A322637 -id:A322637 - cp's OEIS frontend

A322636 Numbers that are sums of consecutive heptagonal numbers (A000566).

0, 1, 7, 8, 18, 25, 26, 34, 52, 55, 59, 60, 81, 89, 107, 112, 114, 115, 136, 148, 170, 188, 189, 193, 195, 196, 235, 248, 260, 282, 286, 300, 307, 308, 337, 341, 342, 396, 403, 424, 430, 448, 449, 455, 456, 469, 521, 530, 540, 572, 585, 616, 619, 628, 637, 644, 645, 684, 697

Offset: 1

Ilya Gutkovskiy, Dec 21 2018

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Heptagonal Number

Cf. A000566, A002413, A034705, A034706, A319184, A319185, A322637.

N:= 1000: # for terms up to N
Hepta:= [seq(n*(5*n-3)/2,n=0..floor((3+sqrt(9+40*N))/10))]:
PS:= ListTools:-PartialSums(Hepta):
S:= select(`<=`,{0,seq(seq(PS[i]-PS[j],j=1..i-1),i=1..nops(PS))},N):
sort(convert(S,list)); # Robert Israel, May 22 2025

terms = 59;
nmax = 17; kmax = 9; (* empirical *)
T = Table[n(5n-3)/2, {n, 0, nmax}];
Union[T, Table[k MovingAverage[T, k], {k, 2, kmax}]//Flatten][[1 ;; terms]] (* Jean-François Alcover, Dec 26 2018 *)

A334012 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero octagonal numbers in exactly n ways.

Original entry on oeis.org

1, 1045, 5985

Offset: 1

Ilya Gutkovskiy, Apr 12 2020

			From _Seiichi Manyama_, May 16 2021: (Start)
Let S(k, m) denote the sum of m octagonal numbers starting from k*(3*k-2). We have
a(1) = S(1, 1);
a(2) = S(19, 1) = S(1, 10);
a(3) = S(45, 1) = S(11, 9) = S(1, 18). (End)

Eric Weisstein's World of Mathematics, Octagonal Number
Index to sequences related to polygonal numbers

Cf. A000567, A054859, A068314, A186337, A298467, A322637, A334007, A334008, A334010, A334011, A344376.

A319184 Numbers that are sums of consecutive pentagonal numbers.

Original entry on oeis.org

0, 1, 5, 6, 12, 17, 18, 22, 34, 35, 39, 40, 51, 57, 69, 70, 74, 75, 86, 92, 108, 117, 120, 121, 125, 126, 145, 156, 162, 176, 178, 190, 195, 196, 209, 210, 213, 247, 248, 262, 270, 279, 282, 287, 288, 321, 330, 354, 365, 376, 386, 387, 399, 404, 405, 424, 425, 438, 457, 475

Offset: 1

Ilya Gutkovskiy, Dec 21 2018

nonn

Eric Weisstein's World of Mathematics, Pentagonal Number

Cf. A000326, A002411, A034705, A034706, A319185, A322636, A322637.

anmax = 1000; nmax = Floor[Sqrt[2*anmax/3]] + 1; Select[Union[Flatten[Table[Sum[k*(3*k-1)/2, {k, i, j}], {i, 0, nmax}, {j, i, nmax}]]], # <= anmax &] (* Vaclav Kotesovec, Dec 21 2018 *)
Module[{nn=20,pn},pn=PolygonalNumber[5,Range[0,nn]];Take[Union[Flatten[Table[Total/@Partition[pn,d,1],{d,nn}]]],60]] (* Harvey P. Dale, Jun 22 2025 *)

A319185 Numbers that are sums of consecutive hexagonal numbers (A000384).

Original entry on oeis.org

0, 1, 6, 7, 15, 21, 22, 28, 43, 45, 49, 50, 66, 73, 88, 91, 94, 95, 111, 120, 139, 153, 154, 157, 160, 161, 190, 202, 211, 230, 231, 245, 251, 252, 273, 276, 277, 322, 325, 343, 350, 364, 365, 371, 372, 378, 421, 430, 435, 463, 475, 496, 503, 507, 518, 524, 525, 554, 561

Offset: 1

Ilya Gutkovskiy, Dec 21 2018

nonn

Eric Weisstein's World of Mathematics, Hexagonal Number

Cf. A000384, A002412, A034705, A034706, A319184, A322636, A322637.

anmax = 1000; nmax = Floor[Sqrt[anmax/2]] + 1; Select[Union[Flatten[Table[Sum[k*(2*k-1), {k, i, j}], {i, 0, nmax}, {j, i, nmax}]]], # <= anmax &] (* Vaclav Kotesovec, Dec 21 2018 *)

A320728 Numbers that are sums of consecutive odd squares (or centered octagonal numbers).

Original entry on oeis.org

0, 1, 9, 10, 25, 34, 35, 49, 74, 81, 83, 84, 121, 130, 155, 164, 165, 169, 202, 225, 251, 276, 285, 286, 289, 290, 361, 371, 394, 420, 441, 445, 454, 455, 514, 515, 529, 596, 625, 645, 650, 670, 679, 680, 683, 729, 802, 804, 841, 875, 885, 934, 959, 961, 968, 969, 970, 1044, 1089

Offset: 1

Ilya Gutkovskiy, Dec 21 2018

nonn

Index entries for sequences related to sums of squares

Cf. A000447, A016754, A034705, A152043, A322610, A322611, A322637, A322638, A322640.

ok(n)={my(i=sqrtint(n)); i=i-(i%2==0); while(i>0, my(a=i^2, j=i); while(j>0 && a<=n, if(a==n, return(1)); j-=2; a=a+j^2); i-=2); 0}
concat([0], select(ok, [1..1200])) \\ Antonio Roldán, Mar 12 2020

A322653 Numbers that are sums of consecutive octagonal pyramidal numbers (A002414).

Original entry on oeis.org

0, 1, 9, 10, 30, 39, 40, 70, 100, 109, 110, 135, 205, 231, 235, 244, 245, 364, 366, 436, 466, 475, 476, 540, 595, 730, 765, 800, 830, 839, 840, 904, 1045, 1135, 1270, 1305, 1340, 1370, 1379, 1380, 1386, 1669, 1794, 1810, 1900, 2035, 2105, 2135, 2144, 2145, 2275, 2350, 2431, 2714

Offset: 1

Ilya Gutkovskiy, Dec 21 2018

nonn

Eric Weisstein's World of Mathematics, Pyramidal Number

Cf. A002414, A002419, A320728, A321450, A322468, A322479, A322637, A322651, A322652.

cp's OEIS Frontend

A322636 Numbers that are sums of consecutive heptagonal numbers (A000566).

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A334012 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero octagonal numbers in exactly n ways.

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A319184 Numbers that are sums of consecutive pentagonal numbers.

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A319185 Numbers that are sums of consecutive hexagonal numbers (A000384).

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Mathematica

A320728 Numbers that are sums of consecutive odd squares (or centered octagonal numbers).

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PARI

A322653 Numbers that are sums of consecutive octagonal pyramidal numbers (A002414).

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