A215972 -id:A215972 - cp's OEIS frontend

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

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

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

Vaclav Kotesovec, Aug 30 2012

Next term > 1900000.

Cf. A215972, A216043, A216044, A216045.

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

Vaclav Kotesovec, Aug 30 2012

Next term > 2000000.

Cf. A215972, A216042, A216044, A216045

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

A216044 Numbers k such that Sum_{j=1..k-1} (4j)!/16^j is an integer.

Original entry on oeis.org

1, 3, 13, 15, 61, 106, 253, 27545, 62785, 218107, 1004593

Offset: 1

Vaclav Kotesovec, Aug 30 2012

Next term > 1200000.

Cf. A215972, A216042, A216043, A216045.

seq={}; sum=0; fak=1; k=0; While[k<10000, sum+=fak; If[Denominator[sum]==1, AppendTo[seq,k+1]]; k++; fak*=k*(4*k-1)*(4*k-2)*(4*k-3)/4;]; 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

Vaclav Kotesovec, Aug 30 2012

Next term is > 1500000.

Cf. A215972, A216042, A216043, A216044.

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

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.

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

A216149 Numbers k such that Sum_{j=1..k-1} (3*j)!/5^j is an integer.

Original entry on oeis.org

1, 3, 16

Offset: 1

Vaclav Kotesovec, Sep 02 2012

The next term, if it exists, is greater than 1000000. - Vaclav Kotesovec, Sep 11 2012

Cf. A215972, A216042, A216043, A216044, A216045.

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 *)

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

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A216042 Numbers k such that Sum_{j=1..k-1} (2*j)!/4^j is an integer.

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A216043 Numbers k such that Sum_{j=1..k-1} (2j)!/3^j is an integer.

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A216044 Numbers k such that Sum_{j=1..k-1} (4j)!/16^j is an integer.

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A216045 Numbers k such that Sum_{j=1..k-1} (4j)!/5^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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A216056 Primes p such that p divides the numerator of Sum( k!/2^k, k=1..p-1 ).

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A216149 Numbers k such that Sum_{j=1..k-1} (3*j)!/5^j is an integer.

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