;
q:= q+2;
od:
a[n]:= add(Z[i], i=1..m);
od:
# alternative
read("transforms") :
end proc:
MATHEMATICA
zf[n_] := (k = 1; ff = {}; While[(fi = Fibonacci[k]) <= n, AppendTo[ff, fi]; k++]; Drop[ff, 1]); zeckRep[n_] := If[n == 0, 0, r = n; s = {}; fr = zf[n]; While[r > 0, lf = Last[fr]; If[lf <= r, r = r - lf; PrependTo[s, lf]]; fr = Drop[fr, -1]]; s]; zeckRepLen[n_] := Length[zeckRep[n]]; Table[zeckRepLen[n], {n, 0, 104}] (* Jean-François Alcover, Sep 27 2011 *) Table[Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5] * # + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]], {n, 0, 143}] (* Alonso del Arte, May 14 2019 *) Flatten[Nest[{Flatten[#], #[[1]] + 1} &, {0, 1}, 9]] (* Paolo Xausa, Jun 17 2024 *)
PROG
(PARI) a(n, mx=0)=if(n<4, n>0, if(!mx, while(fibonacci(mx)<n, mx++)); while(fibonacci(mx)>n, mx--); 1+a(n-fibonacci(mx), mx-2)) \\ Charles R Greathouse IV, Feb 14 2013 (PARI) a(n)=if(n<4, n>0, my(k=2, s, t); while(fibonacci(k++)<=n, ); while(k && n, t=fibonacci(k); if(t<=n, n-=t; s++); k--); s) \\ Charles R Greathouse IV, Sep 02 2015 (Haskell)
(Python)
from sympy import fibonacci
def a(n):
k=0
x=0
while n>0:
k=0
while fibonacci(k)<=n: k+=1
x+=10**(k - 3)
n-=fibonacci(k - 1)
return str(x).count("1")
CROSSREFS
Cf. A000045, A007953, A035514, A035515, A035516, A035517, A105446, A189920, A213676, A000120, A001950, A003714, A007015, A007016, A104324, A182535, A213911, A014417, A003849. Cf. A135817 (lengths of Wythoff representation), A135818 (number of 1's (or A's) in the Wythoff representation).
Nontotients: even numbers k such that phi(m) = k has no solution.
(Formerly M4927)
+10
92
14, 26, 34, 38, 50, 62, 68, 74, 76, 86, 90, 94, 98, 114, 118, 122, 124, 134, 142, 146, 152, 154, 158, 170, 174, 182, 186, 188, 194, 202, 206, 214, 218, 230, 234, 236, 242, 244, 246, 248, 254, 258, 266, 274, 278, 284, 286, 290, 298, 302, 304, 308, 314, 318
COMMENTS
If p is prime then the following two statements are true. I. 2p is in the sequence iff 2p+1 is composite (p is not a Sophie Germain prime). II. 4p is in the sequence iff 2p+1 and 4p+1 are composite. - Farideh Firoozbakht, Dec 30 2005 Another subset of nontotients consists of the numbers j^2 + 1 such that j^2 + 2 is composite. These numbers j are given in A106571. Similarly, let b be 3 or a number such that b == 1 (mod 4). For any j > 0 such that b^j + 2 is composite, b^j + 1 is a nontotient. - T. D. Noe, Sep 13 2007 The Firoozbakht comment can be generalized: Observe that if k is a nontotient and 2k+1 is composite, then 2k is also a nontotient. See A057192 and A076336 for a connection to Sierpiński numbers. This shows that 271129*2^j is a nontotient for all j > 0. - T. D. Noe, Sep 13 2007
REFERENCES
Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 44 at p. 91.
R. K. Guy, Unsolved Problems in Number Theory, B36.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 91.
LINKS
Eric Weisstein's World of Mathematics, Nontotient.
EXAMPLE
There are no values of m such that phi(m)=14, so 14 is a term of the sequence.
MATHEMATICA
searchMax = 320; phiAnsYldList = Table[0, {searchMax}]; Do[phiAns = EulerPhi[m]; If[phiAns <= searchMax, phiAnsYldList[[phiAns]]++ ], {m, 1, searchMax^2}]; Select[Range[searchMax], EvenQ[ # ] && (phiAnsYldList[[ # ]] == 0) &] (* Alonso del Arte, Sep 07 2004 *) totientQ[m_] := Select[ Range[m +1, 2m*Product[(1 - 1/(k*Log[k]))^(-1), {k, 2, DivisorSigma[0, m]}]], EulerPhi[#] == m &, 1] != {}; (* after Jean-François Alcover, May 23 2011 in A002202 *) Select[2 Range@160, ! totientQ@# &] (* Robert G. Wilson v, Mar 20 2023 *)
PROG
(Haskell)
a005277 n = a005277_list !! (n-1)
a005277_list = filter even a007617_list
(Magma) [n: n in [2..400 by 2] | #EulerPhiInverse(n) eq 0]; // Marius A. Burtea, Sep 08 2019
CROSSREFS
See A007617 for all numbers k (odd or even) such that phi(m) = k has no solution.
Untouchable numbers, also called nonaliquot numbers: impossible values for the sum of aliquot parts function ( A001065).
(Formerly M1552)
+10
57
2, 5, 52, 88, 96, 120, 124, 146, 162, 188, 206, 210, 216, 238, 246, 248, 262, 268, 276, 288, 290, 292, 304, 306, 322, 324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, 516, 518, 520, 530, 540, 552, 556, 562, 576, 584, 612, 624, 626, 628, 658
COMMENTS
Chen & Zhao show that the lower density of this sequence is at least 0.06, improving on te Riele. - Charles R Greathouse IV, Dec 28 2013 The term "untouchable number" was coined by Alanen (1972). He found the 570 terms below 5000.
Erdős (1973) proved that the lower asymptotic density of untouchable numbers is positive, te Riele (1976) proved that it is > 0.0324, and Banks and Luca (2004, 2005) proved that it is > 1/48.
Pollack and Pomerance (2016) conjectured that the asymptotic density is ~ 0.17. (End)
The upper asymptotic density is less than 1/2 by the 'almost all' binary Goldbach conjecture, independently proved by Nikolai Chudakov, Johannes van der Corput, and Theodor Estermann. (In this context, this shows that the density of the odd numbers of this form is 0 (consider A001065(p*q) for prime p, q); full Goldbach would prove that 5 is the only odd number in this sequence.) - Charles R Greathouse IV, Dec 05 2022
REFERENCES
Richard K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004, section B10, pp. 100-101.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 65.
József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 93.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 125.
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], p. 840. Yong-Gao Chen and Qing-Qing Zhao, Nonaliquot numbers, Publ. Math. Debrecen, Vol. 78, No. 2 (2011), pp. 439-442.
MATHEMATICA
untouchableQ[n_] := Catch[ Do[ If[n == DivisorSigma[1, k]-k, Throw[True]], {k, 0, (n-1)^2}]] === Null; Reap[ Table[ If[ untouchableQ[n], Print[n]; Sow[n]], {n, 2, 700}]][[2, 1]] (* Jean-François Alcover, Jun 29 2012, after Benoit Cloitre *)
PROG
(PARI) isA078923(n)=if(n==0 || n==1, return(1)); for(m=1, (n-1)^2, if( sigma(m)-m == n, return(1))); 0
isA005114(n)=!isA078923(n)
for(n=1, 700, if (isA005114(n), print(n))) \\ R. J. Mathar, Aug 10 2006 (PARI) is(n)=if(n%2 && n<4e18, return(n==5)); forfactored(m=1, (n-1)^2, if(sigma(m)-m[1]==n, return(0))); 1 \\ Charles R Greathouse IV, Dec 05 2022
Numbers k such that phi(k) = phi(k+1).
(Formerly M2999 N1215)
+10
56
1, 3, 15, 104, 164, 194, 255, 495, 584, 975, 2204, 2625, 2834, 3255, 3705, 5186, 5187, 10604, 11715, 13365, 18315, 22935, 25545, 32864, 38804, 39524, 46215, 48704, 49215, 49335, 56864, 57584, 57645, 64004, 65535, 73124, 105524, 107864, 123824, 131144, 164175, 184635
COMMENTS
Unlike totients, cototient(x + 1) = cototient(x) never holds (except 2 - phi(2) = 3 - phi(3) = 1) because cototient(x) is congruent to x modulo 2. - Labos Elemer, Aug 08 2001 Lal-Gillard and Firoozbakht ask whether there is another pair of consecutive integers in this sequence, apart from a(16) + 1 = a(17) = 5187, see link. - M. F. Hasler, Jan 05 2011 There are 5236 terms less than 10^12. - Jud McCranie, Feb 13 2012 Up to 10^13 there are 10755 terms, and no further consecutive pairs like (5186, 5187). - Giovanni Resta, Feb 27 2014 A051179(k) for k from 0 to 5 are in the sequence. No other members of A051179 are in the sequence, because phi(2^(2^k)-1) = Product_{j=1..k-1} phi(2^(2^j)+1) and phi(2^(2^5)+1) < 2^(2^5) so if k > 5, phi(2^(2^k)-1) < Product_{j=1..k-1} 2^(2^j) = 2^(2^k-1) = phi(2^(2^k)). - Robert Israel, Mar 31 2015
Number of terms < 10^k, k=1,2,3,...: 2, 3, 10, 17, 36, 68, 142, 306, 651, 1267, 2567, 5236, 10755, ..., . - Robert G. Wilson v, Apr 10 2019 Conjecture: Except for the first two terms, all terms are composite and congruent to either 2 or 3 (mod 6). - Robert G. Wilson v, Apr 10 2019 Paul Kinlaw has noticed that up to 10^13 the only counterexample to the above conjecture is a(7424) = 3044760173455. - Giovanni Resta, May 23 2019
REFERENCES
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 15, pp 5, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems Number Theory, Sect. B36.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 66. K. Miller, Solutions of phi(n) = phi(n+1) for 1 <= n <= 500000. Unpublished, 1972. [ Cf. (Untitled), Math. Comp., Vol. 27, p. 447, 1973 ].
FORMULA
Conjecture: a(n) ~ C*n^3*log(n), where C = 9/Pi^2 = 0.91189... - Thomas Ordowski, Oct 21 2014 Sum_{n>=1} 1/a(n) is in the interval (1.4324884, 7.8358) (Kinlaw et al., 2020; an upper bound 441702 was given by Bayless and Kinlaw, 2016). - Amiram Eldar, Oct 15 2020
EXAMPLE
phi(3) = phi(4) = 2, so 3 is in the sequence.
phi(15) = phi(16) = 8, so 15 is in the sequence.
MAPLE
select(n -> numtheory:-phi(n) = numtheory:-phi(n+1), [$1..10^5]); # Robert Israel, Mar 31 2015
MATHEMATICA
Reap[For[n = 1; k = 2; f1 = 1, k <= 10^9, k++, f2 = EulerPhi[k]; If[f1 == f2, Print["a(", n, ") = ", k - 1]; Sow[k - 1]; n++]; f1 = f2]][[2, 1]] (* Jean-François Alcover, Mar 29 2011, revised Dec 26 2013 *) Flatten[Position[Partition[EulerPhi[Range[200000]], 2, 1], {x_, x_}]] (* Harvey P. Dale, Dec 27 2015 *) Select[Range[1000], EulerPhi[#] == EulerPhi[# + 1] &] (* Alonso del Arte, Oct 03 2014 *) SequencePosition[EulerPhi[Range[200000]], {x_, x_}][[All, 1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 01 2018 *) k = 8; lst = {1, 3}; While[k < 200000, If[ !PrimeQ[k +1], ep = EulerPhi[k +1]; If[ EulerPhi[k] == ep, AppendTo[lst, k]]; If[ep == EulerPhi[k +2], AppendTo[lst, k +1]]]; k += 6]; lst (* Robert G. Wilson v, Apr 10 2019 *)
PROG
(Haskell)
import Data.List (elemIndices)
a001274 n = a001274_list !! (n-1)
a001274_list = map (+ 1) $ elemIndices 0 $
zipWith (-) (tail a000010_list) a000010_list
(PARI) list(lim)=my(v=List(), old=1); forfactored(n=2, lim\1+1, my(new=eulerphi(n)); if(old==new, listput(v, n[1]-1)); old=new); Vec(v) \\ Charles R Greathouse IV, Jul 17 2022 (Magma) [n: n in [1..3*10^5] | EulerPhi(n) eq EulerPhi(n+1)]; // Vincenzo Librandi, Apr 14 2015
CROSSREFS
Cf. A000010, A001494, A051953, A003276, A003275, A007015, A179186, A179187, A179188, A179189, A179202, A217139.
Numbers n such that sigma(x) = n has no solution.
(Formerly M1355)
+10
42
2, 5, 9, 10, 11, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 33, 34, 35, 37, 41, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 58, 59, 61, 64, 65, 66, 67, 69, 70, 71, 73, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 88, 89, 92, 94, 95, 97, 99, 100, 101, 103, 105, 106, 107, 109, 111, 113
COMMENTS
With an initial 1, may be constructed inductively in stages from the list L = {1,2,3,....} by the following sieve procedure. Stage 1. Add 1 as the first term of the sequence a(n) and strike off 1 from L. Stage n+1. Add the first (i.e. leftmost) term k of L as a new term of the sequence a(n) and strike off k, sigma(k), sigma(sigma(k)),.... from L. - Joseph L. Pe, May 08 2002 This sieve is a special case of a more general sieve. Let D be a subset of N and let f be an injection on D satisfying f(n) > n. Define the sieve process as follows: 1. Start with empty sequence S. 2. Let E = D. 2. Append the smallest element s of E to S. 3. Remove s, f(s), f(f(s)), f(f(f(s))), ... from E. 4. Go to 2. After this sieving process, S = D - f(D). To get the current sequence, take f = sigma and D = {n | n >= 2}. - Max Alekseyev, Aug 08 2005 By analogy with the untouchable numbers ( A005114), these numbers could be named "sigma-untouchable". - Daniel Lignon, Mar 28 2014 The asymptotic density of this sequence is 1 (Niven, 1951, Rao and Murty, 1979). - Amiram Eldar, Jul 23 2020
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. R. Sita Rama Chandra Rao and G. Sri Rama Chandra Murty, On a theorem of Niven, Canadian Mathematical Bulletin, Vol 22, No. 1 (1979), pp. 113-115.
EXAMPLE
a(4) = 10 because there is no x < 10 whose sigma(x) = 10.
MATHEMATICA
a = {}; Do[s = DivisorSigma[1, n]; a = Append[a, s], {n, 1, 115} ]; Complement[ Table[ n, {n, 1, 115} ], Union[a] ]
PROG
(PARI) list(lim)=my(v=List(), u=vectorsmall(lim\1), t); for(n=1, lim, t=sigma(n); if(t<=lim, u[t]=1)); for(n=2, lim, if(u[n]==0, listput(v, n))); Vec(v) \\ Charles R Greathouse IV, Mar 09 2017 (PARI) A007369_list(LIM, m=0, L=List(), s)={for(n=2, LIM, (s=sigma(n-1))>LIM || bittest(m, s) || m+=1<<s; bittest(m, n)||listput(L, n)); L} \\ A bit slower, but bitmask requires less memory, avoiding stack overflow produced by the earlier code for lim = 1e6 with standard gp setup. - M. F. Hasler, Mar 12 2018
CROSSREFS
See A083532 for the gaps, i.e., first differences. See A048995 for the missed sums of nontrivial divisors.
Smallest k such that sigma(x) = k has exactly n solutions.
(Formerly M4829)
+10
24
2, 1, 12, 24, 96, 72, 168, 240, 336, 360, 504, 576, 1512, 1080, 1008, 720, 2304, 3600, 5376, 2520, 2160, 1440, 10416, 13392, 3360, 4032, 3024, 7056, 6720, 2880, 6480, 10800, 13104, 5040, 6048, 4320, 13440, 5760, 18720, 20736, 19152, 22680, 43680
COMMENTS
It's not obvious that a(n) exists for all n; I'd like to see a proof. - David Wasserman, Jun 07 2002 Note that k-1 is frequently prime. See A115374 for the least prime. For each n, it appears that there are an infinite number of k such that sigma(x)=k has exactly n solutions. - T. D. Noe, Jan 21 2006 According to Sierpiński, H. J. Kanold proved that there is a k such that sigma(x)=k has n or more solutions. Sierpiński states that Erdős proved that if, for some k, sigma(x)=k has exactly n solutions, then there are an infinite number of such k. - T. D. Noe, Oct 18 2006
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
EXAMPLE
a(10) = 504; {204, 220, 224, 246, 284, 286, 334, 415, 451, 503} is the set of x such that sigma(x) = 504.
MATHEMATICA
Needs["Statistics`DataManipulation`"]; s=DivisorSigma[1, Range[10^5]]; f=Frequencies[s]; fs=Sort[f]; tfs=Transpose[fs][[1]]; utfs=Union[tfs]; firstMissing=First[Complement[Range[Last[utfs]], utfs]]; pos=1; Table[While[tfs[[pos]]<n, pos++ ]; fs[[pos, 2]], {n, firstMissing-1}] (* T. D. Noe *) terms = 100; cnt = DivisorSigma[1, Range[terms^3]] // Tally // Sort; a[0] = 2; a[n_] := SelectFirst[cnt, #[[2]] == n&][[1]]; Table[a[n], {n, 0, terms - 1}] (* Jean-François Alcover, Jul 18 2017 *)
CROSSREFS
Cf. A115374 (least prime p such that sigma(x)=sigma(p) has exactly n solutions). Cf. A184393, A184394, A201915 (smallest solution, largest solution, triangle of solutions for sigma(x)=a(n)).
Numbers k such that sigma(x) = k has a unique solution.
(Formerly M2319)
+10
21
1, 3, 4, 6, 7, 8, 13, 14, 15, 20, 28, 30, 36, 38, 39, 40, 44, 57, 62, 63, 68, 74, 78, 91, 93, 102, 110, 112, 121, 127, 133, 138, 150, 158, 160, 162, 164, 171, 174, 176, 183, 194, 195, 198, 200, 204, 212, 217, 222, 230, 242, 255, 256, 258, 260, 266, 278, 282
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
W. Sierpiński, Elementary Theory of Numbers. Państ. Wydaw. Nauk., Warsaw, 1964, p. 165.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
MATHEMATICA
a = Table[ 0, {250} ]; Do[ s = DivisorSigma[ 1, n ]; If[ s < 251, a[ [ s ] ]++ ], {n, 1, 250} ]; Select[ Range[ 250 ], a[ [ # ] ] == 1 & ]
PROG
(PARI) list(lim)=my(v=vectorsmall(lim\1), u=List(), s); for(k=1, #v, s=sigma(k); if(s<=#v, v[s]++)); for(k=1, #v, if(v[k]==1, listput(u, k))); Vec(u) \\ Charles R Greathouse IV, Jun 15 2015
Numbers k such that sigma(k+2) = sigma(k).
(Formerly M5234)
+10
20
33, 54, 284, 366, 834, 848, 918, 1240, 1504, 2910, 2913, 3304, 4148, 4187, 6110, 6902, 7169, 7912, 9359, 10250, 10540, 12565, 15085, 17272, 17814, 19004, 19688, 21410, 21461, 24881, 25019, 26609, 28124, 30592, 30788, 31484, 38210, 38982, 39786, 40310, 45354
COMMENTS
Numbers k such that antisigma(k+2) - antisigma(k) = 2*k + 3, where antisigma(m) = A024816(m) = sum of nondivisors of m. - Jaroslav Krizek, Mar 17 2013
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 33, pp 12, Ellipses, Paris 2008.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
MATHEMATICA
Flatten[Position[Partition[DivisorSigma[1, Range[300000]], 3, 1], {x_, _, x_}]] (* Harvey P. Dale, Aug 08 2011 *) SequencePosition[DivisorSigma[1, Range[300000]], {x_, _, x_}][[All, 1]] (* Harvey P. Dale, Nov 17 2022 *)
PROG
(PARI) je=[]; for(n=1, 10^5, a=sigma(n); b=sigma(n+2); if(a==b, je=concat(je, n))); je
CROSSREFS
Cf. A002961, A015861, A015863, A015865, A015866, A015867, A015876, A015877, A015880, A015881, A015882, A015883, A181647. [From Reinhard Zumkeller, Nov 03 2010]
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