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A329808
Numbers k such that both k and k+1 are sums of a positive square and a positive cube.
2
9, 36, 43, 72, 100, 126, 127, 128, 170, 196, 225, 232, 264, 289, 320, 350, 351, 352, 359, 368, 407, 424, 441, 442, 485, 486, 511, 512, 539, 576, 632, 656, 700, 703, 737, 784, 792, 810, 841, 848, 849, 872, 908, 953, 968, 1000, 1018, 1169, 1183, 1213, 1225, 1240, 1296
OFFSET
1,1
COMMENTS
It is quite easy to give a constructive proof that this sequence is infinite. For example, 64*x^3 + 49*x^2 + 14*x + 1 = (7*x+1)^2 + (4*x)^3 and 64*x^3 + 49*x^2 + 14*x + 2 = (x+1)^2 + (4*x+1)^3. Moreover, if 97*x^2 + 2*x + 1 = y^2, then 64*x^3 + 49*x^2 + 14*x = y^2 + (4*x-1)^3. Obviously there are infinitely many solutions to 97*x^2 + 2*x + 1 = y^2, so there are infinitely many k such that k, k+1 and k+2 are all sums of a positive square and a positive cube.
LINKS
EXAMPLE
43 is a term because 43 = 4^2 + 3^3, 44 = 6^2 + 2^3.
PROG
(PARI) isA329808(n) = is(n)&&is(n+1) \\ is() is defined in A055394.
CROSSREFS
KEYWORD
nonn
AUTHOR
Jianing Song, Nov 21 2019
STATUS
approved