A085559 -id:A085559 - cp's OEIS frontend

A123075 Smallest number expressible as the sum of three 4th powers in exactly n ways.

1, 2673, 811538, 5978882, 137149922, 292965218, 779888018, 5745705602, 105760443698, 49511121842, 1872511131218, 281539574498, 17673688436978, 17873514984962, 253930825318898, 7865870969138, 768054952462322

Offset: 1

Tom Womack (tom(AT)womack.net), Sep 19 2006

nonn

Cf. A085559, A011541, A122540.

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.

A122540 Primes that can be written in n ways as the sum of three fourth powers.

Original entry on oeis.org

2, 137633, 409698593

Offset: 1

Thomas Womack (tom(AT)womack.net), Sep 19 2006

If you replace 'prime' with 'odd number', the sequence is 17, 2673, 16196193, 155129315313, ... Next term is at least 3000^4.

The restriction to primes is quite severe and to odd numbers reasonably so; otherwise x^4+y^4+(x+y)^4=2(x^2+xy+y^2)^2 gives very many examples of the form 2k^2.

			a(2)=137633; 8^4+13^4+18^4 = 9^4+16^4+16^4 and this is the smallest prime N for which this holds
a(3)=409698593; 128^4+109^4+18^4 = 129^4+94^4+86^4 = 142^4+42^4+1

Obvious variant on A085559.

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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A123075 Smallest number expressible as the sum of three 4th powers in exactly 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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A122540 Primes that can be written in n ways as the sum of three fourth powers.

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

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