A024528 a(n) = n-th elementary symmetric function of {1, prime(1), prime(2), ..., prime(n)}.

1, 3, 11, 61, 457, 5237, 70391, 1226677, 23817373, 557499269, 16390571671, 514577415031, 19239924846277, 796257656832167, 34543329507310391, 1636619248175258407, 87355709935877186981, 5186576044693944076609

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Comments

a(0) through a(12) are squarefree. - Gerald McGarvey, Sep 03 2004

For n>0 a(n) is the determinant of the n X n matrix with elements M[i,j] = 1+Prime[i] if i=j and 1 otherwise. - Alexander Adamchuk, Jun 02 2006

Examples

			a(0) = 1
a(1) = 1*2 + A002110(0) = 2 + 1 = 3
a(2) = 3*3 + A002110(1) = 9 + 2 = 11
a(3) = 11*5 + A002110(2) = 55 + 6 = 61
a(4) = 61*7 + A002110(3) = 427 + 30 = 457
a(5) = 457*11 + A002110(4) = 5027 + 210 = 5237
a(6) = 5237*13 + A002110(5) = 68081 + 2310 = 70391
a(7) = 70391*17 + A002110(6) = 1196647 + 30030 = 1226677 - _Philippe Deléham_, Jun 03 2015
		

Links

• Clark Kimberling, Table of n, a(n) for n = 0..500
• Eric Weisstein's World of Mathematics, Harmonic Series of Primes

Crossrefs

Cf. A000040, A002110, A024451.

Cf. A125707 (indices of primes).

Programs

Maple

N:= 30: # to get a(0) to a(N)
Primes:= [seq(ithprime(i),i=1..N)]:
seq(mul(Primes[i],i=1..n)*(1+add(1/Primes[i],i=1..n)),n=0..N); # Robert Israel, Jun 03 2015

Mathematica

Table[ Det[ DiagonalMatrix[ Table[ Prime[i], {i, 1, n} ] ] + 1 ], {n, 1, 20} ] (* Alexander Adamchuk, Jun 02 2006 *)
p[0] = 1; p[n_] := Prime[n];
t[n_] := Table[p[k], {k, 0, n}]
a[n_] := SymmetricPolynomial[n, t[n]]
Table[a[n], {n, 0, 20}]
(* Clark Kimberling, Aug 18 2012 *)

Formula

This sequence is the numerators of the prime harmonic numbers + 1, i.e. a(n)/A002110(n) = Sum_{i=0...n} 1/p(i) where p(0) = 1, p(i) is the i-th prime for n > 0 and A002110 are the primorial numbers. - Gerald McGarvey, Sep 03 2004

a(n) = Det[ DiagonalMatrix[ Table[ Prime[i], {i, 1, n} ] ] + 1 ]. - Alexander Adamchuk, Jun 02 2006

a(0) = 1, a(n) = A000040(n)*a(n-1) + A002110(n-1) for n>=1. - Philippe Deléham, Jun 03 2015

Extensions

More terms from T. D. Noe, Sep 09 2004