A093195 -id:A093195 - cp's OEIS frontend

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.

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
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# see linked program
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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
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# see linked program
```

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

A374095 a(n) is the smallest nonnegative integer k where exactly n solutions to x^2 + 3xy + y^2 = k with 0 < x < y.

Original entry on oeis.org

0, 11, 209, 2299, 6061, 278179, 66671, 5285401, 187891, 1266749, 8067191

Offset: 0

Seiichi Manyama, Jun 28 2024

a(n) is the smallest nonnegative k such that A374093(k) = n.

Cf. A093195, A374094.

Cf. A045468, A374091, A374093.

A374226 a(n) is the smallest number which can be represented as the sum of two distinct positive n-th powers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

3, 65, 87539319

Offset: 1

Ilya Gutkovskiy, Jul 01 2024

			a(3) = 87539319 = 167^3 + 436^3 = 228^3 + 423^3 = 255^3 + 414^3.

Cf. A011541, A093195, A146756, A374227, A374228.

A374287 a(n) is the smallest number which can be represented as the sum of n distinct nonzero squares in exactly 2 ways, or -1 if no such number exists.

Original entry on oeis.org

-1, 65, 62, 90, 103, 136, 200, 276, 376, 481, 625, 806, 975, 1183, 1415, 1688, 1989, 2325, 2698, 3110, 3563, 4059, 4600, 5188, 5825, 6513, 7254, 8050, 8903, 9815, 10788, 11824, 12925, 14093, 15330, 16638, 18019, 19475, 21008, 22620, 24313, 26089, 27950, 29898

Offset: 1

Ilya Gutkovskiy, Jul 02 2024

sign

			a(2) = 65 = 1^2 + 8^2 = 4^2 + 7^2.
a(3) = 62 = 1^2 + 5^2 + 6^2 = 2^2 + 3^2 + 7^2.

Index entries for sequences related to sums of squares

Cf. A000330, A093195, A350241.

a(12) and beyond from Michael S. Branicky, Jul 02 2024

A365295 a(n) is the least positive integer that can be expressed as the sum of two distinct perfect powers (A001597) in exactly n ways.

Original entry on oeis.org

1, 5, 17, 129, 468, 1025, 2628, 12025, 32045, 27625, 138125, 430625, 204425, 160225, 2010025, 2348125, 801125, 1743625, 2082925, 4978025, 4005625, 12325625, 30525625, 73046025, 5928325, 13287625, 46437625, 45177925, 35409725, 120737825, 52073125, 66438125, 29641625, 32846125, 956974625

Offset: 0

Ilya Gutkovskiy, Aug 31 2023

nonn

			For n = 2: a(2) = 17 = 1^2 + 2^4 = 2^3 + 3^2.
a(6) = 2628 via 3^3 + 51^2 = 2^7 + 50^2 = 18^2 + 48^2 = 21^2 + 3^7 = 2^9 + 46^2 = 30^2 + 12^3. - _David A. Corneth_, Sep 09 2023

Karl-Heinz Hofmann and Hugo Pfoertner, Table of n, a(n) for n = 0..77
Karl-Heinz Hofmann, Python program
Hugo Pfoertner, PARI program and results (terms < 2*10^9), Sep 10 2023.

Cf. A001597, A093195, A362424, A363040.

upto(n) = {n = (sqrtint(n) + 1)^2; my(v = vector(n), pows = List([1]), r = -1, res = []); for(j = 2, logint(n, 2), for(i = 2, sqrtnint(n, j), listput(pows, i^j))); pows = Set(pows); for(i = 1, #pows - 1, j = i+1; c = pows[i] + pows[j]; while(c <= n, v[c]++; j++; c = pows[i] + pows[j])); for(i = 1, #v, c = v[i]+1; if(c > #res, res = concat(res, vector(c - #res, j, oo))); if(i < res[c], res[c] = i)); res} \\ David A. Corneth, Sep 08 2023

PARI
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\\ see link
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Python
```
# see link
```

a(8)-a(10) from David Consiglio, Jr., Sep 08 2023

a(9) corrected and a(11)-a(34) from Hugo Pfoertner, Sep 10 2023

cp's OEIS Frontend

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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A374095 a(n) is the smallest nonnegative integer k where exactly n solutions to x^2 + 3*x*y + y^2 = k with 0 < x < y.

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A374226 a(n) is the smallest number which can be represented as the sum of two distinct positive n-th powers in exactly n ways, or -1 if no such number exists.

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A374287 a(n) is the smallest number which can be represented as the sum of n distinct nonzero squares in exactly 2 ways, or -1 if no such number exists.

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Author

Keywords

Examples

Links

Crossrefs

Extensions

A365295 a(n) is the least positive integer that can be expressed as the sum of two distinct perfect powers (A001597) in exactly n ways.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

PARI

Python

Extensions

A374095 a(n) is the smallest nonnegative integer k where exactly n solutions to x^2 + 3xy + y^2 = k with 0 < x < y.