A061002 -id:A061002 - cp's OEIS frontend

A035048 Numerators of alternating sum transform (PSumSIGN) of Harmonic numbers H(n) = A001008/A002805.

1, 1, 4, 3, 23, 11, 176, 25, 563, 137, 6508, 49, 88069, 363, 91072, 761, 1593269, 7129, 31037876, 7381, 31730711, 83711, 744355888, 86021, 3788707301, 1145993, 11552032628, 1171733, 340028535787, 1195757

Offset: 1

N. J. A. Sloane

p^2 divides a(2p-2) for prime p>3. a(2p-2)/p^2 = A061002(n) = A001008(p-1)/p^2 for prime p>2. - Alexander Adamchuk, Jul 07 2006

Robert Israel, Table of n, a(n) for n = 1..2000
N. J. A. Sloane, Transforms

Cf. A035047, A002428, A001008, A058313, A061002.

S:= series(log(1-x)/(x^2-1), x, 101):
seq(numer(coeff(S,x,j)), j=1..100); # Robert Israel, Jun 02 2015

Numerator[Table[Sum[(-1)^(k+1)*Sum[(-1)^(i+1)*1/i,{i,1,k}],{k,1,n}],{n,1,50}]] (* Alexander Adamchuk, Jul 07 2006 *)

a(n)=numerator(polcoeff(log(1-x)/(x^2-1)+O(x^(n+1)),n))

G.f. for A035048(n)/A035047(n) : log(1-x)/(x^2-1). - Benoit Cloitre, Jun 15 2003
a(n) = Numerator[Sum[(-1)^(k+1)*Sum[(-1)^(i+1)*1/i,{i,1,k}],{k,1,n}]]. - Alexander Adamchuk, Jul 07 2006
a(n) = numerator((-1)^(n+1)*1/2*(log(2)+(-1)^(n+1)*(gamma+1/2*(psi(1+n/2)-psi(3/2+n/2))+psi(2+n)))), with gamma the Euler-Mascheroni constant. - - Gerry Martens, Apr 28 2011

A127061 Primes p such that denominator of Sum_{k=1..p-1} 1/k^2 is a square and denominator Sum_{k=1..p-1} 1/k^3 is a cube.

Original entry on oeis.org

2, 3, 5, 17, 29, 31, 37, 41, 97, 439, 443, 449, 457, 461, 463, 1009, 1013, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159, 4283, 4289, 9461, 9463, 9467, 9473, 9479, 9491, 9497, 9511

Offset: 1

Artur Jasinski, Jan 04 2007

nonn

Artur Jasinski & Michel Marcus, Table of n, a(n) for n = 1..60

Cf. A061002, A034602, A127029, A127042, A127043, A127044, A127046, A127047, A127048, A127049, A127051.

With[{nn=10000},Select[Flatten[Position[Denominator[Thread[{Accumulate[1/ Range[ nn]^2], Accumulate[ 1/Range[nn]^3]}]],?(IntegerQ[Sqrt[First[#]]] && IntegerQ[Surd[Last[#],3]]&),{1},Heads->False]],PrimeQ]] (* _Harvey P. Dale, Mar 05 2014 *)

lista(nn) = {forprime(p = 2, nn, if (issquare(denominator(sum(k=1, p-1, 1/k^2))) && ispower(denominator(sum(k=1, p-1, 1/k^3)),3), print1(p, ", ")););} \\ Michel Marcus, Nov 05 2013

Intersection of A127042 and A127046. - Michel Marcus, Nov 05 2013

More terms from Max Alekseyev, Feb 08 2007

Missing terms in the [9461, 9587] range inserted by Michel Marcus, Nov 05 2013

A120285 Numerator of harmonic number H(p-1) = Sum_{k=1..p-1} 1/k for prime p.

Original entry on oeis.org

1, 3, 25, 49, 7381, 86021, 2436559, 14274301, 19093197, 315404588903, 9304682830147, 54801925434709, 2078178381193813, 12309312989335019, 5943339269060627227, 14063600165435720745359, 254381445831833111660789

Offset: 1

Alexander Adamchuk, Jul 07 2006

Prime(n)^2 divides a(n) for n>2.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 22-23.

Robert Israel, Table of n, a(n) for n = 1..342
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem.

Cf. A001008, A061002, A185399.

f3:=proc(n) local p;
p:=ithprime(n);
numer(add(1/i,i=1..p-1));
end proc;
[seq(f3(n),n=1..20)];

Numerator[Table[Sum[1/k,{k,1,Prime[n]-1}],{n,1,20}]]
Table[HarmonicNumber[p],{p,Prime[Range[20]]-1}]//Numerator (* Harvey P. Dale, May 18 2023 *)

a(n) = my(p=prime(n)); numerator(sum(k=1, p-1, 1/k)); \\ Michel Marcus, Dec 25 2018

a(n) = numerator(Sum_{k=1..prime(n)-1} 1/k).
a(n) = A001008(prime(n)-1).
a(n) = A061002(n)*prime(n)^2 for n > 2.

A125551 As p runs through primes >= 5, sequence gives { numerator of Sum_{k=1..p-1} 1/k^2 } / p.

Original entry on oeis.org

41, 767, 178939, 18500393, 48409924397, 12569511639119, 15392144025383, 358066574927343685421, 282108494885353559158399, 911609127797473147741660153, 1128121200256091571107985892349

Offset: 3

Artur Jasinski, Jan 03 2007

nonn

This is an integer by a theorem of Waring and Wolstenholme.

R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057, 2011

Cf. A061002, A034602, A186720, A186722.

f1:=proc(n) local p;
p:=ithprime(n);
(1/p)*numer(add(1/i^2,i=1..p-1));
end proc;
[seq(f1(n),n=3..20)];

a = {}; Do[AppendTo[a, (1/(Prime[x]))Numerator[Sum[1/x^2, {x, 1, Prime[x] - 1}]]], {x, 3, 50}]; a
Table[Sum[1/k^2,{k,p-1}]/p,{p,Prime[Range[3,20]]}]//Numerator (* Harvey P. Dale, Nov 20 2019 *)

A186720 As p runs through the primes, sequence gives denominator of Sum_{k=1..p-1} 1/k^2.

Original entry on oeis.org

1, 4, 144, 3600, 1270080, 153679680, 519437318400, 150117385017600, 221193371393280, 6450247552370862240000, 5424658191543895143840000, 20852386088294732932920960000, 28546916554875489385168794240000, 6855338104106528236638391873920000, 12675520154492970709544386574878080000

Offset: 1

N. J. A. Sloane, Jan 21 2012

nonn

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 22-23.

R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.

Cf. A125551, A061002.

f2:=proc(n) local p;
p:=ithprime(n);
denom(add(1/i^2,i=1..p-1));
end proc;
[seq(f2(n),n=1..20)];

a[n_] := HarmonicNumber[Prime[n] - 1, 2] // Denominator;
Array[a, 15] (* Jean-François Alcover, Nov 25 2017 *)

A076638 Denominators of harmonic numbers when the numerators are divisible by squares of primes >= 5 in the case of Wolstenholme's Theorem.

Original entry on oeis.org

12, 20, 2520, 27720, 720720, 4084080, 5173168, 80313433200, 2329089562800, 13127595717600, 485721041551200, 2844937529085600, 1345655451257488800, 3099044504245996706400, 54749786241679275146400, 3230237388259077233637600

Offset: 1

Michael Gilleland (megilleland(AT)yahoo.com), Oct 23 2002

From Bernard Schott, Dec 28 2018: (Start)
By Wolstenholme's Theorem, if p prime >= 5, the numerator of the harmonic number H_{p-1} is always divisible by p^2. The obtained quotients are in A061002.
The numerators of H_7 and H_{29} are also divisible by prime squares, respectively by 11^2 and 43^2, but not in the case of Wolstenholme's theorem, so the denominators of H_7 and H_{29} are not in this sequence here. (End)

			a(1)=12 because the numerator of H_4 = 25/12 is divisible by the square of 5;
a(2)=20 because the numerator of H_6 = 49/20 is divisible by the square of 7.

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

Cf. A076637, A185399.

a[p_] := Denominator[HarmonicNumber[p - 1]]; a /@ Prime@Range[3, 20] (* Amiram Eldar, Dec 28 2018 *)

More terms added by Amiram Eldar, Dec 04 2018

A127045 Primes p such that denominator of Sum_{k=1..p-1} 1/k^9 is a 9th power.

Original entry on oeis.org

2, 3, 5, 11, 13, 17, 29, 31, 37, 97, 127, 131, 251, 257, 263, 293, 431, 433, 439, 443, 449, 457, 461, 463, 467, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3797, 3803, 3821, 3823, 3833, 3907, 3911, 3917

Offset: 1

Artur Jasinski, Jan 04 2007

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A061002, A034602, A127029, A127042, A127043, A127044, A127046, A127047, A127048, A127049, A127051.

d[n_] := Module[{}, su = 0; a = {}; For[i = 1, i <= n, i++, su = su + 1/ i^9; If[PrimeQ[i + 1], If[IntegerQ[(Denominator[su])^(1/9)], AppendTo[a, i + 1]]]]; a] d[2000]
Select[Flatten[Position[Denominator[Accumulate[1/Range[4000]^9]],?(IntegerQ[ Surd[ #,9]]&)]]+1,PrimeQ] (* _Harvey P. Dale, Aug 06 2022 *)

A127052 Primes p such that denominator of Sum_{k=1..p-1} 1/k^8 is an eighth power.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 37, 41, 53, 67, 71, 73, 97, 101, 127, 131, 197, 199, 211, 251, 367, 373, 379, 773, 787, 797, 809, 811, 1373, 1433, 1439, 2027, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A061002, A034602, A127029, A127042, A127046, A127047, A127048, A127051.

d[n_] := Module[{}, su = 0; a = {}; For[i = 1, i <= n, i++, su = su + 1/ i^8; If[PrimeQ[i + 1], If[IntegerQ[(Denominator[su])^(1/8)], AppendTo[a, i + 1]]]]; a]; d[2000]

A127062 Primes p such that denominator of Sum_{k=1..p-1} 1/k^2 is a square and denominator Sum_{k=1..p-1} 1/k^3 is a cube and denominator Sum_{k=1..p-1} 1/k^4 is a fourth power.

Original entry on oeis.org

2, 3, 5, 17, 29, 31, 97, 439, 443, 449, 457, 461, 463, 1009, 1013, 24391, 24407, 24413, 24419, 24421, 24439, 24443, 24469, 24473, 24481, 117659, 117671, 117673, 117679, 117701, 117703, 117709, 117721, 117727, 117731, 117751, 117757, 117763, 117773

Offset: 1

Artur Jasinski, Jan 04 2007

nonn

Subsequence of A127061. - Max Alekseyev, Feb 08 2007

Cf. A061002, A034602, A127029, A127042, A127043, A127044, A127046, A127047, A127048, A127049, A127051, A127061.

pdenQ[n_]:=Module[{c=Denominator[Table[Sum[1/k^i,{k,n-1}],{i,2,4}]]}, AllTrue[{ Surd[c[[1]],2], Surd[c[[2]],3],Surd[c[[3]],4]},IntegerQ]]; Select[Prime[Range[12000]],pdenQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jun 06 2015 *)

lista(nn) = {forprime(p = 2, nn, if (issquare(denominator(sum(k=1, p-1, 1/k^2))) && ispower(denominator(sum(k=1, p-1, 1/k^3)),3) && ispower(denominator(sum(k=1, p-1, 1/k^4)),4), print1(p, ", ")););} \\ Michel Marcus, Nov 05 2013

Intersection of A127042, A127046 and A127047. - Michel Marcus, Nov 05 2013

More terms from Max Alekseyev, Feb 08 2007

A186722 a(n) = numerator of Sum_{k=1..p-1} 1/k^2 for p the n-th prime.

Original entry on oeis.org

1, 5, 205, 5369, 1968329, 240505109, 822968714749, 238820721143261, 354019312583809, 10383930672892966877209, 8745363341445960333910369, 33729537728506506466441425661, 46252969210499754415427421586309, 11115284554577186575391010113969347, 20577813589884143264711540636313749803

Offset: 1

N. J. A. Sloane, Jan 21 2012

nonn

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 22-23.

R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.

Cf. A125551, A186720, A061002.

f3:=proc(n) local p;
p:=ithprime(n);
numer(add(1/i^2,i=1..p-1));
end proc;
[seq(f3(n),n=1..20)];

Table[Numerator[HarmonicNumber[Prime[n]-1, 2]], {n, 1, 15}] (* Jean-François Alcover, Nov 29 2017 *)

a(n) = my(p=prime(n)); numerator(sum(k=1, p-1, 1/k^2)); \\ Michel Marcus, Apr 05 2015

cp's OEIS Frontend

A035048 Numerators of alternating sum transform (PSumSIGN) of Harmonic numbers H(n) = A001008/A002805.

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A127061 Primes p such that denominator of Sum_{k=1..p-1} 1/k^2 is a square and denominator Sum_{k=1..p-1} 1/k^3 is a cube.

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A120285 Numerator of harmonic number H(p-1) = Sum_{k=1..p-1} 1/k for prime p.

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A125551 As p runs through primes >= 5, sequence gives { numerator of Sum_{k=1..p-1} 1/k^2 } / p.

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Maple

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A186720 As p runs through the primes, sequence gives denominator of Sum_{k=1..p-1} 1/k^2.

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A076638 Denominators of harmonic numbers when the numerators are divisible by squares of primes >= 5 in the case of Wolstenholme's Theorem.

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A127045 Primes p such that denominator of Sum_{k=1..p-1} 1/k^9 is a 9th power.

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A127052 Primes p such that denominator of Sum_{k=1..p-1} 1/k^8 is an eighth power.

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