A300326 Sum of the largest possible permutations that can be written without repetition of digits in each base from binary to n+1.

0, 2, 23, 251, 3181, 47971, 848638, 17283462, 398650506, 10275193716, 292733747621, 9135147415313, 309906954656231, 11356162260536389, 447015900139452604, 18811774444632517324, 842820629057975778516, 40053081963609542635686, 2012366504118798707101875

Offset: 0

Author

R. J. Cano, Mar 03 2018

Keywords

• nonn

Comments

It is seems that {a(1), a(2), a(3), a(4)} are the only primes of this form.

From M. F. Hasler, Mar 04 2018: (Start)

For p = 2 and p = 3, a(n) (mod p) is 8- resp. 9-periodic.

For primes 5 <= p <= 23, a(n) (mod p) is p(p-1) periodic. I conjecture this to hold for all p >= 5.

It also appears that the last 4 terms of these periods are (1, 1, 0, 0) (mod p), for any p >= 2, i.e., a(n) is divisible by p at least for k*P-2 <= n <= k*P for any k >= 0, where P is the period length p(p-1) (resp. 8 or 9 for p = 2 and 3).

These properties might allow a proof that a(1..4) are the only primes. However, a(12) = 14231491*21776141, so there is little hope of finding a reasonably sized finite covering set.

(End)

Examples

			Let us consider the numbers: 0[1], 10[2], 210[3], 3210[4], 43210[5], and 543210[6];
Their respective decimal representations are the first six terms of A062813: 0, 2, 21, 228, 2930, 44790. The partial sums for those terms are 0, 2, 23, 251, 3181, and 47971; after 0, the following 4 sums are primes, but 47971 is not prime. The same is true for subsequent partial sums, whence the conjecture in COMMENTS.
		

Crossrefs

Partial sums of A062813.

Cf. A233783 for the occurrence of the ordered triple (2,23,251) in a different context.

Programs

PARI

A300326_vec(Nmax,s=0)=vector(Nmax,n,s+=A062813(n)) \\ M. F. Hasler, Mar 05 2018

Extensions

Partially edited by M. F. Hasler, Mar 05 2018