A072627 -id:A072627 - cp's OEIS frontend

A322678 Records in A072627.

0, 1, 2, 4, 6, 7, 8, 9, 10, 12, 14, 18, 21, 22, 24, 26, 29, 31, 35, 38, 39, 41, 46, 51, 52, 56, 61, 62, 70, 82, 91, 97, 101, 104, 112, 118, 122, 127, 138, 143, 159, 163, 170, 175, 188, 196, 199, 201, 211, 221, 226, 235, 236, 249, 258, 276, 289, 295, 298, 303

Offset: 1

Daniel Suteu, Dec 23 2018

nonn

Cf. A072627, A202728, A322676.

r=-1; for(n=1, 10^6, t=sumdiv(n, d, isprime(d-1)); if(t>r, r=t; print1(t, ", ")));

a(n) = A072627(A322676(n)).

A129308 a(n) is the number of positive integers k such that k*(k+1) divides n.

Original entry on oeis.org

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

Offset: 1

Leroy Quet, May 26 2007

nonn

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

			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.

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.

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.

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 *)

a(n)=sumdiv(n, d, n%(d+1)==0); \\ Michel Marcus, Jan 06 2015

from itertools import pairwise
from sympy import divisors
def A129308(n): return 0 if n&1 else sum(1 for a, b in pairwise(divisors(n)) if a+1==b) # Chai Wah Wu, Jun 09 2025

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)

More terms from Stefan Steinerberger, May 27 2007

Extended by Ray Chandler, Jun 24 2008

A322977 Number of even divisors d of n such that d-1 is prime.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 3, 0, 1, 0, 2, 0, 2, 0, 2, 0, 0, 0, 5, 0, 0, 0, 2, 0, 2, 0, 3, 0, 0, 0, 4, 0, 1, 0, 3, 0, 3, 0, 2, 0, 0, 0, 6, 0, 0, 0, 1, 0, 3, 0, 3, 0, 0, 0, 6, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 7, 0, 1, 0, 2, 0, 1, 0, 4, 0, 0, 0, 6, 0, 0, 0, 3, 0, 4, 0, 1, 0, 0, 0, 7, 0, 2, 0, 2, 0, 2, 0, 3, 0

Offset: 1

Antti Karttunen, Jan 04 2019

nonn

Equally: Number of divisors d of n such that d-1 is an odd prime.

Antti Karttunen, Table of n, a(n) for n = 1..15120
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A010051, A059841, A072627, A183063, A322975, A322976, A322978 (bisection).

Array[DivisorSum[#, 1 &, And[EvenQ@ #, PrimeQ[# - 1]] &] &, 105] (* Michael De Vlieger, Jan 04 2019 *)

A322977(n) = sumdiv(n, d, (!(d%2))*isprime(d-1));

a(n) = Sum_{d|n, d>1} A059841(d)*A010051(d-1).

a(n) <= A183063(n).

A322976 Number of divisors d of n such that d+2 is prime.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 1, 3, 2, 2, 2, 1, 1, 4, 1, 2, 3, 1, 2, 3, 2, 1, 2, 2, 1, 4, 1, 2, 4, 1, 1, 3, 2, 3, 3, 1, 1, 3, 2, 2, 3, 1, 2, 6, 1, 1, 2, 1, 2, 4, 1, 1, 4, 3, 1, 3, 2, 2, 4, 1, 1, 4, 1, 3, 3, 1, 2, 3, 3, 2, 3, 1, 1, 4, 1, 3, 3, 1, 2, 5, 2, 1, 3, 3, 1, 4, 2, 1, 6, 1, 1, 2, 1, 3, 2, 1, 1, 5, 2, 2, 4, 1, 1, 7

Offset: 1

Antti Karttunen, Jan 04 2019

nonn

			10395 has 32 divisors: [1, 3, 5, 7, 9, 11, 15, 21, 27, 33, 35, 45, 55, 63, 77, 99, 105, 135, 165, 189, 231, 297, 315, 385, 495, 693, 945, 1155, 1485, 2079, 3465, 10395]. When 2 is added to each, as 1+2 = 3, 3+2 = 5, 5+2 = 7, etc, the only sums that are primes are: [3, 5, 7, 11, 13, 17, 23, 29, 37, 47, 79, 101, 107, 137, 167, 191, 233, 317, 947, 1487, 2081, 3467], thus (a10395) = 22.

Antti Karttunen, Table of n, a(n) for n = 1..10395
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A010051, A067513, A072627, A100821, A322358, A322975, A322977, A322978.

Array[DivisorSum[#, 1 &, PrimeQ[# + 2] &] &, 105] (* Michael De Vlieger, Jan 04 2019 *)

A322976(n) = sumdiv(n, d, isprime(d+2));

a(n) = Sum_{d|n} A010051(d+2).

a(A000040(n)) = 1 + A100821(n).

A323156 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = A323155(n) for all n, except f(1) = 0.

Original entry on oeis.org

1, 2, 3, 4, 2, 5, 2, 6, 3, 2, 2, 7, 2, 8, 3, 6, 2, 9, 2, 10, 3, 2, 2, 11, 2, 2, 3, 12, 2, 13, 2, 14, 3, 2, 2, 15, 2, 16, 3, 17, 2, 18, 2, 19, 3, 2, 2, 20, 2, 2, 3, 4, 2, 21, 2, 22, 3, 2, 2, 23, 2, 24, 3, 14, 2, 5, 2, 25, 3, 8, 2, 26, 2, 27, 3, 28, 2, 5, 2, 29, 3, 2, 2, 30, 2, 2, 3, 31, 2, 32, 2, 4, 3, 2, 2, 33, 2, 34, 3, 10, 2, 35, 2, 36, 3

Offset: 1

Antti Karttunen, Jan 09 2019

nonn

For all i, j:
  a(i) = a(j) => A072627(i) = A072627(j),
  a(i) = a(j) => A323157(i) = A323157(j) => A322977(i) = A322977(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A323157, also A322315.

Cf. A072627, A322977.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A323155(n) = { my(m=1); fordiv(n, d, if(isprime(d-1), m *= (d-1)^(1+valuation(n,d-1)))); (m); };
v323156 = rgs_transform(vector(up_to, n, if(1==n,0,A323155(n))));
A323156(n) = v323156[n];

A328450 Numbers that are a smallest number with k pairs of successive divisors, for some k.

Original entry on oeis.org

1, 2, 6, 12, 60, 72, 180, 360, 420, 840, 1260, 2520, 3780, 5040, 13860, 27720, 36960, 41580, 55440, 83160, 166320, 277200, 360360, 471240, 491400, 720720, 1081080, 1113840, 2162160, 2827440, 3341520, 4324320, 5405400, 6126120

Offset: 0

Gus Wiseman, Oct 15 2019

nonn

A sorted version of A287142.

			The divisors of 72 are {1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72}, with pairs of successive divisors {{1, 2}, {2, 3}, {3, 4}, {8, 9}}, and no smaller number has 4 successive pairs, so 72 belongs to the sequence.

Sorted positions of first appearances in A129308.
The longest run of divisors of n has length A055874(n).
Numbers whose divisors > 1 have no non-singleton runs are A088725.
The Heinz number of the multiset of run-lengths of divisors of n is A328166(n).
The smallest number whose divisors have a longest run of length n is A328449(n).
Cf. A000005, A003601, A027750, A033676, A060680, A060681, A072627, A181063, A199970, A287142, A328165.

dat=Table[Count[Differences[Divisors[n]],1],{n,10000}];
Sort[Table[Position[dat,i][[1,1]],{i,Union[dat]}]]

A322975 Number of divisors d of n such that d-2 is prime.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 2, 2, 0, 0, 1, 2, 1, 1, 2, 0, 2, 1, 1, 1, 0, 2, 2, 0, 1, 2, 2, 0, 2, 1, 1, 4, 0, 0, 1, 2, 2, 0, 2, 0, 1, 2, 2, 1, 0, 0, 3, 1, 1, 4, 1, 2, 1, 0, 1, 1, 2, 0, 2, 1, 0, 4, 2, 1, 2, 0, 2, 2, 0, 0, 3, 2, 1, 0, 1, 0, 4, 3, 1, 1, 0, 2, 1, 0, 2, 3, 3, 0, 0, 1, 2, 5

Offset: 1

Antti Karttunen, Jan 04 2019

nonn

			10395 has 32 divisors: [1, 3, 5, 7, 9, 11, 15, 21, 27, 33, 35, 45, 55, 63, 77, 99, 105, 135, 165, 189, 231, 297, 315, 385, 495, 693, 945, 1155, 1485, 2079, 3465, 10395]. When 2 is subtracted from each, as 1-2 = -1, 3-2 = 1, 5-2 = 3, etc, the only differences that are primes are: [3, 5, 7, 13, 19, 31, 43, 53, 61, 97, 103, 163, 229, 313, 383, 691, 1153, 1483, 3463], thus (a10395) = 19.

Antti Karttunen, Table of n, a(n) for n = 1..10395
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A010051, A062301, A067513, A072627, A322358, A322976, A322977, A322978.

Array[DivisorSum[#, 1 &, PrimeQ[# - 2] &] &, 105] (* Michael De Vlieger, Jan 04 2019 *)

A322975(n) = sumdiv(n, d, isprime(d-2));

a(n) = Sum_{d|n, d>2} A010051(d-2).

a(A000040(n)) = A062301(n).

A156190 Number of divisors d of n! such that d-1 is prime.

Original entry on oeis.org

0, 0, 2, 6, 9, 18, 29, 43, 66, 101, 187, 257, 470, 717, 1033, 1293, 2442, 3168, 6011, 7911, 11179, 16888, 32603, 39587, 53063, 79742, 97707, 127679, 246428, 302205, 585129, 670409, 923535, 1385145, 1813935, 2099557, 4091926, 6142573, 8416157

Offset: 1

Ray Chandler, Feb 05 2009

nonn

Ray Chandler, Table of n, a(n) for n=1..42

Cf. A072627, A067847.

Table[Count[Divisors[n!],?(PrimeQ[#-1]&)],{n,40}] (* _Harvey P. Dale, Feb 22 2020 *)

a(n) = A072627(A000142(n)).

A322702 a(n) is the product of primes p such that p+1 divides n.

Original entry on oeis.org

1, 1, 2, 3, 1, 10, 1, 21, 2, 1, 1, 330, 1, 13, 2, 21, 1, 170, 1, 57, 2, 1, 1, 53130, 1, 1, 2, 39, 1, 290, 1, 651, 2, 1, 1, 5610, 1, 37, 2, 399, 1, 5330, 1, 129, 2, 1, 1, 2497110, 1, 1, 2, 3, 1, 9010, 1, 273, 2, 1, 1, 10727970, 1, 61, 2, 651, 1, 10, 1, 201, 2

Offset: 1

Daniel Suteu, Dec 23 2018

nonn

In general, a(n) is the product of A072627(n) distinct prime factors, with a(n) = 1 iff A072627(n) = 0.

			For n=12, the divisors of 12 are {1, 2, 3, 4, 6, 12}. The prime numbers p, such that p+1 is a divisor of 12, are {2, 3, 5, 11}, therefore a(12) = 2 * 3 * 5 * 11 = 330.

Antti Karttunen, Table of n, a(n) for n = 1..10080

Cf. A072627, A027760, A322356, A323156.

a:= n-> mul(`if`(isprime(d-1), d-1, 1), d=numtheory[divisors](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 29 2018

Array[Apply[Times, Select[Divisors@ #, PrimeQ[# - 1] &] - 1 /. {} -> {1}] &, 69] (* Michael De Vlieger, Jan 07 2019 *)

a(n) = my(d=divisors(n)); prod(k=1, #d, if(isprime(d[k]-1), d[k]-1, 1));

a(n) = Product_{p prime, p+1 divides n} p.
a(n) = denominator of Sum_{p prime, p+1 divides n} 1/p.
a(n) = Product_{d|n, d-1 is prime} (d-1), where d runs over the divisors of n.
a(2*n + 1) = 2, iff n == 1 (mod 3), else a(2*n + 1) = 1.
A001221(a(n)) = A072627(n). - Antti Karttunen, Jan 12 2019

A341099 Numbers divisible by at least three terms of A008864.

Original entry on oeis.org

12, 18, 24, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 72, 80, 84, 88, 90, 96, 102, 104, 108, 112, 114, 120, 126, 128, 132, 136, 138, 140, 144, 150, 152, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 248, 252, 256, 258, 264, 270, 272

Offset: 1

J. M. Bergot and Robert Israel, Feb 04 2021

nonn

Numbers divisible by p+1 for at least three primes p.
Numbers k with A072627(k) >= 3.
Every positive multiple of a term is a term.

			a(4) = 30 is a term because it is divisible by 2+1=3, 5+1=6 and 29+1=30.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005101, A005835, A008864, A072627.

filter:= proc(n) local D;
  D:= convert(numtheory:-divisors(n),list);
  numboccur(true, map(t -> isprime(t-1),D))>= 3
end proc:
select(filter, [$1..1000]);

okQ[n_] := DivisorSum[n, Boole[PrimeQ[#-1]]&] >= 3;
Select[Range[1000], okQ] (* Jean-François Alcover, May 16 2023 *)

isok(m) = sumdiv(m, d, isprime(d-1)) >= 3; \\ Michel Marcus, Feb 05 2021

cp's OEIS Frontend

A322678 Records in A072627.

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A129308 a(n) is the number of positive integers k such that k*(k+1) divides n.

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A322977 Number of even divisors d of n such that d-1 is prime.

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A322976 Number of divisors d of n such that d+2 is prime.

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A323156 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = A323155(n) for all n, except f(1) = 0.

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A328450 Numbers that are a smallest number with k pairs of successive divisors, for some k.

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A322975 Number of divisors d of n such that d-2 is prime.

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A156190 Number of divisors d of n! such that d-1 is prime.

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