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 1-2 of 2 results.

A120271 a(n) = numerator(Sum_{k=1..n} 1/(prime(k)-1)).

Original entry on oeis.org

1, 3, 7, 23, 121, 21, 173, 1597, 17927, 127469, 129317, 43619, 44081, 44521, 1033223, 13538159, 395369371, 132680013, 400467919, 402757063, 1214947859, 1221110939, 50305908619, 50529880549, 101470376303, 509322834499, 8691337402883
Offset: 1

Views

Author

Alexander Adamchuk, Jul 01 2006

Keywords

Comments

a(n) is squarefree except for n = 5, 14, 49, ... where squared prime factors are 11, 211, 479, ...
a(n)/A128646(n) is the asymptotic mean over the positive integers of the number of prime divisors that are not greater than prime(n), counted with multiplicity (cf. A007814, A169611, A356006). - Amiram Eldar, Jul 23 2022

Links

Crossrefs

Cf. A128646 (denominators), A119686, A006093, A000040.
Cf. A135212, A078456.
Cf. A007814, A169611, A356006.

Programs

  • Maple
    R:= [seq(1/(ithprime(k)-1),k=1..40)]:
S:= ListTools:-PartialSums(R):
A:= map(numer,S); # Robert Israel, Jan 12 2025
  • Mathematica
    Numerator[Table[Sum[1/(Prime[i]-1),{i,1,n}],{n,1,50}]]
Accumulate[1/(Prime[Range[30]]-1)]//Numerator (* Harvey P. Dale, May 03 2025 *)
  • PARI
    a(n) = numerator(sum(k=1, n, 1/(prime(k)-1))); \\ Michel Marcus, Oct 02 2016

Formula

a(n) = numerator(Sum_{k=1..n} 1/(prime(k)-1)).
a(n) = A078456(n) * A135212(n). - Alexander Adamchuk, Nov 23 2007

A078456 Number of numbers less than prime(1)*...*prime(n) having exactly one prime factor among (prime(1),...,prime(n)) where prime(n) is the n-th prime.

Original entry on oeis.org

1, 3, 14, 92, 968, 12096, 199296, 3679488, 82607616, 2349508608, 71507128320, 2604912721920, 105300128563200, 4466750187110400, 207324589680230400, 10866166392736972800, 634672612705724006400, 38337584554108256256000
Offset: 1

Views

Author

Benoit Cloitre, Dec 31 2002

Keywords

Comments

For n>1 a(n) is the determinant of the (n-1) X (n-1) matrix with elements M[i,j] = Prime[i+1] if i=j and 1 otherwise. (See example lines.) - Alexander Adamchuk, Jun 02 2006
Second column of A096294. - Eric Desbiaux, Jun 20 2013

Examples

			a(2)=3 since 2*3=6 and 2,3,4 have 1 prime factor among (2,3)
3 1 1 1 1 ...
1 5 1 1 1 ...
1 1 7 1 1 ...
1 1 1 11 1 ...
1 1 1 1 13 ...
and so a(2) = 3, a(3) = 3*5 - 1*1 = 14, a(4) = 3*5*7 + 1*1*1 + 1*1*1 - 7*1*1 - 5*1*1 - 3*1*1 = 92, etc.

Links

Crossrefs

Cf. A135212, A120271.
Cf. A117494, A002110.

Programs

  • Mathematica
    Table[ Det[ DiagonalMatrix[ Table[ Prime[i+1]-1, {i, 1, n-1} ] ] + 1 ], {n, 1, 20} ] (* Alexander Adamchuk, Jun 02 2006 *)
  • PARI
    a(n)=sum(k=1,prod(i=1,n, prime(i)),if(isprime(gcd(k,prod(i=1,n, prime(i)))),1,0))
  • PARI
    a(n) = matdet(matrix(n-1, n-1, j, k, if (j==k, prime(j+1), 1))); \\ after Mathematica; Michel Marcus, Oct 02 2016

Formula

a(n) = (prime(n)-1)*a(n-1) + A005867(n). - Matthew Vandermast, Jun 06 2004
a(n) = A120071(n) * A135212(n). - Alexander Adamchuk, Nov 23 2007
a(n) = A117494(A002110(n)). - Ridouane Oudra, Sep 18 2022

Extensions

a(7) from Ralf Stephan, Mar 25 2003
a(8)-a(12) from Matthew Vandermast, Jun 06 2004
More terms from Alexander Adamchuk, Jun 02 2006
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.