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A147655 a(n) is the coefficient of x^n in the polynomial given by Product_{k>=1} (1 + prime(k)*x^k). 21
1, 2, 3, 11, 17, 40, 86, 153, 283, 547, 1069, 1737, 3238, 5340, 9574, 17251, 27897, 45845, 78601, 126725, 207153, 353435, 550422, 881454, 1393870, 2239938, 3473133, 5546789, 8762663, 13341967, 20676253, 31774563, 48248485, 74174759, 111904363, 170184798 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..9000 (first 1001 terms from Harvey P. Dale)

FORMULA

a(n) = [x^n] Product_{k>=1} 1+prime(k)*x^k. - Alois P. Heinz, Sep 05 2014

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

EXAMPLE

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, ....

From Petros Hadjicostas, Apr 10 2020: (Start)

Let f(m) = prime(m). Using the strict partitions of n (see A000009), we get:

a(1) = f(1) = 2,

a(2) = f(2) = 3,

a(3) = f(3) + f(1)*f(2) = 5 + 2*3 = 11,

a(4) = f(4) + f(1)*f(3) = 7 + 2*5 = 17,

a(5) = f(5) + f(1)*f(4) + f(2)*f(3) = 11 + 2*7 + 3*5 = 40,

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,

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)

MAPLE

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

      b(n, i-1) +`if`(i>n, 0, b(n-i, i-1)*ithprime(i))))

    end:

a:= n-> b(n$2):

seq(a(n), n=0..50);  # Alois P. Heinz, Sep 05 2014

MATHEMATICA

nn=40; Take[Rest[CoefficientList[Expand[Times@@Table[1+Prime[n]x^n, {n, nn}]], x]], nn] (* Harvey P. Dale, Jul 01 2012 *)

CROSSREFS

Row sums of A246867 and A258323.

Cf. A000009,A005117, A015723, A022629, A056239, A066189, A112798, A145519, A147541, A325504, A325506, A325537.

Sequence in context: A023870 A025099 A024597 * A051072 A051096 A051078

Adjacent sequences:  A147652 A147653 A147654 * A147656 A147657 A147658

KEYWORD

nonn

AUTHOR

Neil Fernandez, Nov 09 2008

EXTENSIONS

More terms from Harvey P. Dale, Jul 01 2012

a(0)=1 inserted by Alois P. Heinz, Sep 05 2014

Name edited by Petros Hadjicostas, Apr 10 2020

STATUS

approved

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Last modified September 30 02:38 EDT 2022. Contains 357095 sequences. (Running on oeis4.)