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%I #41 Apr 11 2020 05:25:55
%S 1,2,3,11,17,40,86,153,283,547,1069,1737,3238,5340,9574,17251,27897,
%T 45845,78601,126725,207153,353435,550422,881454,1393870,2239938,
%U 3473133,5546789,8762663,13341967,20676253,31774563,48248485,74174759,111904363,170184798
%N a(n) is the coefficient of x^n in the polynomial given by Product_{k>=1} (1 + prime(k)*x^k).
%C Sum of all squarefree numbers whose prime indices sum to n. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. - _Gus Wiseman_, May 09 2019
%H Alois P. Heinz, <a href="/A147655/b147655.txt">Table of n, a(n) for n = 0..9000</a> (first 1001 terms from Harvey P. Dale)
%F a(n) = [x^n] Product_{k>=1} 1+prime(k)*x^k. - _Alois P. Heinz_, Sep 05 2014
%F a(n) = Sum_{(b_1,...,b_n)} f(1)^b_1 * f(2)^b_2 * ... * f(n)^b_n, where f(m) = prime(m), and the sum is taken over all lists (b_1,...,b_n) with b_j in {0,1} and Sum_{j=1..n} j*b_j = n. - _Petros Hadjicostas_, Apr 10 2020
%e Form a product from the primes: (1 + 2*x) * (1 + 3*x^2) * (1 + 5*x^3) * ...* (1 + prime(n)*x^n) * ... Multiplying out gives 1 + 2*x + 3*x^2 + 11*x^3 + ..., so the sequence begins 1, 2, 3, 11, ....
%e From _Petros Hadjicostas_, Apr 10 2020: (Start)
%e Let f(m) = prime(m). Using the strict partitions of n (see A000009), we get:
%e a(1) = f(1) = 2,
%e a(2) = f(2) = 3,
%e a(3) = f(3) + f(1)*f(2) = 5 + 2*3 = 11,
%e a(4) = f(4) + f(1)*f(3) = 7 + 2*5 = 17,
%e a(5) = f(5) + f(1)*f(4) + f(2)*f(3) = 11 + 2*7 + 3*5 = 40,
%e a(6) = f(6) + f(1)*f(5) + f(2)*f(4) + f(1)*f(2)*f(3) = 13 + 2*11 + 3*7 + 2*3*5 = 86,
%e a(7) = f(7) + f(1)*f(6) + f(2)*f(5) + f(3)*f(4) + f(1)*f(2)*f(4) = 17 + 2*13 + 3*11 + 5*7 + 2*3*7 = 153. (End)
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p b(n, i-1) +`if`(i>n, 0, b(n-i, i-1)*ithprime(i))))
%p end:
%p a:= n-> b(n$2):
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Sep 05 2014
%t nn=40;Take[Rest[CoefficientList[Expand[Times@@Table[1+Prime[n]x^n,{n,nn}]],x]],nn] (* _Harvey P. Dale_, Jul 01 2012 *)
%Y Row sums of A246867 and A258323.
%Y Cf. A000009,A005117, A015723, A022629, A056239, A066189, A112798, A145519, A147541, A325504, A325506, A325537.
%K nonn
%O 0,2
%A _Neil Fernandez_, Nov 09 2008
%E More terms from _Harvey P. Dale_, Jul 01 2012
%E a(0)=1 inserted by _Alois P. Heinz_, Sep 05 2014
%E Name edited by _Petros Hadjicostas_, Apr 10 2020
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