A341683 - OEIS

%I #10 Feb 19 2021 03:38:25

%S 0,6,48,97,440,14846,31653,31653,1678739,1678739,122739560,1535115805,

%T 3512442548,58877591352,155766601759,2190435820306,30675804879964,

%U 97141666019166,97141666019166,3353968861840064,48949549603332636,288326348496168639

%N Successive approximations up to 7^n for the 7-adic integer Sum_{k>=0} k!.

%C a(n) == Sum_{k>=0} k! (mod 7^n). Since k! mod 7^n is eventually zero, a(n) is well-defined.

%C In general, for every prime p, the p-adic integer x = Sum_{k>=0} k! is well-defined. To find the approximation up to p^n (n > 0) for x, it is enough to add k! for 0 <= k <= m and then find the remainder of the sum modulo p^n, where m = (p - 1)*(n + floor(log_p((p-1)*n))). This is because p^n divides (m+1)!

%H Jianing Song, <a href="/A341683/b341683.txt">Table of n, a(n) for n = 0..1000</a>

%F For n > 0, a(n) = (Sum_{k=0..m} k!) mod 7^n, where m = 6*(n + floor(log_7(6*n))).

%e For n = 7, since 7^7 divides 49!, we have a(7) = (Sum_{k=0..48} k!) mod 7^7 = 31653.

%e For n = 55, since 7^55 divides 343!, we have a(55) = (Sum_{k=0..342} k!) mod 7^55 = 7563765912082524448071111141811678897409320968.

%o (PARI) a(n) = my(p=7); if(n==0, 0, lift(sum(k=0, (p-1)*(n+logint((p-1)*n, p)), Mod(k!, p^n))))

%Y Cf. A341687 (digits of Sum_{k>=0} k!).

%Y Successive approximations for the p-adic integer Sum_{k>=0} k!: A341680 (p=2), A341681 (p=3), A341682 (p=5), this sequence (p=7).

%Y Cf. A020646 (least positive integer k for which 7^n divides k!).

%K nonn

%O 0,2

%A _Jianing Song_, Feb 17 2021