A123075 -id:A123075 - cp's OEIS frontend

A085559 Smallest number expressible as the sum of three 4th powers in at least n ways.

1, 2673, 811538, 5978882, 137149922, 292965218, 779888018, 5745705602, 49511121842, 49511121842, 281539574498, 281539574498, 7865870969138, 7865870969138, 7865870969138, 7865870969138, 47580188090162

Offset: 1

Stuart Gascoigne (stuart.g(AT)scoigne.com), Jul 04 2003

nonn

The 4th powers used to sum to the numbers in this sequence include some zero terms. Hardy and Wright prove that the sequence is infinite.

			811538 = 29^4 + 17^4 + 12^4 = 28^4 + 21^4 + 7^4 = 27^4 + 23^4 + 4^4.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th Ed (1979) p. 330.

Cf. A011541, A123075.

A343082 a(n) is the smallest number that is the sum of n positive 4th powers in three ways.

Original entry on oeis.org

811538, 16578, 4225, 2676, 2677, 518, 519, 520, 521, 522, 523, 524, 525, 526, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307

Offset: 3

Sean A. Irvine, Apr 04 2021

nonn

This is r(n,4,3) in Alter's notation.

			a(3) = 811538 = 4^4 + 23^4 + 27^4 = 7^4 + 21^4 + 28^4 = 12^4 + 17^4 + 29^4.
a(4) = 16578 = 1^4 + 2^4 + 9^4 + 10^4 = 2^4 + 5^4 + 6^4 + 11^4 = 3^4 + 7^4 + 8^4 + 10^4.

R. Alter, Computations and generalizations on a remark of Ramanujan, pp. 182-196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981. See Table 8, page 192.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A085559, A123075, A342893, A343077, A343081, A343083, A343086.

a(n) = n + 255 for n >= 17.

A343086 a(n) is the smallest number that is the sum of n positive 4th powers in four ways.

Original entry on oeis.org

5978882, 236674, 20995, 6626, 2925, 2925, 2854, 1620, 1621, 777, 778, 779, 780, 781, 782, 528, 529, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 512, 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 530, 531

Offset: 3

Sean A. Irvine, Apr 04 2021

nonn

This is r(n,4,4) in Alter's notation.

			a(3) = 5978882 = 3^4 + 40^4 + 43^4 = 8^4 + 37^4 + 45^4 = 15^4 + 32^4 + 47^4 = 23^4 + 25^4 + 48^4.
a(4) = 236674 = 1^4 + 2^4 + 7^4 + 22^4 = 3^4 + 6^4 + 18^4 + 19^4 = 7^4 + 14^4 + 16^4 + 19^4 = 8^4 + 16^4 + 17^4 + 17^4.

R. Alter, Computations and generalizations on a remark of Ramanujan, pp. 182-196 of "Analytic Number Theory (Philadelphia, 1980)", ed. M. I. Knopp, Lect. Notes Math., Vol. 899, 1981. See Table 12, page 194.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A085559, A123075, A342897, A343077, A343082, A343084, A343085.

a(n) = n + 480 for n >= 32.

A374227 a(n) is the smallest number which can be represented as the sum of three distinct positive n-th powers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

6, 62, 5104, 5978882

Offset: 1

Ilya Gutkovskiy, Jul 01 2024

			a(3) = 5104 = 1^3 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3.

Cf. A025415, A025419, A123075, A146756, A374226, A374228.

A193244 Numbers that are the sum of three biquadrates (fourth powers) in more than one way.

Original entry on oeis.org

16562, 28593, 35378, 43218, 54977, 94178, 106353, 122018, 134162, 137633, 149058, 181202, 195122, 198497, 235298, 235553, 264113, 264992, 300833, 318402, 324818, 364658, 384833, 439922, 457488, 462722, 514098, 522242, 566048, 611618, 624962

Offset: 1

M. F. Hasler, Jan 01 2013

nonn

Erroneous version of A309762. - Ilya Gutkovskiy, Aug 15 2019

A subsequence of A003337. Lists the indices such that A193243(n) > 1.

			a(1) = 16562 = 1^4 + 9^4 + 10^4 = 5^4 + 6^4 + 11^4, and this is the least number having two such decompositions.

Cf. A003337, A193243, A047714, A085559, A122540, A123075.

is_A000404(n)={ for( i=1, #n=factor(n)~%4, n[1, i]==3 & n[2, i]%2 & return); n & ( vecmin(n[1, ])==1 || (n[1, 1]==2 & n[2, 1]%2))} \\ M. F. Hasler, Feb 07 2009

for(n=1,9e9, is_A000404(n) && A193243(n)>1 && print1(n","))

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A085559 Smallest number expressible as the sum of three 4th powers in at least n ways.

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A343082 a(n) is the smallest number that is the sum of n positive 4th powers in three ways.

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A343086 a(n) is the smallest number that is the sum of n positive 4th powers in four ways.

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A374227 a(n) is the smallest number which can be represented as the sum of three distinct positive n-th powers in exactly n ways, or -1 if no such number exists.

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A193244 Numbers that are the sum of three biquadrates (fourth powers) in more than one way.

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