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A215972 Numbers k such that Sum_{j=1..k-1} j!/2^j is an integer.

%I A215972 #29 Feb 14 2021 18:33:26
%S A215972 1,3,6,13,15,26,30,55,61,63,3446,108996,3625183,13951973,28010902,
%T A215972 7165572248,14335792540,114636743487,229264368710,458534096495
%N A215972 Numbers k such that Sum_{j=1..k-1} j!/2^j is an integer.
%H A215972 B. M. M. de Weger, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;5a41ecf9.1208">Sums with factorials</a>, NMBRTHRY list, Aug 28 2012
%F A215972 A215974(n)=A215972(n)-1 for all n. (A215974 is the same with another convention for the upper limit of the sum.)
%e A215972 a(1)=1 is in the sequence because sum(..., 0<k<1)=0 (empty sum) is an integer.
%e A215972 2 is not in the sequence because 1!/2^1 = 1/2 is not an integer.
%e A215972 a(2)=3 is in the sequence because 1!/2^1 + 2!/2^2 = 1 is an integer.
%t A215972 sum = 0; Select[Range[0, 10^4], IntegerQ[sum += #!/2^#] &] + 1 (* _Robert Price_, Apr 04 2019 *)
%o A215972 (PARI) is_A215972(n)=denominator(sum(k=1,n-1,k!/2^k))==1
%o A215972 (PARI) s=0;for(k=1,9e9,denominator(s+=k!/2^k)==1&print1(k+1,","))
%Y A215972 Cf. A215974, A216056, A216042, A216043, A216044, A216045.
%K A215972 nonn,more
%O A215972 1,2
%A A215972 _M. F. Hasler_, Aug 29 2012
%E A215972 Terms through a(20) from Aart Blokhuis and Benne de Weger, Aug 30 2012, who thank Jan Willem Knopper for efficient programming. - _N. J. A. Sloane_, Aug 30 2012