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