A007862 -id:A007862 - cp's OEIS frontend

A329801 Expansion of Sum_{k>=1} x^(k(k + 1)/2) / (1 + x^(k(k + 1)/2)).

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

Offset: 1

Ilya Gutkovskiy, Nov 21 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000217, A007862, A010054, A048272, A304876, A317529.

nmax = 85; CoefficientList[Series[Sum[x^(k (k + 1)/2)/(1 + x^(k (k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[(-1)^(n/d + 1) Boole[IntegerQ[Sqrt[8 d + 1]]], {d, Divisors[n]}], {n, 1, 85}]

A329801(n) = sumdiv(n,d,((-1)^(1+(n/d))) * ispolygonal(d,3)); \\ Antti Karttunen, Jan 15 2025

G.f.: Sum_{k>=1} (-1)^(k + 1) * theta_2(x^(k/2)) / (2 * x^(k/8)).

a(n) = Sum_{d|n} (-1)^(n/d + 1) * A010054(d).

A069470 a(n) = Sum_{k>=1} floor(n/(k*(k+1)/2)).

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 9, 10, 11, 13, 15, 16, 19, 20, 21, 24, 25, 26, 29, 30, 32, 35, 36, 37, 40, 41, 42, 44, 46, 47, 52, 53, 54, 56, 57, 58, 62, 63, 64, 66, 68, 69, 73, 74, 75, 79, 80, 81, 84, 85, 87, 89, 90, 91, 94, 96, 98, 100, 101, 102, 107, 108, 109, 112, 113, 114, 118

Offset: 0

Henry Bottomley, Mar 25 2002

nonn

The summation has floor(1/2 + sqrt(2*n)) = A002024(n) nonzero terms. - Enrique Pérez Herrero, Apr 05 2010

			a(11) = floor(11/1) + floor(11/3) + floor(11/6) + floor(11/10) + floor(11/15) + ... = 11 + 3 + 1 + 1 + 0 + ... = 16.

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A000217, A006218, A007862, A013936, A013937, A013938, A013939.

Cf. A002024, A004275. - Enrique Pérez Herrero, Apr 05 2010

[(&+[Floor(n/(k*(k+1)/2)): k in [1..100]]): n in [0..30]]; // G. C. Greubel, May 23 2018

A069470[n_]:=Sum[Floor[(2*n)/(k*(1 + k))], {k, 1, Floor[1/2 + Sqrt[2*n]]}] (* Enrique Pérez Herrero, Apr 05 2010 *)

for(n=0, 30, print1(sum(k=1, 100, floor(n/(k*(k+1)/2))), ", ")) \\ G. C. Greubel, May 23 2018

a(n) = a(n-1) + A007862(n).
It appears that limit((sum(floor((1/2)*n/(k*(k+1))), k=1..n))/n, n=infinity) = 1/2. - Stephen Crowley, Aug 12 2009
From Enrique Pérez Herrero, Apr 05 2010: (Start)
a(n) <= floor((2*n^2)/(1 + n)) = A004275(n).
a(n) <= floor((2*n*floor((1 + 2*sqrt(2*n))/2))/(1+floor((1+2*sqrt(2*n))/2))). (End)
G.f.: (1/(1 - x)) * Sum_{k>=1} x^(k*(k+1)/2)/(1 - x^(k*(k+1)/2)). - Ilya Gutkovskiy, Jul 11 2019

A069972 a(n) = Sum_{d|2n,d+1|2n} d.

Original entry on oeis.org

1, 1, 3, 1, 1, 6, 1, 1, 3, 5, 1, 6, 1, 1, 8, 1, 1, 6, 1, 5, 9, 1, 1, 6, 1, 1, 3, 8, 1, 15, 1, 1, 3, 1, 1, 14, 1, 1, 3, 5, 1, 12, 1, 1, 17, 1, 1, 6, 1, 5, 3, 1, 1, 6, 11, 8, 3, 1, 1, 15, 1, 1, 9, 1, 1, 17, 1, 1, 3, 5, 1, 14, 1, 1, 8, 1, 1, 18, 1, 5, 3, 1, 1, 19, 1, 1, 3, 1, 1, 24, 14, 1, 3, 1, 1, 6, 1, 1

Offset: 1

Vladeta Jovovic, Apr 29 2002

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A007862.

A069972(n) = { n = 2*n; sumdiv(n,d,if(!(n%(1+d)), d, 0)); }; \\ Antti Karttunen, Mar 02 2023

G.f.: (conjectured) Sum_{n>=1} n*x^(n*(n+1)/2)/(1-x^(n*(n+1)/2)). - Joerg Arndt, Jan 30 2011

A244964 Number of distinct generalized pentagonal numbers dividing n.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jul 10 2014

nonn

For more information about the generalized pentagonal numbers see A001318.

			For n = 10 the generalized pentagonal numbers <= 10 are [0, 1, 2, 5, 7]. There are three generalized pentagonal numbers that divide 10; they are [1, 2, 5], so a(10) = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A001221, A001318, A001511, A005086, A006519, A007862, A027750, A046951, A080995, A147645, A175003, A236103, A238442, A239930.

a[n_] := DivisorSum[n, 1 &, IntegerQ[Sqrt[24*# + 1]] &]; Array[a, 100] (* Amiram Eldar, Dec 31 2023 *)

a(n) = sumdiv(n, d, issquare(24*d + 1)); \\ Amiram Eldar, Dec 31 2023

From Amiram Eldar, Dec 31 2023: (Start)
a(n) = Sum_{d|n} A080995(d).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 6 - 2*Pi/sqrt(3) = 2.372401... . (End)

A279830 a(n) = the least integer that is centered polygonal in exactly n ways.

Original entry on oeis.org

4, 7, 37, 31, 91, 181, 211, 421, 631, 1891, 1261, 2521, 6931, 18481, 20791, 13861, 27721, 41581, 83161, 138601, 245701, 235621, 180181, 556921, 360361, 540541, 1670761, 1081081, 1413721, 2702701, 2162161, 6486481, 3063061, 8288281, 13430341, 6846841, 10270261, 6126121

Offset: 1

Daniel Sterman, Dec 20 2016

nonn

a(n) has exactly n representations as a centered r-gonal number P(r,m) = 1 + r*m*(m+1)/2, with m > 1, r > 0.
a(n) appears n+1 times in A101321, due to the second column containing every positive integer.
a(n)-1 is the first appearance of n+1 in A007862.

			a(4)=31, because 31 is a centered triangular number (A005448), a centered pentagonal number (A005891), a centered decagonal number (A062786), and a central polygonal number (A002061). No number less than 31 has 4 representations.

Lars Blomberg, Table of n, a(n) for n = 1..143

Cf. A007862 (see alternative definition: the number of ways to represent n+1 as a centered polygonal number).
Cf. A063778 (the equivalent for polygonal numbers).
Subset of A275340 (the list of nontrivial centered polygonal numbers).
Subset of A101321 (centered polygonal numbers read by antidiagonals).

f[n_] := Length@Select[Divisors[2 n - 2], IntegerQ@Sqrt[1 + 4 #] &] - 1;
Do[If[IntegerQ[A279830[f[i]]], , A279830[f[i]] = i], {i, 10000}];
A279830 /@ Range[13]
(* Davin Park, Dec 28 2016 *)

Corrected and extended by Davin Park, Dec 27 2016

A294334 Number of partitions of n into triangular numbers dividing n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 1, 1, 4, 2, 1, 9, 1, 1, 7, 1, 1, 16, 1, 3, 9, 1, 1, 25, 1, 1, 10, 2, 1, 74, 1, 1, 12, 1, 1, 50, 1, 1, 14, 5, 1, 85, 1, 1, 35, 1, 1, 81, 1, 6, 18, 1, 1, 100, 2, 3, 20, 1, 1, 544, 1, 1, 46, 1, 1, 145, 1, 1, 24, 8, 1, 219, 1, 1, 81, 1, 1, 197, 1, 9, 28, 1, 1, 628, 1, 1, 30, 1, 1, 2264, 2, 1, 32, 1, 1

Offset: 0

Ilya Gutkovskiy, Oct 28 2017

			a(6) = 4 because 6 has 4 divisors {1, 2, 3, 6} among which 3 are triangular numbers {1, 3, 6} therefore we have [6], [3, 3], [3, 1, 1, 1] and [1, 1, 1, 1, 1, 1].

Cf. A000217, A007294, A007862, A018818, A112886, A284345.

Table[SeriesCoefficient[Product[1/(1 - Boole[Mod[n, k] == 0 && IntegerQ[Sqrt[8 k + 1]]] x^k), {k, 1, n}], {x, 0, n}], {n, 0, 95}]

a(n) = 1 if n in A112886.

A333631 Number of permutations of {1..n} with three consecutive terms in arithmetic progression.

Original entry on oeis.org

0, 0, 0, 2, 6, 40, 238, 1760, 14076, 131732, 1308670, 14678452, 176166906, 2317481348, 32416648496, 490915956484, 7846449011500, 134291298372632, 2416652824505150, 46141903780094080, 922528719841017424, 19456439433050482412, 427837767407051523776, 9873256397944571377332

Offset: 0

Gus Wiseman, Mar 31 2020

nonn

Also permutations whose second differences have at least one zero.

			The a(3) = 2 and a(4) = 6 permutations:
  (1,2,3)  (1,2,3,4)
  (3,2,1)  (1,4,3,2)
           (2,3,4,1)
           (3,2,1,4)
           (4,1,2,3)
           (4,3,2,1)

Wikipedia, Arithmetic progression

The complement is counted by A295370.
The version for prime indices is A333195.
Strict partitions with equal differences are A049980.
Partitions with equal differences are A049988.
Compositions without triples in arithmetic progression are A238423.
Partitions without triples in arithmetic progression are A238424.
Strict partitions without triples in arithmetic progression are A332668.
Cf. A000142, A007862, A175342, A325849, A325850.

Table[Select[Permutations[Range[n]],MatchQ[Differences[#],{_,x_,x_,_}]&]//Length,{n,0,8}]

a(n) = n! - A295370(n).

a(11)-a(21) (using A295370) from Giovanni Resta, Apr 07 2020

a(22)-a(23) (using A295370) from Alois P. Heinz, Jan 27 2024

A334925 G.f.: Sum_{k>=1} x^(k(k^2 + 1)/2) / (1 - x^(k(k^2 + 1)/2)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 2, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2

Offset: 1

Ilya Gutkovskiy, May 16 2020

nonn

Number of divisors of n of the form k*(k^2 + 1)/2 (A006003).

Cf. A006003, A007862, A279495, A300409, A334926.

Cf. A001620, A248177.

nmax = 100; CoefficientList[Series[Sum[x^(k (k^2 + 1)/2)/(1 - x^(k (k^2 + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 * (A248177 + A001620) = 1.343731... . - Amiram Eldar, Jan 02 2024

A343407 Number of proper divisors of n that are triangular numbers.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 14 2021

nonn

Cf. A000217, A007862, A010054, A032741, A293234, A343408.

a:= n-> add(`if`(issqr(8*d+1), 1, 0), d=numtheory[divisors](n) minus {n}):
seq(a(n), n = 1..105);  # Alois P. Heinz, Apr 14 2021

nmax = 105; CoefficientList[Series[Sum[x^(k (k + 1))/(1 - x^(k (k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[If[d < n && IntegerQ[Sqrt[8 d + 1]], 1, 0], {d, Divisors[n]}], {n, 105}]

a(n) = sumdiv(n, d, if ((dMichel Marcus, Apr 14 2021

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

a(n) = Sum_{d|n, d < n} A010054(d).

A327764 Expansion of 1 / (1 - Sum_{i>=1, j>=1} x^(ij(j + 1)/2)).

Original entry on oeis.org

1, 1, 2, 5, 10, 21, 47, 99, 211, 455, 973, 2081, 4464, 9558, 20466, 43848, 93914, 201140, 430844, 922818, 1976553, 4233613, 9067960, 19422576, 41601229, 89105550, 190854784, 408791400, 875589076, 1875421302, 4016959325, 8603912899, 18428694036, 39472363286

Offset: 0

Ilya Gutkovskiy, Sep 24 2019

nonn

Invert transform of A007862.

Cf. A007862, A129921, A327738, A327744, A327745.

nmax = 33; CoefficientList[Series[1/(1 - Sum[x^(k (k + 1)/2)/(1 - x^(k (k + 1)/2)), {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Length[Select[Divisors[k], IntegerQ[Sqrt[8 # + 1]] &]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}]

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

a(0) = 1; a(n) = Sum_{k=1..n} A007862(k) * a(n-k).

cp's OEIS Frontend

A329801 Expansion of Sum_{k>=1} x^(k*(k + 1)/2) / (1 + x^(k*(k + 1)/2)).

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A069470 a(n) = Sum_{k>=1} floor(n/(k*(k+1)/2)).

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A069972 a(n) = Sum_{d|2*n,d+1|2*n} d.

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A244964 Number of distinct generalized pentagonal numbers dividing n.

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A279830 a(n) = the least integer that is centered polygonal in exactly n ways.

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A294334 Number of partitions of n into triangular numbers dividing n.

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A333631 Number of permutations of {1..n} with three consecutive terms in arithmetic progression.

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Extensions

A334925 G.f.: Sum_{k>=1} x^(k*(k^2 + 1)/2) / (1 - x^(k*(k^2 + 1)/2)).

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A329801 Expansion of Sum_{k>=1} x^(k(k + 1)/2) / (1 + x^(k(k + 1)/2)).

A069972 a(n) = Sum_{d|2n,d+1|2n} d.

A334925 G.f.: Sum_{k>=1} x^(k(k^2 + 1)/2) / (1 - x^(k(k^2 + 1)/2)).

A327764 Expansion of 1 / (1 - Sum_{i>=1, j>=1} x^(ij(j + 1)/2)).