A127046 -id:A127046 - cp's OEIS frontend

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

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]

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]

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

A127063 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 and denominator Sum_{k=1..p-1} 1/k^5 is a fifth power.

Original entry on oeis.org

2, 3, 5, 17, 439, 443, 16400183, 16400191, 16400201, 16400203, 16400221, 16400231, 16400233, 16400269, 16400273, 16400299, 16400309, 16400317, 16400347, 16400383, 16400387, 16400389, 16400411, 16400413, 16400429, 16400431

Offset: 1

Artur Jasinski, Jan 04 2007

nonn

Subsequence of A127062 and of A127061. - Max Alekseyev, Feb 08 2007

Cf. A061002, A034602, A127029, A127042, A127043, A127044, A127046, A127047, A127048, A127049, A127051, A127061, A127062.

More terms from Max Alekseyev, Feb 08 2007

cp's OEIS Frontend

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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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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Mathematica

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

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Mathematica

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.

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A127063 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 and denominator Sum_{k=1..p-1} 1/k^5 is a fifth power.

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