A350693 - OEIS

A350693

Number of b > 0 which permit n^3 to be written as a sum of powers of b in n parts. Each exponent c is an integer >= 0, n^3 = b^c_1 + b^c_2 + ... + b^c_n.

3, 5, 8, 7, 10, 13, 17, 19, 12, 20, 16, 18, 18, 25, 25, 21, 14, 28, 31, 34, 19, 22, 29, 34, 28, 33, 29, 38, 19, 33, 30, 31, 34, 51, 44, 30, 20, 41, 38, 44, 18, 37, 42, 52, 27, 30, 37, 59, 39, 50, 28, 35, 37, 82, 64, 44, 19, 36, 27, 36, 27, 52, 85, 65, 35, 40, 29

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OFFSET

2,1

COMMENTS

If n^3 is written in different number bases, a(n) is an upper limit for the count of number bases which allow n^3 to be written as a base-b number with a digit sum of n (generalized Dudeney numbers).

a(n) has an upper limit in the number of divisors of n^3-n. Let d be one of these divisors, then it appears that a lower limit can be found by excluding all divisors d where d+1 does not share all its prime divisors with binomial(n^3, n) (A107444).

LINKS

Table of n, a(n) for n=2..68.

FORMULA

a(n) <= A000005(n^3-n). Conjectured to become a(n) = A000005(n^3-n), if the definition would permit negative values for b and only the absolute value of the sum needs to be equal to n^3.

EXAMPLE

a(2) = 3 because 2^3 = 2^2 + 2^2 = 4^1 + 4^1 = 7^1 + 7^0.

PROG

(PARI) a(n) = sum(d=2, n^3, s=sumdigits(n^3, d); s<=n&&(n-s)%(d-1)==0); \\ Jinyuan Wang, Jan 15 2022

CROSSREFS

Cf. A000005, A046459, A092517, A107444.

Sequence in context: A110641 A121729 A072105 * A364847 A320451 A088509

Adjacent sequences: A350690 A350691 A350692 * A350694 A350695 A350696

KEYWORD

nonn

AUTHOR

Thomas Scheuerle, Jan 12 2022

EXTENSIONS

More terms from Jinyuan Wang, Jan 15 2022

STATUS

approved