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A359328 Maximal coefficient of x^2*(x^2 + x^3)*(x^2 + x^3 + x^5)*...*(x^2 + x^3 + x^5 + ... + x^prime(n)).

%I A359328 #35 Feb 01 2024 16:24:09
%S A359328 1,1,1,2,4,12,46,251,1576,11578,94933,875134,8900088,99276703,
%T A359328 1214131109,16107824706,229757728186,3499486564517,56862172844198,
%U A359328 980725126968577,17899265342632635,345197504845310134,7005723403640260805,149261757412790940113,3329108788695272565243
%N A359328 Maximal coefficient of x^2*(x^2 + x^3)*(x^2 + x^3 + x^5)*...*(x^2 + x^3 + x^5 + ... + x^prime(n)).
%C A359328 Excluding the term 1 from A326178, the exponents of the product x^0*x^2*(x^2 + x^3)*(x^2 + x^3 + x^5)*...*(x^2 + x^3 + x^5 + ... + x^prime(n)) are given by all the other terms of A326178.
%C A359328 a(n) is the number of compositions of the terms of the n-th row of A359337 into n prime parts less than or equal to prime(n), with the first part equal to 2, the second part less than or equal to 3, ..., and the n-th part less than or equal to prime(n).
%H A359328 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%t A359328 Table[Max[CoefficientList[Product[Sum[x^Prime[i],{i,k}],{k,n}],x]],{n,0,24}]
%o A359328 (PARI) a(n) = vecmax(Vec(prod(k=1, n, sum(i=1, k, x^prime(i))))); \\ _Michel Marcus_, Dec 27 2022
%o A359328 (Python)
%o A359328 from collections import Counter
%o A359328 from sympy import prime, primerange
%o A359328 def A359328(n):
%o A359328     if n == 0: return 1
%o A359328     c, p = {0:1}, list(primerange(prime(n)+1))
%o A359328     for k in range(1,n+1):
%o A359328         d = Counter()
%o A359328         for j in c:
%o A359328             a = c[j]
%o A359328             for i in p[:k]:
%o A359328                 d[j+i] += a
%o A359328         c = d
%o A359328     return max(c.values()) # _Chai Wah Wu_, Feb 01 2024
%Y A359328 Cf. A000040, A326178, A350457.
%Y A359328 Cf. A359337 (corresponding exponents), A359338 (minimal corresponding exponent), A359339 (maximal corresponding exponent).
%K A359328 nonn
%O A359328 0,4
%A A359328 _Stefano Spezia_, Dec 26 2022