A007410 -id:A007410 - cp's OEIS frontend

A120287 Numerator of 1/n^3 + 2/(n-1)^3 + 3/(n-2)^3 +...+ (n-1)/2^3 + n.

1, 17, 355, 7715, 203413, 492527, 49601051, 1823359051, 16684019407, 186004308017, 22757931053507, 298630937704541, 50872538998767329, 51223731720255509, 103063783892301061, 7045407930432340853

Offset: 1

Alexander Adamchuk, Jul 07 2006

p divides a(p-1) and a(p-2) for prime p>3.

Numerators of the Eulerian numbers T(-3,k) for k = 0,1..., if T(n,k) is extended to negative n by the recurrence T(n,k) = (k+1)*T(n-1,k) + (n-k)*T(n-1,k-1) (indexed as in A173018). - Michael J. Collins, Oct 10 2024

Cf. A001008, A007406, A007407, A007408, A007410.

Numerator[Table[Sum[Sum[1/i^3,{i,1,k}],{k,1,n}],{n,1,25}]]

a(n) = numerator[Sum[Sum[1/i^3,{i,1,k}],{k,1,n}]].

A120288 Numerator of 1/n^4 + 2/(n-1)^4 + 3/(n-2)^4 +...+ (n-1)/2^4 + n.

Original entry on oeis.org

1, 33, 2033, 87425, 11440331, 82653347, 58026230977, 472474237481, 38806560342253, 431701520479427, 579954645879123307, 7598398013722878661, 16804804720323979155637, 16901141697896969645401

Offset: 1

Alexander Adamchuk, Jul 07 2006

p^2 divides a(p-1) for prime p>3. p divides a(p-2) for prime p>5.

Cf. A001008, A007406, A007407, A007408, A007410.

Numerator[Table[Sum[Sum[1/i^4,{i,1,k}],{k,1,n}],{n,1,20}]]

a(n) = numerator[Sum[Sum[1/i^4,{i,1,k}],{k,1,n}]].

A125194 Numerator of generalized harmonic number H((p-1)/2,2p)= Sum[ 1/k^(2p), {k,1,(p-1)/2}] divided by p^2 for prime p>3.

Original entry on oeis.org

41, 1599366601, 10877829357646990581304675244472669289, 100935935338172297894217692920950359818733561, 9217176064595104612826996436899733706027947436610177335077693637792069056822883934927465549747441

Offset: 3

Alexander Adamchuk, Jan 13 2007

Generalized harmonic number is H(n,m)= Sum[ 1/k^m, {k,1,n} ]. The numerator of generalized harmonic number H(p-1,2p) is divisible by p^2 for prime p>3 (see A120290(n)). The numerator of generalized harmonic number H((p-1)/2,2p) is divisible by p^2 for prime p>3.

			Prime[3] = 5.
a(3) = Numerator[ 1 + 1/2^10 ] / 5^2 = 1025 / 25 = 41.

Eric Weisstein's World of Mathematics, Harmonic Number
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem

Cf. A120290, A119722, A001008, A007406, A007408, A007410.

Do[p=Prime[k];f=0;Do[f=f+1/n^(2p);g=Numerator[f];If[IntegerQ[g/(p)^2],Print[{p,g/p^2}]],{n,1,(p-1)/2}],{k,1,100}]

a(n) = Numerator[ Sum[ 1/k^(2*Prime[n]), {k,1,(Prime[n]-1)/2} ]] / Prime[n]^2 for n>2.

cp's OEIS Frontend

A120287 Numerator of 1/n^3 + 2/(n-1)^3 + 3/(n-2)^3 +...+ (n-1)/2^3 + n.

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A120288 Numerator of 1/n^4 + 2/(n-1)^4 + 3/(n-2)^4 +...+ (n-1)/2^4 + n.

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A125194 Numerator of generalized harmonic number H((p-1)/2,2p)= Sum[ 1/k^(2p), {k,1,(p-1)/2}] divided by p^2 for prime p>3.

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