A127042 -id:A127042 - cp's OEIS frontend

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

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

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]

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

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

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]

cp's OEIS Frontend

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

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

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

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

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

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

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

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

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