cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 18 results. Next

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

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Jan 03 2007

Keywords

Links

Crossrefs

Cf. A061002, A034602, A127029.

Programs

  • Mathematica
    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

Extensions

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

A127031 Let f(n) = exp(Pi*sqrt(n)); sequence gives numbers n such that f(n) - floor(f(n)) < 1/10^6.

Original entry on oeis.org

652, 2608, 880111, 2720885, 4089051, 4619054, 5046630, 5409046, 5433402, 5603556, 5645558, 7278138, 7466589, 10037029, 10730786, 10823358, 11540978, 11860073, 12898258, 14554227, 15107659, 15602035, 15896143, 17070573, 18204473, 19252185, 19425342, 19556500
Offset: 1

Views

Author

Artur Jasinski, Jan 03 2007

Keywords

Comments

a(212)=195246501 is the smallest integer such that the fractional part of e^(Pi*sqrt(n)) begins with exactly 8 zeros. - Anthony Canu, Oct 11 2017

Examples

			5 is not in the sequence since exp(Pi*sqrt(5)) = 1124.186... has fractional part 0.186... which is greater than 1/10^6. But exp(Pi*sqrt(652)) has fractional part 0.0000000001637... which is less than 1/10^6, so 652 is in the sequence. - _Michael B. Porter_, Aug 24 2016

Links

Crossrefs

Cf. A035484, A127022, A127023, A127024, A127025, A127026, A127027, A127028, A127029.

Programs

  • Mathematica
    $MaxExtraPrecision = 1000; a = {}; Do[If[((Exp[Pi Sqrt[x]] - Floor[Exp[Pi Sqrt[x]]]) > 0) && ((Exp[Pi Sqrt[x]] - Floor[Exp[Pi Sqrt[x]]]) < 10^(-6)), AppendTo[a, x]], {x, 1, 100000}]; a
  • PARI
    search(a,b)=my(t,prec=default(realprecision), nprec=round(Pi*sqrt(b)/log(10)+20)); default(realprecision,nprec); for(n=floor(a),b,t=exp(Pi*sqrt(n));if(t-floor(t)<.000001, print(n))); default(realprecision,prec) \\ Charles R Greathouse IV, Jul 28 2009

Extensions

a(3)-a(14) from Charles R Greathouse IV, Jul 28 2009
a(15)-a(18) from Anthony Canu, Aug 24 2016
a(19) from Anthony Canu, Aug 31 2016
a(20)-a(28) from Anthony Canu, Mar 03 2017

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

Views

Author

Artur Jasinski, Jan 03 2007

Keywords

Links

Crossrefs

Cf. A061002, A034602, A127029, A127042.

Programs

  • Mathematica
    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

Views

Author

Artur Jasinski, Jan 03 2007

Keywords

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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

Views

Author

Artur Jasinski, Jan 03 2007

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Artur Jasinski, Jan 03 2007

Keywords

Links

Crossrefs

Cf. A061002, A034602, A127029, A127042.

Programs

  • Maple
    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
  • Mathematica
    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

Extensions

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

Views

Author

Artur Jasinski, Jan 03 2007

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Artur Jasinski, Jan 03 2007

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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

A127026 Let f(n) = exp(Pi*sqrt(n)); sequence gives numbers n such that f(n)-floor(f(n)) < 1/10.

Original entry on oeis.org

2, 30, 33, 42, 56, 96, 100, 103, 140, 148, 164, 171, 172, 182, 187, 188, 205, 211, 223, 226, 239, 241, 243, 253, 268, 300, 318, 328, 359, 361, 364, 379, 387, 394, 408, 410, 421, 423, 425, 426, 436, 455, 465, 467, 490, 492, 509, 529, 536, 546, 579, 583, 587
Offset: 1

Views

Author

Artur Jasinski, Jan 03 2007

Keywords

Crossrefs

Cf. A035484, A127022, A127023, A127024, A127025, A127027, A127028, A127029, A127030.

Programs

  • Mathematica
    a = {}; Do[If[((Exp[Pi Sqrt[x]] - Floor[Exp[Pi Sqrt[x]]]) > 0) && (( Exp[Pi Sqrt[x]] - Floor[Exp[Pi Sqrt[x]]])< 10^(-1)), AppendTo[a, x]], {x, 1, 1000}]; a

A127027 Let f(n) = exp(Pi*sqrt(n)); sequence gives numbers n such that f(n) - floor(f(n)) < 1/10^2.

Original entry on oeis.org

103, 148, 164, 205, 223, 226, 268, 359, 630, 652, 940, 1005, 1194, 1213, 1332, 1353, 1441, 1481, 1519, 1750, 1823, 1825, 1835, 1930, 1951, 1961, 2309, 2339, 2347, 2357, 2498, 2511, 2527, 2554, 2608, 2683, 3086, 3108, 3157, 3377, 3646, 3653, 3656, 3738, 3762
Offset: 1

Views

Author

Artur Jasinski, Jan 03 2007

Keywords

Crossrefs

Cf. A035484, A127022, A127023, A127024, A127025, A127026, A127028, A127029, A127030.

Programs

  • Mathematica
    a = {}; Do[If[((Exp[Pi Sqrt[x]] - Floor[Exp[Pi Sqrt[x]]]) > 0) && (( Exp[Pi Sqrt[x]] - Floor[Exp[Pi Sqrt[x]]])< 10^(-2)), AppendTo[a, x]], {x, 1, 1000}]; a
epsQ[n_]:=Module[{c=Exp[Pi*Sqrt[n]],min=1/10^2},0Harvey P. Dale, Apr 06 2012 *)

Extensions

Corrected by Harvey P. Dale, Apr 06 2012
a(27)-a(45) from Jon E. Schoenfield, Sep 04 2017
Showing 1-10 of 18 results. Next

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