A295787 -id:A295787 - cp's OEIS frontend

A329807 Numbers k such that k, k+1, k+2 and k+3 are all sums of a positive square and a positive cube.

126, 350, 8125, 12742, 19879, 29240, 42974, 76728, 91329, 109241, 140750, 209222, 254681, 258272, 297423, 482958, 744901, 755169, 918601, 986174, 1026214, 1418606, 1515227, 1521233, 1888216, 2082977, 2216080, 2317257, 3510926, 4180848, 4316417, 4330888, 4836895

Offset: 1

Jianing Song, Nov 21 2019

nonn

It is known that there are infinitely many k such that k, k+1, k+2 are all sums of a positive square and a positive cube (see A055394 and A295787). It is natural to ask if this sequence is infinite. There are 243 members here below 10^9.

There are 2 pairs of consecutive numbers below 10^9: (16597502, 16597503) and (593825496, 593825497). Are there infinitely many k such that k, k+1, k+2, k+3 and k+4 are all sums of a positive square and a positive cube?

			350 is here because 350 = 15^2 + 5^3, 351 = 18^2 + 3^3, 352 = 3^2 + 7^3 and 353 = 17^3 + 4^3.

Jianing Song, Table of n, a(n) for n = 1..243 (All terms <= 10^9)

Cf. A055394, A329808, A295787.

isA329807(n) = is(n)&&is(n+1)&&is(n+2)&&is(n+3) \\ is() is defined in A055394.

A329808 Numbers k such that both k and k+1 are sums of a positive square and a positive cube.

Original entry on oeis.org

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

Jianing Song, Nov 21 2019

nonn

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.

			43 is a term because 43 = 4^2 + 3^3, 44 = 6^2 + 2^3.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A055394, A295787, A329807.

isA329808(n) = is(n)&&is(n+1) \\ is() is defined in A055394.

cp's OEIS Frontend

A329807 Numbers k such that k, k+1, k+2 and k+3 are all sums of a positive square and a positive cube.

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