A215974 -id:A215974 - 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

A215976 2-adic valuation of the denominator of sum( k!/2^k, k=1..n ).

Original entry on oeis.org

1, 0, 2, 2, 0, 2, 3, 3, 3, 3, 2, 0, 3, 0, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 0, 3, 4, 4, 0, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 3, 2, 4, 0, 5, 5, 5, 5, 4, 0, 5, 0, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 1

M. F. Hasler, Aug 29 2012

nonn

By construction, this denominator is always a power of 2, the present sequence specifies which power. The sum is an integer iff a(n)=0, the corresponding n are listed in A215974 (= A215972 - 1).

The numerator of the sum is given in A215990.

s=0;for(k=1,199,print1(valuation(denominator(s+=k!/2^k),2),","))

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

a(n)=0 <=> n is in A215974 <=> n+1 is in A215972.

A215990 Numerator of sum( k!/2^k, k=1..n ).

Original entry on oeis.org

0, 1, 1, 7, 13, 7, 73, 461, 1721, 7391, 35741, 95833, 140902, 7208291, 6221977, 738064507, 5846167507, 49265043007, 440034922507, 4152348777757, 41275487330257, 431068442131507, 4718790944945257, 27013799863651691, 322866652557800441, 502628413904332477

Offset: 0

M. F. Hasler, Aug 29 2012

nonn

If a(n) is even, then A215976(n)=0 (and n is listed in A215974); the converse is not necessarily true.

Table[Numerator[Sum[k!/2^k,{k,n}]],{n,0,30}] (* Harvey P. Dale, Jul 02 2017 *)

PARI
```
a(n)=numerator(sum(k=1,n,k!/2^k))
```

s=0;for(k=1,29,print1(numerator(s+=k!/2^k),","))

A216056 Primes p such that p divides the numerator of Sum( k!/2^k, k=1..p-1 ).

Original entry on oeis.org

234781, 115480283

Offset: 1

N. J. A. Sloane, Aug 30 2012

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

Cf. A215972, A215974, A057245

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

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A215976 2-adic valuation of the denominator of sum( k!/2^k, k=1..n ).

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A215990 Numerator of sum( k!/2^k, k=1..n ).

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A216056 Primes p such that p divides the numerator of Sum( k!/2^k, k=1..p-1 ).

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