A332989 -id:A332989 - cp's OEIS frontend

A374141 a(n) is the smallest number which can be represented as the sum of two distinct nonzero hexagonal numbers in exactly n ways, or -1 if no such number exists.

7, 384, 4995, 51106, 204805, 483031, 2443431, 4674256, 10476781, 17272531, 25600656, 60765331, 90406956, 206602126, 332808531, 481676406, 303826656, 435211156, 789949306, 1406495106, 2260173906, 2704798281, 3220562556, 4435869181, 5165053156, 5309576106, 9818788281

Offset: 1

Ilya Gutkovskiy, Jun 28 2024

nonn

			a(2) = 384 = 6 + 378 = 153 + 231.

Michael S. Branicky, Table of n, a(n) for n = 1..39
Michael S. Branicky, Python program for A374141, A374142, and A374143
Eric Weisstein's World of Mathematics, Hexagonal Number

Cf. A000384, A093195, A332989, A342326, A374142, A374143.

Python
```
# see linked program
```

a(9)-a(27) from Michael S. Branicky, Jun 29 2024

A333014 Numbers which can written in exactly four ways as a sum of two distinct nonzero pentagonal numbers.

Original entry on oeis.org

13352, 18877, 45397, 49052, 52027, 53727, 62652, 64182, 73152, 74977, 76677, 79327, 80671, 85177, 87972, 88577, 90702, 91652, 93302, 96669, 98827, 101752, 106036, 106822, 109227, 109487, 116117, 118477, 125347, 133267, 135786, 138087, 138802, 140852, 141532, 144747, 145302, 145641, 147274, 148077, 148927

Offset: 1

Olivier Gérard, Mar 05 2020

nonn

			13352 = P(52) + P(79) = P(29) + P(90) = P(17) + P(93) = P(10) + P(94), where P(n) is the n-th pentagonal number A000326.

Cf. A000326, A332988, A332989, A333011, A333012, A333013.

is(k) = sum(i=1, sqrt(1+12*k)\6, sqrt(1+24*k+12*i-36*i*i)%6==5) == 4; \\ Jinyuan Wang, Mar 06 2020

A374142 a(n) is the smallest number which can be represented as the sum of two distinct nonzero heptagonal numbers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

8, 617, 8726, 255575, 1339801, 2419165, 9402323, 25764500, 35486953, 144568133, 385495261, 735503569, 638279039, 1183071664, 1571634527, 4449717748, 3584182298, 3871587494, 5693954599, 27084640649, 24205505111, 32489035067, 31973745058, 38935021406, 47570693867, 44749048300, 53075499329

Offset: 1

Ilya Gutkovskiy, Jun 28 2024

nonn

			a(2) = 617 = 1 + 616 = 148 + 469.

Michael S. Branicky, Python program for A374141, A374142, and A374143
Eric Weisstein's World of Mathematics, Heptagonal Number

Cf. A000566, A093195, A332989, A342326, A374141, A374143.

Python
```
# see linked program
```

a(8)-a(27) from Michael S. Branicky, Jun 29 2024

A374143 a(n) is the smallest number which can be represented as the sum of two distinct nonzero octagonal numbers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

9, 1053, 12641, 68141, 365641, 953181, 2830641, 6232341, 13969041, 23211261, 104733741, 84994021, 175873641, 159851141, 538547641, 602713041, 810204416, 1019740041, 1053265741, 1972957241, 3339356041, 5914492241, 6886737541, 6388758241, 8902368041, 7858982841, 4942246941, 18439299341, 26639916441

Offset: 1

Ilya Gutkovskiy, Jun 28 2024

nonn

			a(2) = 1053 = 8 + 1045 = 408 + 645.

Michael S. Branicky, Table of n, a(n) for n = 1..38
Michael S. Branicky, Python program for A374141, A374142, and A374143
Eric Weisstein's World of Mathematics, Octagonal Number

Cf. A000567, A093195, A332989, A342326, A374141, A374142.

Python
```
# see linked program
```

a(9)-a(29) from Michael S. Branicky, Jun 29 2024

A374144 a(n) is the smallest number which can be represented as the sum of two distinct nonzero n-gonal numbers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

81, 1105, 205427, 483031, 9402323, 6232341, 79324200, 768459127, 2265692766, 2413112833, 6737406626, 150437989675, 45319359337, 15140186701

Offset: 3

Ilya Gutkovskiy, Jun 28 2024

			a(3) = 81 = 3 + 78 = 15 + 66 = 36 + 45.

Eric Weisstein's World of Mathematics, Polygonal Number
Michael S. Branicky, Python program for A374144

Cf. A057145, A093195, A332989, A342326, A350405, A374141, A374142, A374143.

Python
```
# see linked program
```

a(9)-a(16) from Michael S. Branicky, Jun 30 2024

A333015 Numbers which can be written in exactly five ways as a sum of two distinct nonzero pentagonal numbers.

Original entry on oeis.org

205427, 210552, 230102, 269712, 333802, 346977, 354537, 384802, 397892, 416677, 420077, 426622, 448552, 470902, 471927, 478302, 509752, 520852, 563772, 566177, 569507, 571377, 575202, 580302, 586102, 590162, 599847, 610052, 616552, 618263, 635552, 646177, 647947

Offset: 1

Olivier Gérard, Mar 05 2020

nonn

			205427 = P(234) + P(287) = P(201) + P(311) = P(166) + P(331) = P(56) + P(366) = P(49) + P(367), where P(n) is the n-th pentagonal number (A000326).

Cf. A000326, A332988, A332989, A333011, A333012, A333013, A333014.

is(k) = sum(i=1, sqrt(1+12*k)\6, sqrt(1+24*k+12*i-36*i*i)%6==5) == 5; \\ Jinyuan Wang, Mar 06 2020

More terms from Jinyuan Wang, Mar 06 2020

A374193 a(n) is the smallest number which can be represented as the sum of two nonzero pentagonal pyramidal numbers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

2, 1471, 269406

Offset: 1

Ilya Gutkovskiy, Jun 30 2024

There are no further positive terms <= 10^15. - Michael S. Branicky, Jul 01 2024

			a(2) = 1471 = 1 + 1470 = 288 + 1183.

Eric Weisstein's World of Mathematics, Pentagonal Pyramidal Number

Cf. A002411, A011541, A332989, A334013, A374168, A374194.

cp's OEIS Frontend

A374141 a(n) is the smallest number which can be represented as the sum of two distinct nonzero hexagonal numbers in exactly n ways, or -1 if no such number exists.

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A333014 Numbers which can written in exactly four ways as a sum of two distinct nonzero pentagonal numbers.

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A374142 a(n) is the smallest number which can be represented as the sum of two distinct nonzero heptagonal numbers in exactly n ways, or -1 if no such number exists.

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A374143 a(n) is the smallest number which can be represented as the sum of two distinct nonzero octagonal numbers in exactly n ways, or -1 if no such number exists.

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A374144 a(n) is the smallest number which can be represented as the sum of two distinct nonzero n-gonal numbers in exactly n ways, or -1 if no such number exists.

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A333015 Numbers which can be written in exactly five ways as a sum of two distinct nonzero pentagonal numbers.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Extensions

A374193 a(n) is the smallest number which can be represented as the sum of two nonzero pentagonal pyramidal numbers in exactly n ways, or -1 if no such number exists.

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