A173795 Smallest prime that is the sum of a square and a positive cube in n different ways.

3, 2, 17, 2089, 65537, 3193361, 445341529, 4190216689, 25140740257, 813368268793, 333413867957257, 1057543811051633, 1448734752622601

Offset

0,1

Comments

From Kevin T. Acres Sep 22 2012 (Start)

_Noam D. Elkies_ has determined, after an exhaustive search, to 7.5 * 10^15, that 1057543811051633 and 1448734752622601 are the lowest primes such that they are sums of a square and a positive cube in 11 and 12 different ways respectively.

107122676734733201 remains a potential, but unproven, candidate for n = 13 and 14.

107122676734733201 = 18076^3 + 327286985^2 = 56276^3 + 327023625^2 = 83413^3 + 326408198^2 = 128726^3 + 324021045^2 = 180440^3 + 318194601^2 = 319330^3 + 273056899^2 = 339826^3 + 260535965^2 = 344065^3 + 257666476^2 = 385333^3 + 223400642^2 = 403688^3 + 203312727^2 = 415601^3 + 187984920^2 = 447428^3 + 132481143^2 = 457750^3 + 105867851^2 = 460826^3 + 96236115^2

(End)

Links

Table of n, a(n) for n=0..12.

Example

a(0) = 3 (smallest prime not of the form a^2 + b^3).
a(1) = 2 = 1^2 + 1^3.
a(2) = 17 = 4^2 + 1^3 = 3^2 + 2^3.
a(3) = 2089 = 45^2 + 4^3 = 33^2 + 10^3 = 19^2 + 12^3.
a(4) = 65537 = 256^2 + 1^3 = 255^2 + 8^3 = 219^2 + 26^3 = 122^2 + 37^3.
a(5) = 3193361 = 1769^2 + 40^3 = 1606^2 + 85^3 = 1481^2 + 100^3 = 1047^2 + 128^3 = 285^2 + 146^3.
a(6) = 445341529 = 21023^2 + 150^3 = 20955^2 + 184^3 = 20898^2 + 205^3 = 20773^2 + 240^3 = 11195^2 + 684^3 = 2523^2 + 760^3.
a(7) = 4190216689 = 64729^2 + 72^3 = 64005^2 + 454^3 = 61219^2 + 762^3 = 42867^2 + 1330^3 = 36008^2 + 1425^3 = 20915^2 + 1554^3 = 17479^2 + 1572^3.
a(8) = 25140740257 = 155951^2 + 936^3 = 155440^2 + 993^3 = 153739^2 + 1146^3 = 151371^2 + 1306^3 = 126172^2 + 2097^3 = 121809^2 + 2176^3 = 116477^2 + 2262^3 = 38097^2 + 2872^3.
a(9) = 813368268793 = 901707^2 + 664^3 = 900233^2 + 1434^3 = 808084^2 + 5433^3 = 693429^2 + 6928^3 = 610741^2 + 7608^3 = 432210^2 + 8557^3 = 392373^2 + 8704^3 = 379349^2 + 8748^3 = 275817^2 + 9034^3.
a(10) = 333413867957257 = 18202887^2 + 12742^3 = 18190720^2 + 13593^3 = 16205565^2 + 41368^3 = 15621373^2 + 44712^3 = 14905630^2 + 48093^3 = 12187395^2 + 56968^3 = 11330919^2 + 58966^3 = 10486383^2 + 60682^3 = 9216035^2 + 62868^3 = 3854589^2 + 68296^3.
a(11) = 1057543811051633 = 7534^3 + 32513323^2 = 33184^3 + 31953127^2 = 46552^3 + 30929945^2 = 57377^3 + 29472900^2 = 69374^3 + 26901003^2 = 87989^3 + 19399158^2 = 94369^3 + 14735668^2 = 94874^3 + 14267997^2 = 95114^3 + 14038467^2 = 97952^3 + 10850535^2 = 101828^3 + 1302009^2
a(12) = 1448734752622601 = 30668^3 + 37681437^2 = 42326^3 + 37052775^2 = 49498^3 + 36434353^2 = 55000^3 + 35810051^2 = 68585^3 + 33557676^2 = 68890^3 + 33493199^2 = 78020^3 + 31206051^2 = 85838^3 + 28570377^2 = 88258^3 + 27590783^2 = 94820^3 + 24417699^2 = 105368^3 + 16700163^2 = 111901^3 + 6894130^2

Crossrefs

Cf. A000040, A000290, A000578, A055394, A066649, A122956.

Sequence in context: A126354 A361084 A158939 * A249769 A026345 A092644

Adjacent sequences: A173792 A173793 A173794 * A173796 A173797 A173798

Keyword

nonn

Author

_Donovan Johnson_, Mar 01 2010

Extensions

More terms (n=11 and 12) from _Noam D Elkies_.

Outdated comments removed by _Kevin T. Acres_, Sep 22 2012

Status

approved