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.

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A115963 Numerator of Sum_{i=1..n} 1/prime(i)^3.

Original entry on oeis.org

1, 35, 4591, 1601713, 2141141003, 4716413174591, 23198819007792583, 159253748925534977797, 1938552948676080555065099, 47290471293028435532185602511, 1409101231790431848106470385672201
Offset: 1

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Author

Jonathan Vos Post, Mar 14 2006

Keywords

Comments

Denominators = A115964. See also: A024451 Numerator of Sum_{i=1..n} 1/prime(i). A002110 Primorial [denominator of Numerator of Sum_{i=1..n} 1/prime(i)]. A061015 Numerator of Sum_{i=1..n} 1/prime(i)^2.

Examples

			1/8, 35/216, 4591/27000, 1601713/9261000, 2141141003/12326391000, 4716413174591/27081081027000.
		

Crossrefs

Programs

  • Mathematica
    Accumulate[1/Prime[Range[20]]^3]//Numerator (* Harvey P. Dale, Dec 30 2024 *)

Formula

a(n) = Numerator of Sum_{i=1..n} 1/A000040(i)^3.

A354417 a(n) is the numerator of the sum of the reciprocals of the first n squarefree numbers.

Original entry on oeis.org

1, 3, 11, 61, 11, 82, 171, 1951, 26133, 13424, 41273, 716656, 13871719, 4700888, 9548741, 222854273, 112857219, 3310041496, 20075905417, 628822761157, 19239404599, 9709078632, 1959180271, 73097429088, 147378388979, 445594718515, 18404305970657, 3089336006908, 133763418792581
Offset: 1

Views

Author

Ilya Gutkovskiy, May 26 2022

Keywords

Examples

			1, 3/2, 11/6, 61/30, 11/5, 82/35, 171/70, 1951/770, 26133/10010, 13424/5005, 41273/15015, ...
		

Crossrefs

Programs

  • Maple
    s:= 0: R:= NULL: count:= 0:
    for x from 1 while count < 40 do
      if numtheory:-issqrfree(x) then
        s:= s + 1/x;
        v:= numer(s);
        R:= R, v;
        count:= count+1;
      fi;
    od:
    R; # Robert Israel, Mar 05 2023
  • Mathematica
    Accumulate[1/Select[Range[43], SquareFreeQ]] // Numerator
  • PARI
    a(n) = my(i=0, s=0); for(x=1, oo, if(core(x)==x, s+=1/x; i++; if(i==n, return(numerator(s))))) \\ Felix Fröhlich, May 26 2022

Formula

a(n)/A354418(n) ~ (6/Pi^2) * (log(n) + c) + O*(1.044/sqrt(n)), where f = O*(g) means |f| <= g and c = gamma + 2 * Sum_{p prime} log(p)/(p^2-1) = A001620 + 2 * A306016 = 1.71713765109059847340... (Ramaré, 2019; Alterman, 2022). - Amiram Eldar, Oct 29 2022
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