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

%I A300326 #32 Jan 07 2022 19:32:51
%S A300326 0,2,23,251,3181,47971,848638,17283462,398650506,10275193716,
%T A300326 292733747621,9135147415313,309906954656231,11356162260536389,
%U A300326 447015900139452604,18811774444632517324,842820629057975778516,40053081963609542635686,2012366504118798707101875
%N A300326 Sum of the largest possible permutations that can be written without repetition of digits in each base from binary to n+1.
%C A300326 It is seems that {a(1), a(2), a(3), a(4)} are the only primes of this form.
%C A300326 From _M. F. Hasler_, Mar 04 2018: (Start)
%C A300326 For p = 2 and p = 3, a(n) (mod p) is 8- resp. 9-periodic.
%C A300326 For primes 5 <= p <= 23, a(n) (mod p) is p(p-1) periodic. I conjecture this to hold for all p >= 5.
%C A300326 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).
%C A300326 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.
%C A300326 (End)
%e A300326 Let us consider the numbers: 0[1], 10[2], 210[3], 3210[4], 43210[5], and 543210[6];
%e A300326 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.
%o A300326 (PARI) A300326_vec(Nmax,s=0)=vector(Nmax,n,s+=A062813(n)) \\ _M. F. Hasler_, Mar 05 2018
%Y A300326 Partial sums of A062813.
%Y A300326 Cf. A233783 for the occurrence of the ordered triple (2,23,251) in a different context.
%K A300326 nonn
%O A300326 0,2
%A A300326 _R. J. Cano_, Mar 03 2018
%E A300326 Partially edited by _M. F. Hasler_, Mar 05 2018