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A023626 Self-convolution of (1, p(1), p(2), ...). 6
1, 4, 10, 22, 43, 80, 137, 222, 343, 508, 737, 1030, 1411, 1888, 2477, 3198, 4059, 5096, 6297, 7702, 9327, 11176, 13301, 15682, 18355, 21344, 24673, 28358, 32411, 36896, 41769, 47082, 52883, 59148, 65937, 73298, 81251, 89776, 98957 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

p(1),p(2),p(3)... are the prime numbers (A000040). The analogous sequence for the partition numbers is A048574.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

FORMULA

G.f: x*(1+b(x))^2 = (c(x)^2)/x, where b(x) and c(x) are respectively the g.f. of A000040 and A008578. - Mario C. Enriquez, Dec 10 2016

EXAMPLE

G.f. = x + 4*x^2 + 10*x^3 + 22*x^4 + 43*x^5 + 80*x^6 + 137*x^7 + ...

MATHEMATICA

z = 100; p = Join[{1}, Prime[Range[z]]];

a[n_] := Sum[p[[i]] p[[n - i + 1]], {i, 1, n}];

Table[a[n], {n, 1, z}]  (* Clark Kimberling, Dec 01 2016 *)

a[ n_] := If[ n < 1, 0, SeriesCoefficient[ (1 + O[x]^n + Sum[ Prime[k] x^k, {k, n - 1}])^2, {x, 0, n - 1}]]; (* Michael Somos, Dec 01 2016 *)

Table[With[{c=Join[{1}, Prime[Range[n]]]}, ListConvolve[c, c]], {n, 0, 40}]// Flatten (* Harvey P. Dale, Oct 19 2018 *)

PROG

(Haskell)

a023626 n = a023626_list !! (n-2)

a023626_list = f a000040_list [1] where

   f (p:ps) rs = (sum $ zipWith (*) rs a008578_list) : f ps (p : rs)

-- Reinhard Zumkeller, Nov 09 2015

CROSSREFS

Cf. A000040, A008578, A014342, A048574.

Sequence in context: A008256 A006001 A034357 * A048574 A052837 A052821

Adjacent sequences:  A023623 A023624 A023625 * A023627 A023628 A023629

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified July 4 11:31 EDT 2020. Contains 335448 sequences. (Running on oeis4.)