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Search: a115964 -id:a115964
Displaying 1-10 of 11 results found. page 1 2
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A024451 a(n) is the numerator of Sum_{i = 1..n} 1/prime(i). +10
61
0, 1, 5, 31, 247, 2927, 40361, 716167, 14117683, 334406399, 9920878441, 314016924901, 11819186711467, 492007393304957, 21460568175640361, 1021729465586766997, 54766551458687142251, 3263815694539731437539, 201015517717077830328949, 13585328068403621603022853 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Arithmetic derivative of p#: a(n) = A003415(A002110(n)). - Reinhard Zumkeller, Feb 25 2002
(n-1)-st elementary symmetric functions of first n primes; see Mathematica section. - Clark Kimberling, Dec 29 2011
Denominators of the harmonic mean of the first n primes; A250130 gives the numerators. - Colin Barker, Nov 14 2014
Let Pn(n) = A002110 denote the primorial function. The average number of distinct prime factors <= prime(n) in the natural numbers up to Pn(n) is equal to Sum_{i = 1..n} 1/prime(i). - Jamie Morken, Sep 17 2018
Conjecture: All terms are squarefree numbers. - Nicolas Bělohoubek, Apr 13 2022
The above conjecture would imply that for n > 0, gcd(a(n), A369651(n)) = 1. See corollary 2 on the page 4 of Ufnarovski-Åhlander paper. - Antti Karttunen, Jan 31 2024
REFERENCES
S. R. Finch, Mathematical Constants, Cambridge, 2003, Sect. 2.2.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Sect. VII.28.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..350 (terms n = 1..100 from T. D. Noe)
Victor Ufnarovski and Bo Åhlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.
FORMULA
Limit_{n->oo} (Sum_{p <= n} 1/p - log log n) = 0.2614972... = A077761.
a(n) = (Product_{i=1..n} prime(i))*(Sum_{i=1..n} 1/prime(i)). - Benoit Cloitre, Jan 30 2002
(n+1)-st elementary symmetric function of the first n primes.
a(n) = a(n-1)*A000040(n) + A002110(n-1). - Henry Bottomley, Sep 27 2006
From Antti Karttunen, Jan 31 2024 and Feb 08 2024: (Start)
a(0) = 0, for n > 0, a(n) = 2*A203008(n-1) + A070826(n).
For n > 0, a(n) = A327860(A143293(n-1)).
For n > 0, a(n) = A348301(n) + A002110(n).
For n = 3..175, a(n) = A356253(A002110(n)). [See comments in A356253.]
(End)
EXAMPLE
0/1, 1/2, 5/6, 31/30, 247/210, 2927/2310, 40361/30030, 716167/510510, 14117683/9699690, ...
MAPLE
h:= n-> add(1/(ithprime(i)), i=1..n);
t1:=[seq(h(n), n=0..50)];
t1a:=map(numer, t1); # A024451
t1b:=map(denom, t1); # A002110 - N. J. A. Sloane, Apr 25 2014
MATHEMATICA
a[n_] := Numerator @ Sum[1/Prime[i], {i, n}]; Array[a, 18] (* Jean-François Alcover, Apr 11 2011 *)
f[k_] := Prime[k]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}] (* A024451 *)
(* Clark Kimberling, Dec 29 2011 *)
Numerator[Accumulate[1/Prime[Range[20]]]] (* Harvey P. Dale, Apr 11 2012 *)
PROG
(Magma) [ Numerator(&+[ NthPrime(k)^-1: k in [1..n]]): n in [1..18] ]; // Bruno Berselli, Apr 11 2011
(PARI) a(n) = numerator(sum(i=1, n, 1/prime(i))); \\ Michel Marcus, Sep 18 2018
(Python)
from sympy import prime
from fractions import Fraction
def a(n): return sum(Fraction(1, prime(k)) for k in range(1, n+1)).numerator
print([a(n) for n in range(20)]) # Michael S. Branicky, Feb 12 2021
(Python)
from math import prod
from sympy import prime
def A024451(n):
q = prod(plist:=tuple(prime(i) for i in range(1, n+1)))
return sum(q//p for p in plist) # Chai Wah Wu, Nov 03 2022
CROSSREFS
Denominators are A002110.
See also A106830/A034386, A241189/A241190, A241191/A241192, A061015/A061742, A115963/A115964, A250133/A296358, and A096795/A051451, A354417/A354418, A354859/A354860.
Row sums of A077011 and A258566.
Cf. A003415, A002110, A070826, A143293, A203008, A250130, A327860, A348301, A369651.
Cf. A109628 (indices k where a(k) is prime), A244622 (corresponding primes), A244621 (a(n) mod 12).
Cf. A369972 (k where prime(1+k)|a(k)), A369973 (corresponding primorials), A293457 (corresponding primes).
Cf. also A223037, A260615, A274070, A327978, A353299, A353534, A356253.
KEYWORD
nonn,frac,easy,nice
AUTHOR
N. J. A. Sloane, Clark Kimberling
EXTENSIONS
a(0)=0 prepended by Alois P. Heinz, Jun 26 2015
STATUS
approved
A014701 Number of multiplications to compute n-th power by the Chandah-sutra method. +10
14
0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 5, 6, 6, 7, 5, 6, 6, 7, 6, 7, 7, 8, 5, 6, 6, 7, 6, 7, 7, 8, 6, 7, 7, 8, 7, 8, 8, 9, 6, 7, 7, 8, 7, 8, 8, 9, 7, 8, 8, 9, 8, 9, 9, 10, 6, 7, 7, 8, 7, 8, 8, 9, 7, 8, 8, 9, 8, 9, 9, 10, 7, 8, 8, 9, 8, 9, 9 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
In other words, number of steps to reach 1 starting from n and using the process: x -> x-1 if n is odd and x -> x/2 otherwise.
a(n) = number of 0's + twice number of 1's (disregarding the leading digit 1) in the binary expansion of n, i.e., A007088(n). - Lekraj Beedassy, May 28 2010
From Daniel Forgues, Jul 31 2012: (Start)
For the binary Fibonacci rabbits sequence (A036299) (cf. OEIS Wiki link below) we have the substitution/concatenation rule: a(n), n >= 3, may be obtained by the concatenation of a(n-1) and a(n-2), with a(1) = 0, a(2) = 1. Thus, using . (dot) as the concatenation operator, we have the recursive substitution/concatenation
a(n) = a(n-0)
a(n) = a(n-1).a(n-2)
a(n) = a(n-2).a(n-3).a(n-3).a(n-4)
a(n) = a(n-3).a(n-4).a(n-4).a(n-5).a(n-4).a(n-5).a(n-5).a(n-6)
which suggests the sequence
{0}
{1, 2}
{2, 3, 3, 4}
{3, 4, 4, 5, 4, 5, 5, 6}
whose concatenation gives A014701 (this sequence).
Number of multiplications to compute n-th power by the Chandah-sutra method, also called left-to-right binary exponentiation:
x^1 = x^( 1_2) = (x) (0 prod)
x^2 = x^( 10_2) = (x^2) (1 prod)
x^3 = x^( 11_2) = (x^2) * (x) (2 prod)
x^4 = x^( 100_2) = (x^2)^2 (2 prod)
x^5 = x^( 101_2) = (x^2)^2 * (x) (3 prod)
x^6 = x^( 110_2) = (x^2)^2 * (x^2) (3 prod)
x^7 = x^( 111_2) = (x^2)^2 * (x^2) * (x) (4 prod)
x^8 = x^(1000_2) = ((x^2)^2)^2 (3 prod) (End)
From Ya-Ping Lu, Mar 03 2021: (Start)
Index at which record m occurs is A052955(m).
First appearance of m in the sequence (or the record value m) is at n = 2^(m/2 + 1) - 1 for even m, and at n = 3*2^((m - 1)/2) - 1 for odd m.
The last appearance of m in the sequence is at n = 2^m. (End)
a(n) is the digit sum of n-1 in bijective base-2. Since the Fibonacci number F(m) can be defined as the number of ways to compose m as the sum of 1s and 2s, we get that m appears F(m) times in the sequence. - Oscar Cunningham, Apr 14 2024
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..16384 (first 1000 terms from T. D. Noe)
C. K. Caldwell, The Prime Glossary, binary exponentiation
Hermann Gruber and Markus Holzer, Optimal Regular Expressions for Palindromes of Given Length, Proceedings of the 46th International Symposium on Mathematical Foundations of Computer Science, Article No. 53, pp. 53:1-53:15, 2021.
J. Jordan and R. Southwell, Further Properties of Reproducing Graphs, Applied Mathematics, Vol. 1 No. 5, 2010, pp. 344-350. - From N. J. A. Sloane, Feb 03 2013
Szymon Łukaszyk and Wawrzyniec Bieniawski, Assembly Theory of Binary Messages (How to Assemble a Black Hole and Use it to Assemble New Binary Information?), Preprints (2024).
OEIS Wiki, Binary Fibonacci rabbits sequence
Index to sequences related to the complexity of n
Index entries for sequences related to binary expansion of n
FORMULA
a(n) = A056792(n) - 1 = A056791(n) - 2.
a(n) = floor(log_2(n)) + (number of 1's in binary representation of n) - 1. - Corrected (- 1 at end) by Daniel Forgues, Aug 01 2012
a(2^n) = n, a(2^n-1) = 2*(n-1), and for n >= 2, log_2(n) <= a(n) <= 2*log_2(n) - 1. - Robert FERREOL, Oct 01 2014
Let u(1) = 1, u(2*n) = u(n)+1, u(2*n+1) = u(2*n)+1; then a(1) = 0 and a(n) = u(n-1). - Benoit Cloitre, Dec 19 2002
G.f.: -2/(1-x) + (1/(1-x)) * Sum_{k>=0} (2*x^2^k + x^2^(k+1))/(1+x^2^k). - Ralf Stephan, Aug 15 2003
From {0}, apply the substitution rule (n -> n+1, n+2) repeatedly, giving {{0}, {1, 2}, {2, 3, 3, 4}, {3, 4, 4, 5, 4, 5, 5, 6}, ...} and concatenate. - Daniel Forgues, Jul 31 2012
For n > 1: a(n) = A007953(A007931(n-1)). - Reinhard Zumkeller, Oct 26 2012
a(n) >= A003313(n). - Charles R Greathouse IV, Jan 03 2018
a(n) = a(floor(n/2)) + 1 + (n mod 2) for n > 1. - Pablo Hueso Merino, Oct 28 2020
a(n+1) = max_{1<=i<=n} (H(i) + H(n-i)) where H(n) denotes the Hamming weight of n (A000120(n)). See Lemma 8 in Gruber/Holzer 2021 article. - Hermann Gruber, Jun 26 2024
EXAMPLE
5 -> 4 -> 2 -> 1 so 3 steps are needed to reach 1 hence a(5)=3; 9 -> 8 -> 4 -> 2 -> 1 hence a(9)=4.
MAPLE
A014701 := proc(n) local j, k; j := n; k := 0; while(j>1) do if j mod 2=1 then j := j-1 else j := j/2 fi; k := k+1 od end;
# second Maple program:
a:= n-> add(i+1, i=Bits[Split](n))-2:
seq(a(n), n=1..128); # Alois P. Heinz, Aug 30 2021
MATHEMATICA
a[n_] := DigitCount[n, 2] /. {x_, y_} -> 2x + y - 2; Array[a, 100] (* Robert G. Wilson v, Jul 31 2012 *)
PROG
(Haskell)
a014701 1 = 0
a014701 n = a007953 $ a007931 (n - 1)
-- Reinhard Zumkeller, Oct 26 2012
(PARI) a(n)=hammingweight(n)+logint(n, 2)-1 \\ Charles R Greathouse IV, Dec 29 2016
(Python)
def a(n):
if n==1:
return 0
return a(n//2)+1+n%2
for i in range(1, 60):
print(a(i), end=", ")
# Pablo Hueso Merino, Oct 28 2020
CROSSREFS
Cf. A003313, A056791, A056792, A115964, A052955.
KEYWORD
easy,nonn
AUTHOR
James Kilfiger (jamesk(AT)maths.warwick.ac.uk)
STATUS
approved
A115963 Numerator of Sum_{i=1..n} 1/prime(i)^3. +10
5
1, 35, 4591, 1601713, 2141141003, 4716413174591, 23198819007792583, 159253748925534977797, 1938552948676080555065099, 47290471293028435532185602511, 1409101231790431848106470385672201 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Denominators = A115964. See also: A024451 Numerator of Sum_{i=1..n} 1/prime(i). A002110 Primorial [denominator of Numerator of Sum_{i=1..n} 1/prime(i)]. A061015 Numerator of Sum_{i=1..n} 1/prime(i)^2.
LINKS
Table of n, a(n) for n=1..11.
FORMULA
a(n) = Numerator of Sum_{i=1..n} 1/A000040(i)^3.
EXAMPLE
1/8, 35/216, 4591/27000, 1601713/9261000, 2141141003/12326391000, 4716413174591/27081081027000.
CROSSREFS
Cf. A000040, A075986/A075987, A106830/A034386.
KEYWORD
easy,frac,nonn
AUTHOR
Jonathan Vos Post, Mar 14 2006
STATUS
approved
A241189 Numerator of Sum_{i=1..n} 1/(prime(i)*prime(i+1)). +10
5
1, 7, 11, 127, 1693, 29243, 561623, 13019431, 379503437, 11809225121, 438235268123, 18007758091069, 775817745542929, 36524284093223105, 1938403609207158571, 2160165866032831207, 131893095784520401909, 8844093116997411126541, 628373208972323386101329, 45900898298568589325230523 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
a(371) has 1002 decimal digits. - Michael De Vlieger, Jan 27 2016
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..370
EXAMPLE
1/6, 7/30, 11/42, 127/462, 1693/6006, 29243/102102, 561623/1939938, 13019431/44618574, 379503437/1293938646, 11809225121/40112098026, 438235268123/1484147626962, ...
MAPLE
g:= n-> add(1/(ithprime(i)*ithprime(i+1)), i=1..n);
t1:=[seq(g(n), n=1..20)];
t1a:=map(numer, t1); # A241189
t1b:=map(denom, t1); # A241190
MATHEMATICA
Table[Numerator@ Sum[1/(Prime[i + 1] Prime@ i), {i, n}], {n, 20}] (* Michael De Vlieger, Jan 27 2016 *)
Accumulate[1/#&/@(Times@@@Partition[Prime[Range[25]], 2, 1])]//Numerator (* Harvey P. Dale, Mar 14 2023 *)
CROSSREFS
Cf. A024451/A002110, A241189/A241190, A241191/A241192.
See also A061015/A061742, A115963/A115964.
KEYWORD
nonn,frac
AUTHOR
N. J. A. Sloane, Apr 25 2014, based on a suggestion from Timothy Varghese.
STATUS
approved
A304117 If n = Product (p_j^k_j) then a(n) = Product (pi(p_j)*k_j), where pi() = A000720. +10
4
1, 1, 2, 2, 3, 2, 4, 3, 4, 3, 5, 4, 6, 4, 6, 4, 7, 4, 8, 6, 8, 5, 9, 6, 6, 6, 6, 8, 10, 6, 11, 5, 10, 7, 12, 8, 12, 8, 12, 9, 13, 8, 14, 10, 12, 9, 15, 8, 8, 6, 14, 12, 16, 6, 15, 12, 16, 10, 17, 12, 18, 11, 16, 6, 18, 10, 19, 14, 18, 12, 20, 12, 21, 12, 12, 16, 20, 12, 22, 12 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Extended graphical example
Index entries for sequences computed from indices in prime factorization
Index entries for sequences computed from exponents in factorization of n
FORMULA
a(n) = A005361(n)*A156061(n).
a(p^k) = A000720(p)*k where p is a prime.
a(A002110(m)^k) = k^m*m!.
As an example:
a(A000040(k)) = k.
a(A006450(k)) = A000040(k).
a(A001248(k)) = a(A031215(k)) = A005843(k).
a(A030078(k)) = a(A031336(k)) = A008585(k)
a(A061742(k)) = A000165(k).
a(A115964(k)) = A032031(k).
a(A002110(k)) = A000142(k).
a(A080696(k)) = A002110(k).
EXAMPLE
a(36) = 8 because 36 = 2^2*3^2 = prime(1)^2*prime(2)^2 and 1*2*2*2 = 8.
MATHEMATICA
a[n_] := Times @@ (PrimePi[#[[1]]] #[[2]] & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 1, 80}]
PROG
(PARI) a(n) = my(f=factor(n)); for (k=1, #f~, f[k, 1] = primepi(f[k, 1])*f[k, 2]; f[k, 2] = 1); factorback(f); \\ Michel Marcus, May 06 2018
CROSSREFS
Cf. A000026, A000040, A000720, A001414, A002110, A003963, A005361, A056239, A156061, A304037.
KEYWORD
nonn,mult
AUTHOR
Ilya Gutkovskiy, May 06 2018
STATUS
approved
A304037 If n = Product (p_j^k_j) then a(n) = Sum (pi(p_j)^k_j), where pi() = A000720. +10
3
0, 1, 2, 1, 3, 3, 4, 1, 4, 4, 5, 3, 6, 5, 5, 1, 7, 5, 8, 4, 6, 6, 9, 3, 9, 7, 8, 5, 10, 6, 11, 1, 7, 8, 7, 5, 12, 9, 8, 4, 13, 7, 14, 6, 7, 10, 15, 3, 16, 10, 9, 7, 16, 9, 8, 5, 10, 11, 17, 6, 18, 12, 8, 1, 9, 8, 19, 8, 11, 8, 20, 5, 21, 13, 11, 9, 9, 9, 22, 4, 16, 14, 23, 7, 10, 15, 12, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Table of n, a(n) for n=1..88.
Ilya Gutkovskiy, Extended graphical example
Index entries for sequences computed from indices in prime factorization
FORMULA
If gcd(u,v) = 1 then a(u*v) = a(u) + a(v).
a(p^k) = A000720(p)^k where p is a prime.
a(A002110(m)^k) = 1^k + 2^k + ... + m^k.
As an example:
a(A000040(k)) = k.
a(A006450(k)) = A000040(k).
a(A038580(k)) = A006450(k).
a(A001248(k)) = a(A011757(k)) = A000290(k).
a(A030078(k)) = a(A055875(k)) = A000578(k).
a(A002110(k)) = a(A011756(k)) = A000217(k).
a(A061742(k)) = A000330(k).
a(A115964(k)) = A000537(k).
a(A080696(k)) = A007504(k).
a(A076954(k)) = A001923(k).
EXAMPLE
a(72) = 5 because 72 = 2^3*3^2 = prime(1)^3*prime(2)^2 and 1^3 + 2^2 = 5.
MATHEMATICA
a[n_] := Plus @@ (PrimePi[#[[1]]]^#[[2]]& /@ FactorInteger[n]); a[1] = 0; Table[a[n], {n, 1, 88}]
CROSSREFS
Cf. A000040, A000720, A002110, A003963, A008475, A056239, A066328, A156061, A222416.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, May 05 2018
STATUS
approved
A356014 Consider the exponents in the prime factorization of n, and replace each run of k consecutive e's by a unique e; the resulting list corresponds to the exponents in the prime factorization of a(n). +10
3
1, 2, 3, 4, 3, 2, 3, 8, 9, 10, 3, 12, 3, 10, 3, 16, 3, 18, 3, 20, 21, 10, 3, 24, 9, 10, 27, 20, 3, 2, 3, 32, 21, 10, 3, 4, 3, 10, 21, 40, 3, 10, 3, 20, 45, 10, 3, 48, 9, 50, 21, 20, 3, 54, 21, 40, 21, 10, 3, 12, 3, 10, 63, 64, 21, 10, 3, 20, 21, 10, 3, 72, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
We ignore the exponents (all 0's) for the prime numbers beyond the greatest prime factor of n.
This sequence operates on prime exponents as A090079 and A337864 operate on binary and decimal digits, respectively.
LINKS
Table of n, a(n) for n=1..73.
FORMULA
a(a(n)) = a(n).
a(n^k) = a(n)^k for any k >= 0.
a(n) = A319521(A356008(n)).
A007814(a(n)) = A007814(n).
a(n) = 3 iff n belongs to A294674 \ {1}.
a(n) = 4 iff n belongs to A061742 \ {1}.
a(n) = 8 iff n belongs to A115964.
EXAMPLE
For n = 99:
- 99 = 11^1 * 7^0 * 5^0 * 3^2 * 2^0,
- the list of exponents is: 1 0 0 2 0,
- compressing consecutive values, we obtain: 1 0 2 0,
- so a(99) = 7^1 * 5^0 * 3^2 * 2^0 = 63.
PROG
(PARI) a(n) = { my (v=1, e=-1, k=0); forprime (p=2, oo, if (n==1, return (v), if (e!=e=valuation(n, p), v*=prime(k++)^e); n/=p^e)) }
CROSSREFS
Cf. A007814, A061742, A067255, A090079, A115964, A294674, A319521, A337864, A356008, A356021 (fixed points).
KEYWORD
nonn
AUTHOR
Rémy Sigrist, Jul 23 2022
STATUS
approved
A361810 a(n) is the sum of divisors of n that are both infinitary and exponential. +10
3
1, 2, 3, 4, 5, 6, 7, 10, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 30, 25, 26, 30, 28, 29, 30, 31, 34, 33, 34, 35, 36, 37, 38, 39, 50, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 60, 55, 70, 57, 58, 59, 60, 61, 62, 63, 68, 65, 66, 67, 68 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The number of these divisors is A359411(n).
The indices of records of a(n)/n are the primorials (A002110) cubed, i.e., 1 and the terms of A115964.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
Multiplicative with a(p^e) = Sum_{d|e, bitor(d, e) == e} p^d.
a(n) >= n, with equality if and only if n is in A138302.
limsup_{n->oo} a(n)/n = Product_{p prime} (1 + 1/p^2) = 15/Pi^2 (A082020).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((1 - 1/p)*(1 + Sum_{e>=1} Sum_{d|e, bitor(d, e) == e} p^(d-2*e))) = 0.51015879911178031024... .
EXAMPLE
a(8) = 10 since 8 has 2 divisors that are both infinitary and exponential, 2 and 8, and 2 + 8 = 10.
MATHEMATICA
f[p_, e_] := DivisorSum[e, p^# &, BitOr[#, e] == e &]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) s(p, e) = sumdiv(e, d, p^d*(bitor(d, e) == e));
a(n) = {my(f = factor(n)); prod(i = 1, #f~, s(f[i, 1], f[i, 2])); }
CROSSREFS
Cf. A002110, A049417, A051377, A082020, A115964, A138302, A359411.
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Mar 25 2023
STATUS
approved
A359412 Numbers with a record number of divisors that are both infinitary and exponential. +10
2
1, 8, 216, 27000, 9261000, 12326391000, 27081081027000, 110924107886592000, 544970142046826496000, 3737950204299182936064000, 45479640135708158783090688000, 1109202943269786284560798789632000, 33044264882950203203350756741926912000, 1673791149116076642859325881248823873536000 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Indices of records in A359411.
a(2)-a(7) are the first 6 terms of A115964.
The first 15 terms are cubes. Are there noncubes in this sequence?
The corresponding record values are 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, ... . Apparently, this sequence of records is the powers of 2 (A000079).
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..15
MATHEMATICA
s[n_] := DivisorSum[n, 1 &, BitAnd[n, #] == # &]; f[p_, e_] := s[e]; d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n];
v = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {_, _}][[;; , 2]];
seq = {}; dm = 0; Do[If[(dk = d[v[[k]]]) > dm, dm = dk; AppendTo[seq, v[[k]]]], {k, 1, Length[v]}]; seq
CROSSREFS
Cf. A000079, A000578, A115964, A359411.
Subsequence of A025487.
Similar sequences: A037992, A318278.
KEYWORD
nonn
AUTHOR
Amiram Eldar, Dec 30 2022
STATUS
approved
A362854 The sum of the divisors of n that are both bi-unitary and exponential. +10
2
1, 2, 3, 4, 5, 6, 7, 10, 9, 10, 11, 12, 13, 14, 15, 18, 17, 18, 19, 20, 21, 22, 23, 30, 25, 26, 30, 28, 29, 30, 31, 34, 33, 34, 35, 36, 37, 38, 39, 50, 41, 42, 43, 44, 45, 46, 47, 54, 49, 50, 51, 52, 53, 60, 55, 70, 57, 58, 59, 60, 61, 62, 63, 70, 65, 66, 67, 68 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The number of these divisors is A362852(n).
The indices of records of a(n)/n are the primorials (A002110) cubed, i.e., 1 and the terms of A115964.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
Multiplicative with a(p^e) = Sum_{d|e} p^d if e is odd, and (Sum_{d|e} p^d) - p^(e/2) if e is even.
a(n) >= n, with equality if and only if n is cubefree (A004709).
limsup_{n->oo} a(n)/n = Product_{p prime} (1 + 1/p^2) = 15/Pi^2 (A082020).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((1 - 1/p)*(1 + Sum_{e>=1} Sum_{d|e, d != e/2}, p^(d-2*e))) = 0.5124353304539905... .
EXAMPLE
a(8) = 10 since 8 has 2 divisors that are both bi-unitary and exponential, 2 and 8, and 2 + 8 = 10.
MATHEMATICA
f[p_, e_] := DivisorSum[e, p^# &] - If[OddQ[e], 0, p^(e/2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) s(p, e) = sumdiv(e, d, p^d*(2*d != e));
a(n) = {my(f = factor(n)); prod(i = 1, #f~, s(f[i, 1], f[i, 2])); }
CROSSREFS
Cf. A002110, A004709, A051377, A082020, A115964, A188999, A222266, A322791, A361810, A362852.
KEYWORD
nonn,mult
AUTHOR
Amiram Eldar, May 05 2023
STATUS
approved
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Last modified August 30 13:06 EDT 2024. Contains 375543 sequences. (Running on oeis4.)