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Search: a088722 -id:a088722
Displaying 1-10 of 11 results found. page 1 2
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A129308 a(n) is the number of positive integers k such that k*(k+1) divides n. +10
29
0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 5, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 4, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 4, 0, 1, 0, 1, 0, 4, 0, 1, 0, 1, 0, 3, 0, 1, 0, 2, 0, 2, 0, 1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,6
COMMENTS
The usual OEIS policy is not to include sequences like this where alternate terms are zero; this is an exception.
In other words, a(n) is the number of oblong numbers (A002378) dividing n. - Bernard Schott, Jul 29 2022
LINKS
Ray Chandler, Table of n, a(n) for n = 1..10000
P. Erdős and R. R. Hall, On some unconventional problems on the divisors of integers, J. Austral. Math. Soc., Ser. A, 25, 479-485 (1978).
MathOverflow, On the number of consecutive divisors of an integer.
FORMULA
a(2n-1) = 0; a(2n) = A007862(n). - Ray Chandler, Jun 24 2008
G.f.: Sum_{n>=1} x^(n*(n+1))/(1-x^(n*(n+1))). - Joerg Arndt, Jan 30 2011 [modified by Ilya Gutkovskiy, Apr 14 2021]
a(n) = A000005(n) - A137921(n), where A137921(n) is the number of maximal runs of successive divisors of n. - Gus Wiseman, Oct 15 2019
a(n) = Sum_{d|n} A005369(d). - Ridouane Oudra, Jan 22 2021
a(n) = A195155(n)-1. - Antti Karttunen, Feb 21 2023
From Amiram Eldar, Dec 31 2023: (Start)
a(n) = A088722(n) + A059841(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1. (End)
EXAMPLE
The divisors of 20 are 1,2,4,5,10,20. Of these there are two that are of the form k(k+1): 2 = 1*2 and 20 = 4*5. So a(2) = 2.
MATHEMATICA
a = {}; For[n = 1, n < 90, n++, k = 1; co = 0; While[k < Sqrt[n], If[IntegerQ[ n/(k*(k + 1))], co++ ]; k++ ]; AppendTo[a, co]]; a (* Stefan Steinerberger, May 27 2007 *)
Table[Count[Differences[Divisors[n]], 1], {n, 30}] (* Gus Wiseman, Oct 15 2019 *)
PROG
(PARI) a(n)=sumdiv(n, d, n%(d+1)==0); \\ Michel Marcus, Jan 06 2015
CROSSREFS
Positions of 0's and 1's are A088725, whose characteristic function is A360128.
First appearance of n is A287142(n), with sorted version A328450.
The longest run of divisors of n has length A055874(n).
One less than A195155.
Cf. A000005, A002378, A003601, A007862, A027750, A033676, A060680, A060681, A072627, A088722, A181063, A199970, A328026, A328165, A328166.
Cf. A005369, A059841, A287142, A344005.
KEYWORD
nonn
AUTHOR
Leroy Quet, May 26 2007
EXTENSIONS
More terms from Stefan Steinerberger, May 27 2007
Extended by Ray Chandler, Jun 24 2008
STATUS
approved
A088725 Numbers having no divisors d>1 such that also d+1 is a divisor. +10
21
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 88, 89, 91 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Complement of A088723.
Union of A132895 and A005408, the odd numbers. - Ray Chandler, May 29 2008
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 9, 79, 778, 7782, 77813, 778055, 7780548, 77805234, 778052138, 7780519314, ... . Apparently, the asymptotic density of this sequence exists and equals 0.77805... . - Amiram Eldar, Jun 14 2022
LINKS
Table of n, a(n) for n=1..72.
FORMULA
A088722(a(n)) = 0.
EXAMPLE
From Gus Wiseman, Oct 16 2019: (Start)
The sequence of terms together with their divisors > 1 begins:
1: {}
2: {2}
3: {3}
4: {2,4}
5: {5}
7: {7}
8: {2,4,8}
9: {3,9}
10: {2,5,10}
11: {11}
13: {13}
14: {2,7,14}
15: {3,5,15}
16: {2,4,8,16}
17: {17}
19: {19}
21: {3,7,21}
22: {2,11,22}
23: {23}
25: {5,25}
(End)
MATHEMATICA
Select[Range[100], FreeQ[Differences[Rest[Divisors[#]]], 1]&] (* Harvey P. Dale, Sep 16 2017 *)
PROG
(PARI) isok(n) = {my(d=setminus(divisors(n), [1])); #setintersect(d, apply(x->x+1, d)) == 0; } \\ Michel Marcus, Oct 28 2019
CROSSREFS
Positions of 0's and 1's in A129308.
Positions of 0's and 1's in A328457 (also).
Numbers whose divisors (including 1) have no non-singleton runs are A005408.
The number of runs of divisors of n is A137921(n).
The longest run of divisors of n has length A055874(n).
Cf. A000005, A027750, A060680, A088722, A088723 (complement), A088724, A088726, A328166, A328450.
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Oct 12 2003
EXTENSIONS
Extended by Ray Chandler, May 29 2008
STATUS
approved
A088723 Numbers k with at least one divisor d>1 such that d+1 also divides k. +10
11
6, 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 72, 78, 80, 84, 90, 96, 100, 102, 108, 110, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 180, 182, 186, 192, 198, 200, 204, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252, 258, 260 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
A088722(a(n))>0; complement of A088725.
Complement of A132895 relative to A005843, the even numbers. - Chandler
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
MATHEMATICA
Select[Range[300], MemberQ[Differences[Select[Divisors[#], #>1&]], 1]&] (* Harvey P. Dale, Apr 03 2011 *)
PROG
(Haskell)
a088723 n = a088723_list !! (n-1)
a088723_list = filter f [2..] where
f x = 1 `elem` (zipWith (-) (tail divs) divs)
where divs = tail $ a027750_row x
-- Reinhard Zumkeller, Dec 16 2013
CROSSREFS
Cf. A088722, A088724, A088725, A088726.
Cf. A027750, A008588 (subsequence).
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Oct 12 2003
EXTENSIONS
Extended by Ray Chandler, May 29 2008
STATUS
approved
A328457 Length of the longest run of divisors > 1 of n. +10
10
0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 5, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,6
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..100000
MATHEMATICA
Table[If[n==1, 0, Max@@Length/@Split[Rest[Divisors[n]], #2==#1+1&]], {n, 100}]
PROG
(PARI) A328457(n) = { my(rl=0, pd=0, m=0); fordiv(n, d, if(d>1, if(d>(1+pd), m = max(m, rl); rl=0); pd=d; rl++)); max(m, rl); }; \\ Antti Karttunen, Feb 23 2023
CROSSREFS
Records occur at A328448.
Positions of 0's and 1's are A088725.
The version that looks at all divisors (including 1) is A055874.
The number of successive pairs of divisors > 1 of n is A088722(n).
The Heinz number of the multiset of run-lengths of divisors of n is A328166(n).
The longest run of nontrivial divisors of n is A328458(n).
Cf. A000005, A027750, A129308, A181063, A199970, A328162, A328195, A328449, A360128.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 16 2019
EXTENSIONS
Data section extended up to a(105) by Antti Karttunen, Feb 23 2023
STATUS
approved
A088726 Smallest numbers having exactly n divisors d>1 such that also d+1 is a divisor. +10
9
1, 6, 12, 72, 60, 180, 360, 420, 840, 1260, 3780, 2520, 5040, 13860, 36960, 41580, 27720, 55440, 83160, 166320, 277200, 491400, 471240, 360360, 1113840, 720720, 1081080, 3341520, 2162160, 2827440, 5405400, 4324320, 12972960, 6126120 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
A088722(a(n))=n, A088722(k)<>n, for n<a(n).
LINKS
Ray Chandler, Table of n, a(n) for n=0..57
FORMULA
a(n) = 2*A130317(n+1) for n>0. - Chandler
CROSSREFS
Cf. A088722, A088723, A088724, A088725.
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Oct 12 2003
EXTENSIONS
Extended by Ray Chandler, May 29 2008
STATUS
approved
A088724 Numbers having exactly one divisor d>1 such that also d+1 is a divisor. +10
5
6, 18, 20, 40, 54, 56, 66, 78, 80, 100, 102, 110, 112, 114, 138, 140, 160, 162, 174, 182, 186, 198, 200, 222, 224, 234, 246, 258, 260, 272, 282, 318, 320, 340, 354, 364, 366, 392, 400, 402, 414, 426, 438, 448, 460, 474, 486, 498, 500, 506, 520, 522, 534, 544 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Subsequence of A088723.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 1, 10, 100, 976, 9712, 97140, 971139, 9711054, 97109111, 971091745, ... . Apparently, the asymptotic density of this sequence exists and equals 0.097109... . - Amiram Eldar, Jul 09 2022
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..1000
FORMULA
A088722(a(n)) = 1.
MATHEMATICA
Select[Range[600], Count[Differences[Rest[Divisors[#]]], 1]==1&] (* Harvey P. Dale, Sep 05 2015 *)
CROSSREFS
Cf. A088722, A088723, A088725, A088726.
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Oct 12 2003
EXTENSIONS
Extended by Ray Chandler, May 29 2008
STATUS
approved
A328458 Maximum run-length of the nontrivial divisors (greater than 1 and less than n) of n. +10
4
1, 0, 0, 1, 0, 2, 0, 1, 1, 1, 0, 3, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 3, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 3, 0, 1, 1, 2, 0, 2, 0, 1, 1, 1, 0, 3, 1, 1, 1, 1, 0, 2, 1, 2, 1, 1, 0, 5, 0, 1, 1, 1, 1, 2, 0, 1, 1, 1, 0, 3, 0, 1, 1, 1, 1, 2, 0, 2, 1, 1, 0, 3, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 3, 0, 1, 1, 2, 0, 2, 0, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,6
COMMENTS
By convention, a(1) = 1, and a(p) = 0 for p prime.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..100000
EXAMPLE
The non-singleton runs of the nontrivial divisors of 1260 are: {2,3,4,5,6,7} {9,10} {14,15} {20,21} {35,36}, so a(1260) = 6.
MATHEMATICA
Table[Switch[n, 1, 1, _?PrimeQ, 0, _, Max@@Length/@Split[DeleteCases[Divisors[n], 1|n], #2==#1+1&]], {n, 100}]
PROG
(PARI) A328458(n) = if(1==n, n, my(rl=0, pd=0, m=0); fordiv(n, d, if(1<d && d<n, if(d>(1+pd), m = max(m, rl); rl=0); pd=d; rl++)); max(m, rl)); \\ Antti Karttunen, Feb 23 2023
CROSSREFS
Positions of first appearances are A328459.
Positions of 0's and 1's are A088723.
The version that looks at all divisors is A055874.
The number of successive pairs of divisors > 1 of n is A088722(n).
The Heinz number of the multiset of run-lengths of divisors of n is A328166(n).
Cf. A033676, A060681, A060775, A070824, A088725, A129308, A181063, A199970, A328165, A163870, A328194, A328448, A328449.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 17 2019
EXTENSIONS
Data section extended up to a(105) by Antti Karttunen, Feb 23 2023
STATUS
approved
A360128 a(n) = 1 if there are no divisors d>1 of n such that also d+1 is a divisor of n, otherwise 0. +10
4
1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for characteristic functions
FORMULA
a(n) = [A088722(n) == 0], where [ ] is the Iverson bracket.
a(n) = [A129308(n) < 2] = [A328457(n) < 2].
PROG
(PARI) A360128(n) = !sumdiv(n, d, (d>1)&&!(n%(d+1)));
CROSSREFS
Characteristic function of A088725.
Cf. A088722, A129308, A328457, A360129.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Feb 20 2023
STATUS
approved
A132895 Even numbers for which all divisors, with the exception of 1 and 2, are isolated. A positive divisor, k, of n is isolated if neither (k-1) nor (k+1) divides n. +10
3
2, 4, 8, 10, 14, 16, 22, 26, 28, 32, 34, 38, 44, 46, 50, 52, 58, 62, 64, 68, 70, 74, 76, 82, 86, 88, 92, 94, 98, 104, 106, 116, 118, 122, 124, 128, 130, 134, 136, 142, 146, 148, 152, 154, 158, 164, 166, 170, 172, 176, 178, 184, 188, 190, 194, 196, 202, 206, 208, 212 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Obviously, all divisors of an odd number are isolated.
a(n) = 2*A112886(n). - Chandler
LINKS
Table of n, a(n) for n=1..60.
EXAMPLE
28 is a term of the sequence because its divisors are 1,2,4,7,14, 28 and only 1 and 2 are non-isolated. 30 does not belong to the sequence because its divisors are 1,2,3,4,6,8,12, 24 and 1,2,3,4 are non-isolated.
MAPLE
with(numtheory): b:=proc(n) local div, ISO, i: div:=divisors(n): ISO:={}: for i to tau(n) do if member(div[i]-1, div)=false and member(div[i]+1, div)=false then ISO:=`union`(ISO, {div[i]}) end if end do end proc: a:=proc(n) if nops(b(n))= tau(n)-2 then n else end if end proc: seq(a(n), n=4..200);
MATHEMATICA
Select[2*Range[120], Min[Differences[Rest[Divisors[#]]]]>1&] (* Harvey P. Dale, Jul 13 2022 *)
CROSSREFS
Cf. A112886, A133950, A133951, A088722-A088726.
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Oct 16 2007, Oct 19 2007
EXTENSIONS
Corrected and extended by Ray Chandler, May 29 2008
STATUS
approved
A360119 Number of divisors of n which are not also differences between consecutive divisors, minus the number of differences between consecutive divisors of n which are not also divisors of n. Here the differences are counted with repetition if they occur more than once. +10
3
1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 1, 4, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 5, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 6, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 6, 1, 1, 1, 3, 1, 3, 1, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,6
COMMENTS
Because the algorithm for computing this sequence (see the PARI program) starts with s set to the number of divisors, and s is decremented at most once on each iteration in the loop over the first differences of the divisors, and because there is one less difference than there are divisors, it implies that a(n) >= 1 for all n.
Note that if a(n) = 1, then A088722(n) = 0, but not vice versa, i.e., the positions of 1's in this sequence is just a subsequence of A088725. See A360129 for the exceptions.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537
FORMULA
a(n) = A060764(n) - A360118(n).
a(n) <= A000005(n).
PROG
(PARI) A360119(n) = { my(d=divisors(n), erot=vecsort(vector(#d-1, k, d[k+1] - d[k])), s=#d); for(i=1, #erot, if(n%erot[i], s--, if(1==i || erot[i]!=erot[i-1], s--))); (s); };
CROSSREFS
Cf. A000005, A060764, A088722, A088725, A360118, A360129.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Feb 20 2023
STATUS
approved
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Last modified August 29 09:09 EDT 2024. Contains 375511 sequences. (Running on oeis4.)