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

0,2

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)

Links

Table of n, a(n) for n=0..18.

Example

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.

Prog

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

Crossrefs

Partial sums of A062813.

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

Sequence in context: A309221 A168128 A233783 * A068983 A083427 A083470

Adjacent sequences: A300323 A300324 A300325 * A300327 A300328 A300329

Keyword

nonn

Author

R. J. Cano, Mar 03 2018

Extensions

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

Status

approved