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Numbers m such that the least s >= 0 such that Sum_{k=0..m} (k+s)!/C(m,k) is an integer satisfies s = m - 1.
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%I #12 Apr 03 2020 07:51:36

%S 1,2,4,5,13,17,19,23,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,

%T 103,107,109,113,127,131,137,139,149,151,163,167,173,179,181,191,193,

%U 197,199,223,227,229,239,241,251,257,263,269,271,277,281,283,293,307

%N Numbers m such that the least s >= 0 such that Sum_{k=0..m} (k+s)!/C(m,k) is an integer satisfies s = m - 1.

%C The prime numbers that are not in this sequence are 3, 7, 11, 29, 157, 211, 233, 419, ... - _Jinyuan Wang_, Apr 03 2020

%o (PARI) for(n=1,150,s=0; while(frac(sum(k=0,n,(k+s)!/binomial(n,k)))>0,s++); if(n-s==1,print1(n,",")))

%K nonn

%O 1,2

%A _Benoit Cloitre_, Feb 16 2003

%E More terms from _Jinyuan Wang_, Apr 03 2020