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

%I A024528 #31 Feb 16 2025 08:32:34
%S A024528 1,3,11,61,457,5237,70391,1226677,23817373,557499269,16390571671,
%T A024528 514577415031,19239924846277,796257656832167,34543329507310391,
%U A024528 1636619248175258407,87355709935877186981,5186576044693944076609
%N A024528 a(n) = n-th elementary symmetric function of {1, prime(1), prime(2), ..., prime(n)}.
%C A024528 a(0) through a(12) are squarefree. - _Gerald McGarvey_, Sep 03 2004
%C A024528 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
%H A024528 Clark Kimberling, <a href="/A024528/b024528.txt">Table of n, a(n) for n = 0..500</a>
%H A024528 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicSeriesofPrimes.html">Harmonic Series of Primes</a>
%F A024528 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
%F A024528 a(n) = Det[ DiagonalMatrix[ Table[ Prime[i], {i, 1, n} ] ] + 1 ]. - _Alexander Adamchuk_, Jun 02 2006
%F A024528 a(0) = 1, a(n) = A000040(n)*a(n-1) + A002110(n-1) for n>=1. - _Philippe Deléham_, Jun 03 2015
%e A024528 a(0) = 1
%e A024528 a(1) = 1*2 + A002110(0) = 2 + 1 = 3
%e A024528 a(2) = 3*3 + A002110(1) = 9 + 2 = 11
%e A024528 a(3) = 11*5 + A002110(2) = 55 + 6 = 61
%e A024528 a(4) = 61*7 + A002110(3) = 427 + 30 = 457
%e A024528 a(5) = 457*11 + A002110(4) = 5027 + 210 = 5237
%e A024528 a(6) = 5237*13 + A002110(5) = 68081 + 2310 = 70391
%e A024528 a(7) = 70391*17 + A002110(6) = 1196647 + 30030 = 1226677 - _Philippe Deléham_, Jun 03 2015
%p A024528 N:= 30: # to get a(0) to a(N)
%p A024528 Primes:= [seq(ithprime(i),i=1..N)]:
%p A024528 seq(mul(Primes[i],i=1..n)*(1+add(1/Primes[i],i=1..n)),n=0..N); # _Robert Israel_, Jun 03 2015
%t A024528 Table[ Det[ DiagonalMatrix[ Table[ Prime[i], {i, 1, n} ] ] + 1 ], {n, 1, 20} ] (* _Alexander Adamchuk_, Jun 02 2006 *)
%t A024528 p[0] = 1; p[n_] := Prime[n];
%t A024528 t[n_] := Table[p[k], {k, 0, n}]
%t A024528 a[n_] := SymmetricPolynomial[n, t[n]]
%t A024528 Table[a[n], {n, 0, 20}]
%t A024528 (* _Clark Kimberling_, Aug 18 2012 *)
%Y A024528 Cf. A000040, A002110, A024451.
%Y A024528 Cf. A125707 (indices of primes).
%K A024528 nonn
%O A024528 0,2
%A A024528 _Clark Kimberling_
%E A024528 More terms from _T. D. Noe_, Sep 09 2004