A359338 -id:A359338 - cp's OEIS frontend

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)).

1, 1, 1, 2, 4, 12, 46, 251, 1576, 11578, 94933, 875134, 8900088, 99276703, 1214131109, 16107824706, 229757728186, 3499486564517, 56862172844198, 980725126968577, 17899265342632635, 345197504845310134, 7005723403640260805, 149261757412790940113, 3329108788695272565243

Offset: 0

Stefano Spezia, Dec 26 2022

nonn

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.

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).

Index entries for sequences related to compositions

Cf. A000040, A326178, A350457.

Cf. A359337 (corresponding exponents), A359338 (minimal corresponding exponent), A359339 (maximal corresponding exponent).

Table[Max[CoefficientList[Product[Sum[x^Prime[i],{i,k}],{k,n}],x]],{n,0,24}]

a(n) = vecmax(Vec(prod(k=1, n, sum(i=1, k, x^prime(i))))); \\ Michel Marcus, Dec 27 2022

from collections import Counter
from sympy import prime, primerange
def A359328(n):
    if n == 0: return 1
    c, p = {0:1}, list(primerange(prime(n)+1))
    for k in range(1,n+1):
        d = Counter()
        for j in c:
            a = c[j]
            for i in p[:k]:
                d[j+i] += a
        c = d
    return max(c.values()) # Chai Wah Wu, Feb 01 2024

A359337 Irregular triangle read by rows: the n-th row gives the exponents of the powers of x corresponding to the maximal coefficient of the product x^2(x^2 + x^3)(x^2 + x^3 + x^5)...(x^2 + x^3 + x^5 + ... + x^prime(n)).

Original entry on oeis.org

0, 2, 4, 5, 7, 12, 16, 17, 22, 24, 32, 42, 53, 65, 79, 96, 114, 134, 155, 180, 205, 233, 263, 294, 329, 364, 403, 442, 485, 529, 576, 625, 676, 729, 785, 842, 902, 964, 1029, 1097, 1167, 1238, 1313, 1390, 1469, 1552, 1636, 1723, 1813, 1904, 1999, 2096, 2195, 2298

Offset: 0

Stefano Spezia, Dec 27 2022

Conjecture: except for n = 2, 5, and 6, the rows have length equal to 1.

			The irregular triangle begins:
    0;
    2;
    4, 5;
    7;
   12;
   16, 17;
   22, 24;
   32;
   42;
   53;
   65;
   ...

Cf. A000040, A359328.

Cf. A359338 (minimal exponent), A359339 (maximal exponent).

b[n_]:=CoefficientList[Product[Sum[x^Prime[i],{i,k}],{k,n}],x]; Table[Position[b[n],Max[b[n]]]-1,{n,0,50}]//Flatten

A359339 Maximal exponent of the powers of x corresponding to the maximal coefficient of the product x^2(x^2 + x^3)(x^2 + x^3 + x^5)...(x^2 + x^3 + x^5 + ... + x^prime(n)).

Original entry on oeis.org

0, 2, 5, 7, 12, 17, 24, 32, 42, 53, 65, 79, 96, 114, 134, 155, 180, 205, 233, 263, 294, 329, 364, 403, 442, 485, 529, 576, 625, 676, 729, 785, 842, 902, 964, 1029, 1097, 1167, 1238, 1313, 1390, 1469, 1552, 1636, 1723, 1813, 1904, 1999, 2096, 2195, 2298, 2402, 2510

Offset: 0

Stefano Spezia, Dec 27 2022

nonn

Cf. A000040, A359328.

Cf. A359337 (exponents), A359338 (minimal exponent).

b[n_]:=CoefficientList[Product[Sum[x^Prime[i],{i,k}],{k,n}],x]; Table[Max[Position[b[n],Max[b[n]]]]-1,{n,0,52}]

cp's OEIS Frontend

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)).

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A359337 Irregular triangle read by rows: the n-th row gives the exponents of the powers of x corresponding to the maximal coefficient of the product x^2(x^2 + x^3)(x^2 + x^3 + x^5)...(x^2 + x^3 + x^5 + ... + x^prime(n)).

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A359339 Maximal exponent of the powers of x corresponding to the maximal coefficient of the product x^2(x^2 + x^3)(x^2 + x^3 + x^5)...(x^2 + x^3 + x^5 + ... + x^prime(n)).

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cp's OEIS Frontend

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)).

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PARI

Python

A359337 Irregular triangle read by rows: the n-th row gives the exponents of the powers of x corresponding to the maximal coefficient of the product x^2*(x^2 + x^3)*(x^2 + x^3 + x^5)*...*(x^2 + x^3 + x^5 + ... + x^prime(n)).

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Mathematica

A359339 Maximal exponent of the powers of x corresponding to the maximal coefficient of the product x^2*(x^2 + x^3)*(x^2 + x^3 + x^5)*...*(x^2 + x^3 + x^5 + ... + x^prime(n)).

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Mathematica

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)).

A359337 Irregular triangle read by rows: the n-th row gives the exponents of the powers of x corresponding to the maximal coefficient of the product x^2(x^2 + x^3)(x^2 + x^3 + x^5)...(x^2 + x^3 + x^5 + ... + x^prime(n)).

A359339 Maximal exponent of the powers of x corresponding to the maximal coefficient of the product x^2(x^2 + x^3)(x^2 + x^3 + x^5)...(x^2 + x^3 + x^5 + ... + x^prime(n)).