A362424 -id:A362424 - cp's OEIS frontend

A362425 Number of partitions of n into 3 distinct perfect powers (A001597).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 3, 0, 3, 3, 1, 1, 1, 4, 1, 1, 2, 3, 1, 0, 3, 1, 2, 1, 3, 4, 2, 1, 1, 2, 3, 2, 2, 4, 1, 1, 2, 3, 2, 2, 2, 4, 1, 1, 1, 2, 2, 1, 4, 2, 2, 0, 2, 3, 4, 1

Offset: 0

Ilya Gutkovskiy, Apr 19 2023

nonn

Eric Weisstein's World of Mathematics, Perfect Power.

Cf. A001597, A362424.

perfectPowerQ[n_] := n == 1 || GCD @@ FactorInteger[n][[;; , 2]] > 1; a[n_] := Count[IntegerPartitions[n, {3}], ?(AllTrue[#, perfectPowerQ] && UnsameQ @@ # &)]; Array[a, 100, 0] (* _Amiram Eldar, May 05 2023 *)

A363040 a(n) is the smallest number which can be represented as the sum of n distinct perfect powers (A001597) in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

1, 17, 37, 53, 86, 119, 177, 215, 275, 331, 424, 516, 632, 764, 928, 1057, 1247, 1427, 1635, 1879, 2119, 2409, 2715, 3008, 3395, 3760, 4189, 4667, 5171, 5617, 6178, 6786, 7438, 8071, 8836, 9572, 10456, 11333, 12396, 13266, 14214, 15379, 16518, 17703, 19018, 20275

Offset: 1

Ilya Gutkovskiy, May 14 2023

nonn

			For n = 2: 17 = 1 + 16 = 8 + 9.

Eric Weisstein's World of Mathematics, Perfect Power

Cf. A001597, A362424, A362425, A362428.

a(13) and beyond from Michael S. Branicky, May 24 2023

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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Python
```
# see link
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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

A362425 Number of partitions of n into 3 distinct perfect powers (A001597).

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A363040 a(n) is the smallest number which can be represented as the sum of n distinct perfect powers (A001597) in exactly n ways, or -1 if no such number exists.

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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.

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Author

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PARI

PARI

Python

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