cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A334007 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero triangular numbers in exactly n ways.

Original entry on oeis.org

1, 10, 2180, 10053736, 13291443468940
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 12 2020

Keywords

Examples

			Let S(k, m) denote the sum of m triangular numbers starting from k(k+1)/2. We have
a(1) = S(1, 1);
a(2) = S(4, 1) = S(1, 3);
a(3) = S(31, 4) = S(27, 5) = S(9, 15);
a(4) = S(945, 22) = S(571, 56) = S(968, 21) = S(131, 266);
a(5) = S(4109, 38947) = S(25213, 20540) = S(10296, 32943) = S(32801, 15834) = S(31654, 16472).

Links

Crossrefs

Cf. A000217, A054859, A068314, A034706, A050943, A186337, A298467, A307666, A309783, A334008, A334010, A334011, A334012.

Extensions

a(5) from Giovanni Resta, Apr 13 2020

A344376 Numbers that are both octagonal numbers (A000567) and octagonal pyramidal numbers (A002414).

Original entry on oeis.org

0, 1, 1045, 5985, 123395663059845, 774611255177760
Offset: 1

Views

Author

Seiichi Manyama, May 16 2021

Keywords

Comments

Terms correspond to integral points on an elliptic curve, which allows all of them to be found efficiently. - Max Alekseyev, Feb 19 2024

Crossrefs

Intersection of A000567 and A002414.
Cf. A027568, A334012.
Cf. A378361 (octagonal indices), A378918 (octagonal pyramidal indices).

Programs

  • PARI
    for(k=0, 1e5, if(ispolygonal(m=k*(k+1)*(2*k-1)/2, 8), print1(m", ")))

Extensions

Keywords 'fini' and 'full' added by Max Alekseyev, Feb 19 2024

A334008 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero pentagonal numbers in exactly n ways.

Original entry on oeis.org

1, 287, 472320, 89051435880
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 12 2020

Keywords

Examples

			Let S(k, m) denote the sum of m pentagonal numbers starting from the k-th. We have
a(1) = S(1, 1);
a(2) = S(14, 1) = S(2, 7);
a(3) = S(103, 24) = S(19, 80) = S(67, 41);
a(4) = S(10833, 484) = S(4542, 1936) = S(9153, 660) = S(2817, 3036);

Links

Crossrefs

Cf. A000326, A054859, A068314, A186337, A298467, A319184, A334007, A334010, A334011, A334012.

Extensions

a(4) from Giovanni Resta, Apr 13 2020

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

Views

Author

Ilya Gutkovskiy, Apr 12 2020

Keywords

Examples

			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).

Links

Crossrefs

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

Extensions

a(4) from Giovanni Resta, Apr 13 2020

A334011 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero heptagonal numbers in exactly n ways.

Original entry on oeis.org

1, 872, 8240232, 263346158075
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 12 2020

Keywords

Examples

			Let S(k, m) denote the sum of m heptagonal numbers starting from the k-th. We have
a(1) = S(1, 1);
a(2) = S(13, 2) = S(3, 8);
a(3) = S(133, 98) = S(479, 14) = S(168, 77);
a(4) = S(6773, 1785) = S(810, 6006) = S(7467, 1547) = S(38758, 70).

Links

Crossrefs

Cf. A000566, A054859, A068314, A186337, A298467, A322636, A334007, A334008, A334010, A334012.

Extensions

a(4) from Giovanni Resta, Apr 14 2020

A334037 a(n) is the least integer that can be expressed as the difference of two octagonal numbers in exactly n ways.

Original entry on oeis.org

1, 133, 560, 1729, 4160, 10640, 14560, 22400, 44800, 58240, 138320, 98560, 123200, 203840, 246400, 394240, 320320, 492800, 800800, 640640, 1047200, 1823360, 1724800, 1281280, 2094400, 1601600, 2475200, 2722720, 4484480, 3203200, 5532800, 6697600, 5445440, 7958720
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 12 2020

Keywords

Comments

The least integer that can be expressed as the sum of one or more consecutive numbers congruent to 1 mod 6 in exactly n ways.
Index of first occurrence of n in A333818.

Links

Crossrefs

Cf. A000567, A016921, A038547, A068314, A333818, A334012, A334034, A334035, A334036.

Extensions

More terms from Jinyuan Wang, Apr 13 2020

A343777 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero n-gonal numbers in exactly n ways.

Original entry on oeis.org

2180, 554503705
Offset: 3

Views

Author

Ilya Gutkovskiy, Apr 30 2021

Keywords

Links

Crossrefs

Cf. A298467, A334007, A334008, A334010, A334011, A334012, A338105.
Showing 1-7 of 7 results.

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