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%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
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