A319185 -id:A319185 - 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 *)

A322637 Numbers that are sums of consecutive octagonal numbers (A000567).

Original entry on oeis.org

0, 1, 8, 9, 21, 29, 30, 40, 61, 65, 69, 70, 96, 105, 126, 133, 134, 135, 161, 176, 201, 222, 225, 229, 230, 231, 280, 294, 309, 334, 341, 355, 363, 364, 401, 405, 408, 470, 481, 505, 510, 531, 534, 539, 540, 560, 621, 630, 645, 681, 695, 735, 736, 749, 756, 764, 765, 814, 833, 846

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, Octagonal Number

Cf. A000567, A002414, A034705, A034706, A319184, A319185, A322636.

N:= 1000: # for terms up to N
Octa:= [seq(n*(3*n-2),n=0..floor((1+sqrt(1+3*N))/3))]:
PS:= ListTools:-PartialSums(Octa):
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 = 60;
nmax = 17; kmax = 9; (* empirical *)
T = Table[n(3n-2), {n, 0, nmax}];
Union[T, Table[k MovingAverage[T, k], {k, 2, kmax}]//Flatten][[1 ;; terms]] (* Jean-François Alcover, Dec 26 2018 *)

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 *)

A334010 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero hexagonal numbers in exactly n ways.

Original entry on oeis.org

1, 703, 274550, 11132303325

Offset: 1

Ilya Gutkovskiy, Apr 12 2020

			Let S(k, m) denote the sum of m hexagonal numbers starting from the k-th. We have
a(1) = S(1, 1);
a(2) = S(19, 1) = S(13, 2);
a(3) = S(62, 25) = S(184, 4) = S(25, 51);
a(4) = S(3065, 505) = S(22490, 11) = S(1215, 1430) = S(1938, 946).

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

Cf. A000384, A054859, A068314, A186337, A298467, A319185, A334007, A334008, A334011, A334012.

a(4) from Giovanni Resta, Apr 13 2020

A322651 Numbers that are sums of consecutive hexagonal pyramidal numbers (A002412).

Original entry on oeis.org

0, 1, 7, 8, 22, 29, 30, 50, 72, 79, 80, 95, 145, 161, 167, 174, 175, 252, 256, 306, 328, 335, 336, 372, 413, 508, 525, 558, 580, 587, 588, 624, 715, 785, 880, 897, 930, 946, 952, 959, 960, 1149, 1222, 1240, 1310, 1405, 1455, 1477, 1484, 1485, 1547, 1612, 1661, 1864, 1925

Offset: 1

Ilya Gutkovskiy, Dec 21 2018

nonn

Eric Weisstein's World of Mathematics, Hexagonal Pyramidal Number

Cf. A002412, A002417, A152043, A319185, A321450, A322468, A322479, A322652, A322653.

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

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A322637 Numbers that are sums of consecutive octagonal numbers (A000567).

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A319184 Numbers that are sums of consecutive pentagonal numbers.

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A334010 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero hexagonal numbers in exactly n ways.

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A322651 Numbers that are sums of consecutive hexagonal pyramidal numbers (A002412).

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