A359328 -id:A359328 - cp's OEIS frontend

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)).

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

A359338 Minimal 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, 4, 7, 12, 16, 22, 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), A359339 (maximal exponent).

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

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}]

A359388 a(n) is the number of compositions of n into prime parts, with the 1st part equal to 2, the 2nd part less than or equal to 3, ..., and the k-th part less than or equal to prime(k), and so on.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 2, 4, 5, 7, 11, 15, 24, 33, 50, 73, 105, 159, 229, 342, 501, 738, 1094, 1604, 2378, 3499, 5166, 7627, 11243, 16610, 24494, 36165, 53376, 78775, 116301, 171642, 253398, 374034, 552139, 815079, 1203166, 1776174, 2621938, 3870572, 5713798, 8434744

Offset: 0

Stefano Spezia, Dec 29 2022

nonn

			The 7 such compositions of n = 11 are:
[ 1]  (2, 2, 2, 2, 3);
[ 2]  (2, 2, 2, 3, 2);
[ 3]  (2, 2, 3, 2, 2);
[ 4]  (2, 3, 2, 2, 2);
[ 5]  (2, 2, 2, 5);
[ 6]  (2, 2, 5, 2);
[ 7]  (2, 3, 3, 3).

Alois P. Heinz, Table of n, a(n) for n = 0..4000
Index entries for sequences related to compositions

Cf. A000040, A004526, A023360, A078974, A326178, A359328.

b:= proc(n, i) option remember; `if`(n=0, 1, add(
      b(n-ithprime(j), i+1), j=1..min(i, numtheory[pi](n))))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 29 2022

a[n_]:=Coefficient[Expand[Sum[Product[Sum[x^Prime[i], {i, k}], {k,m}], {m, 0,Floor[n/2]}]],x,n]; Array[a,48,0]

G.f.: Sum_{m>=0} Product_{k=1..m} Sum_{i=1..k} x^prime(i).

a(n) ~ c*A078974^n, where c = 0.094587447... .

A369775 Maximal coefficient of (1 + x^2) * (1 + x^2 + x^3) * (1 + x^2 + x^3 + x^5) * ... * (1 + x^2 + ... + x^prime(n)).

Original entry on oeis.org

1, 1, 2, 5, 16, 65, 293, 1807, 12946, 106475, 972260, 9858553, 109451903, 1323071345, 17398667717, 247055196932, 3753507625272, 60680317203979, 1043036844360792, 18969267205680868, 364107881070036688, 7366172106829696356, 156467911373737550264

Offset: 0

Ilya Gutkovskiy, Jan 31 2024

nonn

Cf. A000040, A000140, A039831, A350457, A359328.

Table[Max[CoefficientList[Product[(1 + Sum[x^Prime[j], {j, 1, i}]), {i, 1, n}], x]], {n, 0, 22}]

a(n) = vecmax(Vec(prod(k=1, n, 1 + sum(i=1, k, x^prime(i))))); \\ Michel Marcus, Feb 01 2024

from collections import Counter
from sympy import prime, primerange
def A369775(n):
    if n == 0: return 1
    c, p = {0:1}, list(primerange(prime(n)+1))
    for k in range(1,n+1):
        d = Counter(c)
        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

cp's OEIS Frontend

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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A359338 Minimal 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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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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A359388 a(n) is the number of compositions of n into prime parts, with the 1st part equal to 2, the 2nd part less than or equal to 3, ..., and the k-th part less than or equal to prime(k), and so on.

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A369775 Maximal coefficient of (1 + x^2) * (1 + x^2 + x^3) * (1 + x^2 + x^3 + x^5) * ... * (1 + x^2 + ... + 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)).

A359338 Minimal 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)).

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)).