A007862 -id:A007862 - cp's OEIS frontend

A195155 Number of divisors d of n such that d-1 also divides n or d-1 = 0.

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

Offset: 1

Omar E. Pol, Sep 19 2011

First differs from A055874 at a(20).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000005, A007862, A049820, A055874, A129308, A137921, A195150, A195153.

with(numtheory):
a:= n-> add(`if`(d=1 or irem(n, d-1)=0, 1, 0), d=divisors(n)):
seq(a(n), n=1..200);  # Alois P. Heinz, Oct 17 2011

d1[n_]:=Module[{d=Rest[Divisors[n]]},Count[d,?(Divisible[n,#-1]&)]+1]; Array[d1, 90] (* _Harvey P. Dale, Oct 31 2013 *)

from itertools import pairwise
from sympy import divisors
def A195155(n): return 1 if n&1 else 1+sum(1 for a, b in pairwise(divisors(n)) if a+1==b) # Chai Wah Wu, Jun 09 2025

a(n) = A000005(n) - A195150(n).
a(n) = 1 + A129308(n).
a(2n-1) = 1; a(2n) = 1 + A007862(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2. - Amiram Eldar, Dec 31 2023
a(n) <= A038548(n) <= A000005(n). - Charles R Greathouse IV, Jun 09 2025

A325335 Number of integer partitions of n with adjusted frequency depth 4 whose parts cover an initial interval of positive integers.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 1, 3, 3, 3, 5, 8, 6, 13, 12, 14, 17, 22, 17, 28, 29, 30, 38, 50, 46, 67, 64, 75, 81, 104, 99, 127, 128, 150, 155, 201, 189, 236, 244, 293, 302, 363, 372, 437, 457, 548, 547, 638, 671, 754, 809, 922, 947, 1074, 1144, 1290, 1342, 1515, 1574

Offset: 0

Gus Wiseman, May 01 2019

nonn

The adjusted frequency depth of an integer partition (A325280) is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).

The Heinz numbers of these partitions are given by A325387.

			The a(4) = 1 through a(10) = 5 partitions:
  (211)  (221)   (21111)  (2221)    (22211)    (22221)     (222211)
         (2111)           (22111)   (221111)   (2211111)   (322111)
                          (211111)  (2111111)  (21111111)  (2221111)
                                                           (22111111)
                                                           (211111111)

Column k = 4 of A325336.

Cf. A000009, A007862, A181819, A182850, A317081, A320348, A323014, A325280, A325326, A325334, A325387.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#1]]&,ptn,Length[#1]>1&]]];
Table[Length[Select[IntegerPartitions[n],normQ[#]&&fdadj[#]==4&]],{n,0,30}]

A327744 Expansion of Product_{i>=1, j>=1} 1 / (1 - x^(ij(j + 1)/2)).

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 17, 23, 35, 54, 77, 108, 163, 221, 309, 436, 593, 800, 1109, 1470, 1968, 2642, 3482, 4566, 6052, 7848, 10204, 13276, 17092, 21924, 28245, 35949, 45762, 58231, 73609, 92789, 117140, 146799, 183826, 229995, 286483, 356040, 442566, 547489

Offset: 0

Ilya Gutkovskiy, Sep 23 2019

nonn

Euler transform of A007862.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A004101, A007862, A327745.

nmax = 43; CoefficientList[Series[Product[1/(1 - x^k)^Length[Select[Divisors[k], IntegerQ[Sqrt[8 # + 1]] &]], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Length[Select[Divisors[d], IntegerQ[Sqrt[8 # + 1]] &]], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 43}]
nmax = 50; CoefficientList[Series[Product[1/QPochhammer[x^(k*(k + 1)/2)], {k, 1, Sqrt[2*nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 24 2019 *)

G.f.: Product_{k>=1} 1 / (1 - x^k)^A007862(k).

A327745 Expansion of Product_{i>=1, j>=1} (1 + x^(ij(j + 1)/2)).

Original entry on oeis.org

1, 1, 1, 3, 3, 4, 8, 9, 11, 19, 23, 28, 42, 51, 62, 89, 108, 130, 178, 215, 260, 344, 413, 496, 639, 766, 916, 1155, 1380, 1641, 2040, 2426, 2870, 3520, 4166, 4912, 5960, 7023, 8246, 9911, 11634, 13610, 16224, 18972, 22111, 26183, 30507, 35430, 41698

Offset: 0

Ilya Gutkovskiy, Sep 23 2019

nonn

Weigh transform of A007862.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A007862, A280451, A327744.

nmax = 48; CoefficientList[Series[Product[(1 + x^k)^Length[Select[Divisors[k], IntegerQ[Sqrt[8 # + 1]] &]], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d Length[Select[Divisors[d], IntegerQ[Sqrt[8 # + 1]] &]], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 48}]
nmax = 50; CoefficientList[Series[Product[QPochhammer[-1, x^(k*(k + 1)/2)]/2, {k, 1, Sqrt[2*nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 24 2019 *)

G.f.: Product_{k>=1} (1 + x^k)^A007862(k).

A333195 Numbers with three consecutive prime indices in arithmetic progression.

Original entry on oeis.org

8, 16, 24, 27, 30, 32, 40, 48, 54, 56, 60, 64, 72, 80, 81, 88, 96, 104, 105, 108, 110, 112, 120, 125, 128, 135, 136, 144, 150, 152, 160, 162, 168, 176, 184, 189, 192, 200, 208, 210, 216, 220, 224, 232, 238, 240, 243, 248, 250, 256, 264, 270, 272, 273, 280, 288

Offset: 1

Gus Wiseman, Mar 29 2020

nonn

Also numbers whose first differences of prime indices do not form an anti-run, meaning there are adjacent equal differences.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices begins:
    8: {1,1,1}          105: {2,3,4}
   16: {1,1,1,1}        108: {1,1,2,2,2}
   24: {1,1,1,2}        110: {1,3,5}
   27: {2,2,2}          112: {1,1,1,1,4}
   30: {1,2,3}          120: {1,1,1,2,3}
   32: {1,1,1,1,1}      125: {3,3,3}
   40: {1,1,1,3}        128: {1,1,1,1,1,1,1}
   48: {1,1,1,1,2}      135: {2,2,2,3}
   54: {1,2,2,2}        136: {1,1,1,7}
   56: {1,1,1,4}        144: {1,1,1,1,2,2}
   60: {1,1,2,3}        150: {1,2,3,3}
   64: {1,1,1,1,1,1}    152: {1,1,1,8}
   72: {1,1,1,2,2}      160: {1,1,1,1,1,3}
   80: {1,1,1,1,3}      162: {1,2,2,2,2}
   81: {2,2,2,2}        168: {1,1,1,2,4}
   88: {1,1,1,5}        176: {1,1,1,1,5}
   96: {1,1,1,1,1,2}    184: {1,1,1,9}
  104: {1,1,1,6}        189: {2,2,2,4}

Wikipedia, Arithmetic progression

Anti-run compositions are counted by A003242.
Normal anti-runs of length n + 1 are counted by A005649.
Strict partitions with equal differences are A049980.
Partitions with equal differences are A049988.
These are the Heinz numbers of the partitions *not* counted by A238424.
Permutations avoiding triples in arithmetic progression are A295370.
Strict partitions avoiding triples in arithmetic progression are A332668.
Anti-run compositions are ranked by A333489.
Cf. A006560, A007862, A238423, A307824, A325328, A325849, A325852.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],MatchQ[Differences[primeMS[#]],{_,x_,x_,_}]&]

A049993 a(n) is the number of arithmetic progressions of 3 or more positive integers, nondecreasing with sum <= n.

Original entry on oeis.org

0, 0, 1, 2, 3, 6, 7, 9, 13, 16, 17, 24, 25, 28, 36, 40, 41, 51, 52, 58, 68, 72, 73, 87, 91, 95, 107, 114, 115, 134, 135, 141, 155, 160, 167, 189, 190, 195, 211, 223, 224, 248, 249, 257, 282, 288, 289, 316, 320, 332, 353, 362, 363, 392, 401, 413, 436, 443, 444, 484, 485, 492, 522, 533, 543, 578

Offset: 1

Clark Kimberling

nonn

Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions, Rostok. Math. Kolloq. 64 (2009), 11-16.
Sadek Bouroubi and Nesrine Benyahia Tani, Integer partitions into arithmetic progressions with an odd common difference, Integers 9(1) (2009), 77-81.
Graeme McRae, Counting arithmetic sequences whose sum is n.
Graeme McRae, Counting arithmetic sequences whose sum is n [Cached copy]
Augustine O. Munagi, Combinatorics of integer partitions in arithmetic progression, Integers 10(1) (2010), 73-82.
Augustine O. Munagi and Temba Shonhiwa, On the partitions of a number into arithmetic progressions, Journal of Integer Sequences 11 (2008), Article 08.5.4.
A. N. Pacheco Pulido, Extensiones lineales de un poset y composiciones de números multipartitos, Maestría thesis, Universidad Nacional de Colombia, 2012.
Wikipedia, Arithmetic progression.

Cf. A007862, A014405, A014406, A049980, A049981, A049982, A049983, A049986, A049987, A049988, A049989, A049990, A049991, A049992, A240026, A240027, A307824, A320466, A325325.

From Petros Hadjicostas, Sep 29 2019: (Start)
a(n) = Sum_{k = 1..n} A049992(k).
G.f.: (g.f. of A049992)/(1-x). (End)

More terms from Petros Hadjicostas, Sep 29 2019

A195307 Where records occur in A129308 and also in A195155.

Original entry on oeis.org

1, 2, 6, 12, 60, 180, 360, 420, 840, 1260, 2520, 5040, 13860, 27720, 55440, 83160, 166320, 277200, 360360, 720720, 1081080, 2162160, 2827440, 4324320, 6126120, 12252240, 24504480, 36756720, 73513440, 147026880, 183783600, 232792560, 367567200, 465585120, 698377680

Offset: 1

Omar E. Pol, Oct 16 2011

nonn

Observation: a(n) ending at 0, if 5 <= n <= 24 and possibly more.
From David A. Corneth, Apr 14 2021: (Start)
Conjecture: for each term k > 1 in the sequence there exists prime p such that k/p is in the sequence.
From the first 35 terms only a(23) = 2827440 is not in A025487.
In the list of conjectured terms, if actual terms <= 10^16 are 97-smooth and have the following property: a(n+1) = a(n) + k*gcd(a(n), a(n-1), ..., a(n-20)) setting a(n) = 1 for n < 1 then those terms are actual terms.
The conjectured terms are 41-smooth and satisfy a(n+1) = a(n) + k*gcd(a(n), a(n-1), ..., a(n-13)). (End)
From Bernard Schott, Jul 30 2022: (Start)
Equivalently, integers whose number of oblong divisors (A129308) sets a new record.
Corresponding records of number of oblong divisors are 0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, ... (End)

			a(4) = 12 is in the sequence because A129308(12) = 3 is larger than any earlier value in A129308. - _Bernard Schott_, Jul 30 2022

David A. Corneth, Conjectured terms <= 10^16 (a(1)..a(35) are certain)

Cf. A002378, A007862, A088726, A093687, A097212, A129308, A181826, A195155, A287142.

More terms a(6)-a(24) from Alois P. Heinz, Oct 16 2011

a(25)-a(35) from David A. Corneth, Apr 14 2021

A281615 Expansion of Sum_{i>=1} x^(i(i+1)/2)/(1 - x^(i(i+1)/2)) / Product_{j>=1} (1 - x^(j*(j+1)/2)).

Original entry on oeis.org

1, 2, 4, 6, 8, 13, 17, 21, 30, 37, 44, 60, 72, 83, 107, 127, 144, 181, 210, 236, 289, 333, 371, 446, 507, 562, 664, 750, 825, 965, 1083, 1187, 1371, 1530, 1668, 1912, 2122, 2307, 2618, 2896, 3138, 3540, 3897, 4211, 4717, 5180, 5581, 6222, 6803, 7317, 8116, 8853, 9497, 10486, 11401, 12215, 13430, 14572, 15576, 17067

Offset: 1

Ilya Gutkovskiy, Jan 25 2017

nonn

Total number of parts in all partitions of n into nonzero triangular numbers (A000217).

Convolution of A007294 and A007862.

			a(6) = 13 because we have [6], [3, 3], [3, 1, 1, 1], [1, 1, 1, 1, 1, 1] and 1 + 2 + 4 + 6 = 13.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Index entries for related partition-counting sequences

Cf. A000217, A007294, A007862, A281541.

nmax = 60; Rest[CoefficientList[Series[Sum[x^(i (i + 1)/2)/(1 - x^(i (i + 1)/2)), {i, 1, nmax}]/Product[1 - x^(j (j + 1)/2), {j, 1, nmax}], {x, 0, nmax}], x]]

G.f.: Sum_{i>=1} x^(i*(i+1)/2)/(1 - x^(i*(i+1)/2)) / Product_{j>=1} (1 - x^(j*(j+1)/2)).

a(n) ~ exp(3*zeta(3/2)^(2/3) * (Pi*n)^(1/3)/2) * zeta(3/2)^(1/3) / (2^(3/2) * sqrt(3) * Pi^(4/3) * n^(5/6)). - Vaclav Kotesovec, Sep 15 2021

A325333 Number of integer partitions of n whose multiplicities all appear the same number of times.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 14, 19, 23, 32, 39, 50, 63, 82, 96, 125, 152, 186, 226, 271, 326, 392, 473, 552, 663, 771, 918, 1065, 1261, 1448, 1710, 1953, 2283, 2608, 3062, 3455, 4013, 4552, 5271, 5974, 6884, 7774, 8937, 10065, 11570, 12953, 14838, 16710, 18979

Offset: 0

Gus Wiseman, May 01 2019

nonn

			The a(0) = 1 through a(7) = 14 partitions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)
           (11)  (21)   (22)    (32)     (33)      (43)
                 (111)  (31)    (41)     (42)      (52)
                        (211)   (221)    (51)      (61)
                        (1111)  (311)    (222)     (322)
                                (2111)   (321)     (331)
                                (11111)  (411)     (421)
                                         (2211)    (511)
                                         (3111)    (2221)
                                         (21111)   (4111)
                                         (111111)  (22111)
                                                   (31111)
                                                   (211111)
                                                   (1111111)
For example, the partition (4,3,3,3,2,2,2,1) has multiplicities (1,3,3,1), and since both multiplicities 1 and 3 appear twice, (4,3,3,3,2,2,2,1) is counted under a(20).

Cf. A000041, A007862, A047966, A049988, A098859, A323022, A325329, A325369.

Table[Length[Select[IntegerPartitions[n],SameQ@@Length/@Split[Sort[Length/@Split[#]]]&]],{n,0,30}]

A327629 Expansion of Sum_{k>=1} x^(k(k + 1)/2) / (1 - x^(k(k + 1)/2))^2.

Original entry on oeis.org

1, 2, 4, 4, 5, 9, 7, 8, 12, 11, 11, 18, 13, 14, 21, 16, 17, 27, 19, 22, 29, 22, 23, 36, 25, 26, 36, 29, 29, 50, 31, 32, 44, 34, 35, 55, 37, 38, 52, 44, 41, 65, 43, 44, 64, 46, 47, 72, 49, 55, 68, 52, 53, 81, 56, 58, 76, 58, 59, 100, 61, 62, 87, 64, 65, 100, 67, 68, 92, 77

Offset: 1

Ilya Gutkovskiy, Sep 19 2019

nonn

Sum of divisors d of n such that n/d is triangular number.

Cf. A000217, A006463, A007862, A010054, A076752, A112886 (fixed points), A185027.

nmax = 70; CoefficientList[Series[Sum[x^(k (k + 1)/2)/(1 - x^(k (k + 1)/2))^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := DivisorSum[n, # &, IntegerQ[Sqrt[8 n/# + 1]] &]; Table[a[n], {n, 1, 70}]

a(n)={sumdiv(n, d, if(ispolygonal(d,3), n/d))} \\ Andrew Howroyd, Sep 19 2019

from sympy import divisors
from sympy.ntheory.primetest import is_square
def A327629(n): return sum(n//d for d in divisors(n,generator=True) if is_square((d<<3)+1)) # Chai Wah Wu, Jun 07 2025

a(n) = Sum_{d|n} A010054(n/d) * d.

cp's OEIS Frontend

A195155 Number of divisors d of n such that d-1 also divides n or d-1 = 0.

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A325335 Number of integer partitions of n with adjusted frequency depth 4 whose parts cover an initial interval of positive integers.

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A327744 Expansion of Product_{i>=1, j>=1} 1 / (1 - x^(i*j*(j + 1)/2)).

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A327745 Expansion of Product_{i>=1, j>=1} (1 + x^(i*j*(j + 1)/2)).

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A333195 Numbers with three consecutive prime indices in arithmetic progression.

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A049993 a(n) is the number of arithmetic progressions of 3 or more positive integers, nondecreasing with sum <= n.

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A195307 Where records occur in A129308 and also in A195155.

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A281615 Expansion of Sum_{i>=1} x^(i*(i+1)/2)/(1 - x^(i*(i+1)/2)) / Product_{j>=1} (1 - x^(j*(j+1)/2)).

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A327744 Expansion of Product_{i>=1, j>=1} 1 / (1 - x^(ij(j + 1)/2)).

A327745 Expansion of Product_{i>=1, j>=1} (1 + x^(ij(j + 1)/2)).

A281615 Expansion of Sum_{i>=1} x^(i(i+1)/2)/(1 - x^(i(i+1)/2)) / Product_{j>=1} (1 - x^(j*(j+1)/2)).

A327629 Expansion of Sum_{k>=1} x^(k(k + 1)/2) / (1 - x^(k(k + 1)/2))^2.