A215972
Numbers k such that Sum_{j=1..k-1} j!/2^j is an integer.
Original entry on oeis.org
1, 3, 6, 13, 15, 26, 30, 55, 61, 63, 3446, 108996, 3625183, 13951973, 28010902, 7165572248, 14335792540, 114636743487, 229264368710, 458534096495
Offset: 1
a(1)=1 is in the sequence because sum(..., 0<k<1)=0 (empty sum) is an integer.
2 is not in the sequence because 1!/2^1 = 1/2 is not an integer.
a(2)=3 is in the sequence because 1!/2^1 + 2!/2^2 = 1 is an integer.
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sum = 0; Select[Range[0, 10^4], IntegerQ[sum += #!/2^#] &] + 1 (* Robert Price, Apr 04 2019 *)
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is_A215972(n)=denominator(sum(k=1,n-1,k!/2^k))==1
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s=0;for(k=1,9e9,denominator(s+=k!/2^k)==1&print1(k+1,","))
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
A216042
Numbers k such that Sum_{j=1..k-1} (2*j)!/4^j is an integer.
Original entry on oeis.org
1, 3, 53, 106, 427, 3416, 6806, 13665, 27330, 130030, 220227, 437666
Offset: 1
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seq={}; sum=0; fak=1; k=0; While[k<10000, sum+=fak; If[Denominator[sum]==1, AppendTo[seq,k+1]]; k++; fak*=k*(2k-1)/2;]; seq
A216043
Numbers k such that Sum_{j=1..k-1} (2j)!/3^j is an integer.
Original entry on oeis.org
1, 4, 12, 105, 112, 322, 8807, 8831
Offset: 1
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seq={}; sum=0; fak=1; k=0; While[k<10000, sum+=fak; If[Denominator[sum]==1, AppendTo[seq,k+1]]; k++; fak*=2*k*(2k-1)/3;]; seq
A216045
Numbers k such that Sum_{j=1..k-1} (4j)!/5^j is an integer.
Original entry on oeis.org
1, 6, 30, 145, 151, 592, 732, 3895, 87806, 292432
Offset: 1
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seq={}; sum=0; fak=1; k=0; While[k<10000, sum+=fak; If[Denominator[sum]==1, AppendTo[seq,k+1]]; k++; fak*=4*k*(4*k-1)*(4*k-2)*(4*k-3)/5;]; seq
A216149
Numbers k such that Sum_{j=1..k-1} (3*j)!/5^j is an integer.
Original entry on oeis.org
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seq={}; sum=0; fak=1; k=0; While[k<10000, sum+=fak; If[Denominator[sum]==1, AppendTo[seq,k+1]]; k++; fak*=3*k*(3*k-1)*(3*k-2)/5;]; seq
Select[Range[20],IntegerQ[Sum[(3k)!/5^k,{k,#-1}]]&] (* Harvey P. Dale, Apr 07 2019 *)
Showing 1-5 of 5 results.
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