A303375 Numbers of the form a^5 + b^6, with integers a, b > 0.

2, 33, 65, 96, 244, 307, 730, 761, 972, 1025, 1088, 1753, 3126, 3189, 3854, 4097, 4128, 4339, 5120, 7221, 7777, 7840, 8505, 11872, 15626, 15657, 15868, 16649, 16808, 16871, 17536, 18750, 20903, 23401, 32432, 32769, 32832, 33497, 36864, 46657, 46688, 46899, 47680, 48393

Offset: 1

Author

M. F. Hasler, Apr 22 2018

Keywords

• nonn
• easy

Comments

Although it is easy to produce many terms of this sequence, it is nontrivial to check whether a very large number is of this form.

This sequence is among others motivated by the hard-to-compute sequence A300567 = numbers z such that z^7 = x^5 + y^6 for some x, y >= 1.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000404 (a^2 + b^2), A055394 (a^2 + b^3), A111925 (a^2 + b^4), A100291 (a^4 + b^3), A100292 (a^5 + b^2), A100293 (a^5 + b^3), A100294 (a^5 + b^4).

Cf. A303372 (a^2 + b^6), A303373 (a^3 + b^6), A303374 (a^4 + b^6).

See also A300567: numbers z such that z^7 = x^5 + y^6 for some x, y >= 1.

Programs

PARI

is(n,k=5,m=6)=for(b=1,sqrtnint(n-1,m),ispower(n-b^m,n)&&return(b)) \\ Returns b > 0 if n is in the sequence, else 0.
A303375_vec(L=10^5,k=5,m=6,S=List())={for(a=1,sqrtnint(L-1,m),for(b=1,sqrtnint(L-a^m,k), listput(S,a^m+b^k)));Set(S)} \\ all terms up to limit L

Formula

a(n) >> n^(30/11). Probably this is the correct asymptotic order. - Charles R Greathouse IV, Jan 23 2025