A280559 - OEIS

%I #47 Aug 16 2021 03:24:47

%S 1,14,233,244,331,889,2725,5219,6746

%N Numbers m that divide Sum_{k=1..m} binomial(m,k) mod k.

%C Ratios are 0, 2, 44, 47, 58, 162, 529, 1004, 1318.

%e C(14,1) mod 1 + C(14,2) mod 2 + ... + C(14,13) mod 13 + C(14,14) mod 14 = 0 + 1 + 1 + 1 + 2 + 3 + 2 + 3 + 4 + 1 + 1 + 7 + 1 + 1 = 28 and 28/14 = 2 so 14 is a term.

%p P:=proc(q) local k,n; for n from 1 to q do

%p if type(add(binomial(n,k) mod k,k=1..n)/n,integer) then print(n); fi; od; end: P(10^6);

%t Select[Range[10^3], Divisible[Sum[Mod[Binomial[#, k], k], {k, #}], #] &] (* _Michael De Vlieger_, Feb 07 2017 *)

%o (PARI) isok(n) = (sum(k=1, n, binomial(n,k) % k) % n) == 0; \\ _Michel Marcus_, Jul 16 2017

%Y Cf. A076541.

%K nonn,more

%O 1,2

%A _Paolo P. Lava_, Jan 31 2017