A216056 -id:A216056 - cp's OEIS frontend

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

1, 3, 6, 13, 15, 26, 30, 55, 61, 63, 3446, 108996, 3625183, 13951973, 28010902, 7165572248, 14335792540, 114636743487, 229264368710, 458534096495

Offset: 1

M. F. Hasler, Aug 29 2012

			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.

B. M. M. de Weger, Sums with factorials, NMBRTHRY list, Aug 28 2012

Cf. A215974, A216056, A216042, A216043, A216044, A216045.

sum = 0; Select[Range[0, 10^4], IntegerQ[sum += #!/2^#] &] + 1 (* Robert Price, Apr 04 2019 *)

is_A215972(n)=denominator(sum(k=1,n-1,k!/2^k))==1

s=0;for(k=1,9e9,denominator(s+=k!/2^k)==1&print1(k+1,","))

A215974(n)=A215972(n)-1 for all n. (A215974 is the same with another convention for the upper limit of the sum.)

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

A215974 Numbers n such that Sum_{k=1..n} k!/2^k is an integer.

Original entry on oeis.org

0, 2, 5, 12, 14, 25, 29, 54, 60, 62, 3445, 108995, 3625182, 13951972, 28010901, 7165572247, 14335792539, 114636743486, 229264368709, 458534096494

Offset: 1

M. F. Hasler, Aug 29 2012

This sequence lists the indices n for which A215976(n)=0 (power of 2 in denominator) and for which A215990 (numerator of the sum) may be even.

			a(1)=0 is in the sequence because sum(..., 1 <= k <= 0)=0 (empty sum) is an integer.
1 is not in the sequence because 1!/2^1 = 1/2 is not an integer.
a(2)=2 is in the sequence because 1!/2^1 + 2!/2^2 = 1 is an integer.

B. M. M. de Weger, Sums with factorials, NMBRTHRY list, Aug 28 2012

Cf. A215972, A216056.

sum = 0; Select[Range[0, 10^4], IntegerQ[sum += #!/2^#] &] (* Robert Price, Apr 04 2019 *)

is_A215974(n)=denominator(sum(k=1,n,k!/2^k))==1

s=0;for(k=1,9e9,denominator(s+=k!/2^k)==1&print1(k,","))

A215974(n) = A215972(n)-1 for all n. (The two sequences differ only in the use of the upper limit. The present convention seems more natural, the other one was used in the post on the NmbrThry list.)

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

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

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A215974 Numbers n such that Sum_{k=1..n} k!/2^k is an integer.

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Examples

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Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions