A061002 -id:A061002 - cp's OEIS frontend

A127042 Primes p such that denominator of Sum_{k=1..p-1} 1/k^2 is a square.

2, 3, 5, 7, 17, 19, 29, 31, 37, 41, 97, 127, 131, 211, 223, 227, 229, 233, 239, 241, 439, 443, 449, 457, 461, 463, 727, 733, 739, 743, 751, 757, 761, 769, 773, 863, 877, 881, 883, 887, 967, 971, 977, 983, 991, 997, 1009, 1013, 1187, 1193, 1201, 1901, 1907, 1913, 1931, 1933

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

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

Cf. A061002, A034602, A127029.

a = {}; Do[If[Sqrt[Denominator[Sum[1/x^2, {x, 1, Prime[x] - 1}]]] == Floor[Sqrt[Denominator[Sum[1/x^2, {x, 1, Prime[x] - 1}]]]], AppendTo[a, Prime[x]]], {x, 1, 50}]; a

More terms from Franklin T. Adams-Watters, Jan 21 2012

A127046 Primes p such that denominator of Sum_{k=1..p-1} 1/k^3 is a cube.

Original entry on oeis.org

2, 3, 5, 11, 13, 17, 29, 31, 37, 41, 83, 89, 97, 137, 139, 293, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

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

Cf. A061002, A034602, A127029, A127042.

d[n_] := Module[{}, su = 0; a = {}; For[i = 1, i <= n, i++, su = su + 1/ i^3; If[PrimeQ[i + 1], If[IntegerQ[(Denominator[su])^(1/3)], AppendTo[a, i + 1]]]]; a]; d[2000]
Select[Prime[Range[200]], IntegerQ[Surd[Denominator[Sum[1/k^3, {k,#-1}]], 3]]&] (* Harvey P. Dale, Mar 13 2013 *)

A127047 Primes p such that denominator of Sum_{k=1..p-1} 1/k^4 is a fourth power.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 53, 67, 71, 73, 97, 101, 103, 107, 109, 127, 131, 197, 199, 211, 223, 227, 229, 233, 293, 367, 373, 379, 383, 389, 397, 401, 439, 443, 449, 457, 461, 463, 557, 563, 569, 571, 577, 877, 881, 883, 967, 971, 977, 983, 991, 997

Offset: 1

Artur Jasinski, Jan 03 2007

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..665 from Robert Israel)

Cf. A061002, A034602, A127029, A127042, A127046.

S:= 0: R:= NULL: count:= 0:
for k from 1 while count < 100 do
  S:= S + 1/k^4;
  if isprime(k+1) and surd(denom(S),4)::integer then R:= R,k+1; count:= count+1 fi
od:
R; # Robert Israel, Oct 25 2019

d[n_] := Module[{}, su = 0; a = {}; For[i = 1, i <= n, i++, su = su + 1/ i^4; If[PrimeQ[i + 1], If[IntegerQ[(Denominator[su])^(1/4)], AppendTo[a, i + 1]]]]; a]; d[10000]
Select[Flatten[Position[Denominator[Accumulate[1/Range[1000]^4]],?(IntegerQ[ Surd[ #,4]]&)]],PrimeQ] (* _Harvey P. Dale, Feb 08 2015 *)

A127048 Primes p such that denominator of Sum_{k=1..p-1} 1/k^5 is a fifth power.

Original entry on oeis.org

2, 3, 5, 11, 13, 17, 37, 41, 53, 83, 127, 131, 137, 139, 149, 151, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 853, 857, 859, 863, 877, 881, 883, 887, 929, 967, 1091, 1093, 1097, 1103, 1109, 1151

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.

d[n_] := Module[{}, su = 0; a = {}; For[i = 1, i <= n, i++, su = su + 1/ i^5; If[PrimeQ[i + 1], If[IntegerQ[(Denominator[su])^(1/5)], AppendTo[a, i + 1]]]]; a]; d[2000]

A127043 Primes p such that denominator of Sum_{k=1..p-1} 1/k^2 is not a square.

Original entry on oeis.org

11, 13, 23, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 101, 103, 107, 109, 113, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A061002, A034602, A127029, A127042.

S:= 0: R:= NULL: count:= 0:
for k from 1 while count < 100 do
  S:= S + 1/k^2;
  if isprime(k+1) and not issqr(denom(S)) then
       R:= R,k+1; count:= count+1;
  fi
od:
R; # Robert Israel, Oct 25 2019

a = {}; Do[If[Sqrt[Denominator[Sum[1/x^2, {x, 1, Prime[x] - 1}]]] == Floor[Sqrt[Denominator[Sum[1/x^2, {x, 1, Prime[x] - 1}]]]], 1,AppendTo[a, Prime[x]]], {x, 1, 50}]; a

More terms from Robert Israel, Oct 25 2019

A127051 Primes p such that denominator of Sum_{k=1..p-1} 1/k^7 is a seventh power.

Original entry on oeis.org

2, 3, 5, 11, 13, 17, 29, 31, 37, 41, 83, 131, 251, 257, 263, 269, 271, 293, 419, 421, 479, 1163, 1171, 1181, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 3137, 3163, 3167, 3169, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607

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.

d[n_] := Module[{}, su = 0; a = {}; For[i = 1, i <= n, i++, su = su + 1/ i^7; If[PrimeQ[i + 1], If[IntegerQ[(Denominator[su])^(1/7)], AppendTo[a, i + 1]]]]; a]; d[2000]

A127044 Squares of denominators of Sum_{k=1..p-1} 1/k^2 for p in A127042.

Original entry on oeis.org

1, 2, 12, 60, 720720, 12252240, 80313433200, 2329089562800, 144403552893600, 5342931457063200, 718766754945489455304472257065075294400, 52573842877942565273243107104095419458814459401768000

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..65

Cf. A061002, A034602, A127029, A127042, A127043.

a = {}; Do[If[Sqrt[Denominator[Sum[1/x^2, {x, 1, Prime[x] - 1}]]] == Floor[Sqrt[Denominator[Sum[1/x^2, {x, 1, Prime[x] - 1}]]]], AppendTo[a, Sqrt[Denominator[Sum[1/x^2, {x, 1, Prime[x] - 1}]]]]], {x, 1, 50}]; a

A076637 Numerators of harmonic numbers when these numerators are divisible by squares of primes >= 5 in the case of Wolstenholme's Theorem.

Original entry on oeis.org

25, 49, 7381, 86021, 2436559, 14274301, 19093197, 315404588903, 9304682830147, 54801925434709, 2078178381193813, 12309312989335019, 5943339269060627227, 14063600165435720745359, 254381445831833111660789, 15117092380124150817026911

Offset: 1

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

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. - Bernard Schott, Dec 02 2018

The numbers 363, numerator of H_7 and 9227046511387, numerator of H_{29}, which have been found by Amiram Eldar and Michel Marcus, are also divisible by prime squares, respectively by 11^2 and 43^2, but not in the case of Wolstenholme's Theorem. So, a new sequence A322434 is created with all the numerators of Harmonic numbers which are divisible by any prime square >= 5. - Bernard Schott, Dec 08 2018

			25 is a term because the numerator of the harmonic number H_4 = 1 + 1/2+ 1/3 + 1/4 = 25/12 is divisible by the square of 5;
49 is a term because the numerator of the harmonic number H_6 = 1 + 1/2+ 1/3 + 1/4 + 1/5 + 1/6 = 49/20 is divisible by the square of 7.

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

Cf. A076638, A001008.

a[p_] := Numerator[HarmonicNumber[p - 1]]; a /@ Prime@Range[3, 20] (* Amiram Eldar, Dec 08 2018 *)

More terms from Amiram Eldar, Dec 04 2018

A127049 Primes p such that denominator of Sum_{k=1..p-1} 1/k^6 is a sixth power.

Original entry on oeis.org

2, 3, 5, 7, 17, 19, 41, 43, 47, 97, 127, 191, 193, 197, 199, 211, 223, 227, 229, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 991, 997, 1009, 1013, 1187, 1193, 1201, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3613, 3617, 3623, 3631

Offset: 1

Artur Jasinski, Jan 03 2007, Jan 04 2007

nonn

Amiram Eldar, 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^6; If[PrimeQ[i + 1] && IntegerQ[(Denominator[su])^(1/6)], AppendTo[a, i + 1]]]; a]; d[2000]
Select[Prime[Range[600]],IntegerQ[Surd[Denominator[Sum[1/k^6,{k,#-1}]], 6]]&] (* Harvey P. Dale, Aug 04 2019 *)

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

A185399 As p runs through the primes, sequence gives denominator of Sum_{k=1..p-1} 1/k.

Original entry on oeis.org

1, 2, 12, 20, 2520, 27720, 720720, 4084080, 5173168, 80313433200, 2329089562800, 13127595717600, 485721041551200, 2844937529085600, 1345655451257488800, 3099044504245996706400, 54749786241679275146400, 3230237388259077233637600

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.

T. D. Noe, Table of n, a(n) for n = 1..100
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. A001008, A002805 (numerators and denominators of harmonic numbers).

Cf. A061002, A193758.

[Denominator(HarmonicNumber(NthPrime(n)-1)): n in [1..40]]; // Vincenzo Librandi, Dec 05 2018

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

nn = 20; sm = 0; t = Table[sm = sm + 1/k; Denominator[sm], {k, Prime[nn]}]; Table[t[[p - 1]], {p, Prime[Range[nn]]}] (* T. D. Noe, Apr 23 2013 *)

a(n) = denominator(sum(k=1, prime(n)-1, 1/k)); \\ Michel Marcus, Dec 05 2018

a(n) = denominator(sum((k+1)/(p-k-1), k=0..p-2)), where p = the n-th prime. - Gary Detlefs, Jan 12 2012

a(n) = numerator(H(p)/H(p-1)) - denominator(H(p)/H(p-1)), where p is the n-th prime and H(n) is the n-th harmonic number. - Gary Detlefs, Apr 21 2013

cp's OEIS Frontend

A127042 Primes p such that denominator of Sum_{k=1..p-1} 1/k^2 is a square.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A127046 Primes p such that denominator of Sum_{k=1..p-1} 1/k^3 is a cube.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A127047 Primes p such that denominator of Sum_{k=1..p-1} 1/k^4 is a fourth power.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A127048 Primes p such that denominator of Sum_{k=1..p-1} 1/k^5 is a fifth power.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A127043 Primes p such that denominator of Sum_{k=1..p-1} 1/k^2 is not a square.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A127051 Primes p such that denominator of Sum_{k=1..p-1} 1/k^7 is a seventh power.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A127044 Squares of denominators of Sum_{k=1..p-1} 1/k^2 for p in A127042.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A076637 Numerators of harmonic numbers when these numerators are divisible by squares of primes >= 5 in the case of Wolstenholme's Theorem.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A127049 Primes p such that denominator of Sum_{k=1..p-1} 1/k^6 is a sixth power.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica