A350457 -id:A350457 - 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

A350514 Maximal coefficient of Product_{j=1..n} (1 - x^prime(j)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 6, 8, 12, 17, 30, 41, 70, 107, 186, 307, 531, 887, 1561, 2701, 4817, 8514, 15030, 26490, 47200, 84622, 151809, 273912, 496807, 900595, 1643185, 2999837, 5498916, 10111429, 18596096, 34306158, 63585519, 118215700

Offset: 0

Alois P. Heinz, Jan 02 2022

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A000040, A007504, A046675, A086376, A350457.

b:= proc(n) option remember; `if`(n=0, 1,
      expand((1-x^ithprime(n))*b(n-1)))
    end:
a:= n-> max(coeffs(b(n))):
seq(a(n), n=0..60);

a[n_] := Times@@(1-x^Prime[Range[n]])//Expand//CoefficientList[#, x]&//Max;
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Jun 02 2022 *)

a(n) = vecmax(Vec(prod(j=1, n, 1-'x^prime(j)))); \\ Michel Marcus, Jan 04 2022

from sympy.abc import x
from sympy import prime, prod
def A350514(n): return 1 if n == 0 else max(prod(1-x**prime(i) for i in range(1,n+1)).as_poly().coeffs()) # Chai Wah Wu, Jan 04 2022

A367843 Maximum of the absolute value of the coefficients of (1 - x^2) * (1 - x^3) * (1 - x^5) * ... * (1 - x^prime(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 6, 8, 12, 18, 30, 46, 70, 113, 186, 314, 531, 894, 1561, 2705, 4817, 8514, 15030, 26502, 47200, 84698, 151809, 273961, 496807, 900596, 1643185, 2999837, 5498916, 10111429, 18596096, 34306158, 63585519, 118215700

Offset: 0

Ilya Gutkovskiy, Feb 06 2024

nonn

Cf. A046675, A160089, A350457, A350514.

Table[Max[Abs[CoefficientList[Product[(1 - x^Prime[k]), {k, 1, n}], x]]], {n, 0, 46}]

from collections import Counter
from sympy import prime
def A367843(n):
    c = {0:1}
    for k in range(1,n+1):
        p, b = prime(k), Counter(c)
        for j in c:
            b[j+p] -= c[j]
        c = b
    return max(map(abs,c.values())) # Chai Wah Wu, Feb 06 2024

A369708 Maximal coefficient of (1 + x^3) * (1 + x^5) * (1 + x^7) * ... * (1 + x^prime(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 4, 5, 8, 14, 23, 40, 70, 126, 221, 394, 711, 1290, 2354, 4344, 8015, 14868, 27585, 51094, 95160, 178436, 335645, 634568, 1202236, 2261052, 4267640, 8067296, 15318171, 29031484, 55248527, 105251904, 200711160, 383580180, 733704990

Offset: 0

Ilya Gutkovskiy, Jan 29 2024

nonn

Cf. A024939, A025591, A350457.

b:= proc(n) option remember; `if`(n<2, 1, expand(b(n-1)*(1+x^ithprime(n)))) end:
a:= n-> max(coeffs(b(n))):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 29 2024

Table[Max[CoefficientList[Product[(1 + x^Prime[k]), {k, 2, n}], x]], {n, 0, 40}]

a(n) = vecmax(Vec(prod(i=2, n, 1+x^prime(i)))); \\ Michel Marcus, Jan 29 2024

from collections import Counter
from sympy import prime
def A369708(n):
    c = {0:1}
    for i in range(2,n+1):
        p, d = prime(i), Counter(c)
        for k in c:
            d[k+p] += c[k]
        c = d
    return max(c.values()) # Chai Wah Wu, Jan 31 2024

A350504 Maximal coefficient of (1 + x) * (1 + x^3) * (1 + x^5) * ... * (1 + x^(2*n-1)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 5, 8, 13, 22, 38, 68, 118, 211, 380, 692, 1262, 2316, 4277, 7930, 14745, 27517, 51541, 96792, 182182, 343711, 650095, 1231932, 2338706, 4447510, 8472697, 16164914, 30884150, 59086618, 113189168, 217091832, 416839177, 801247614, 1541726967, 2969432270

Offset: 0

Ilya Gutkovskiy, Jan 28 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A025591, A039828, A160235, A350457.

b:= proc(n) option remember; `if`(n=0, 1,
      expand((1+x^(2*n-1))*b(n-1)))
    end:
a:= n-> max(coeffs(b(n))):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 28 2022

b[n_] := b[n] = If[n == 0, 1, Expand[(1 + x^(2*n - 1))*b[n - 1]]];
a[n_] := Max[CoefficientList[b[n], x]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 01 2022, after Alois P. Heinz *)

a(n) = vecmax(Vec(prod(k=1, n, 1+x^(2*k-1)))); \\ Seiichi Manyama, Jan 28 2021

a(n) ~ sqrt(3) * 2^(n - 1/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 04 2022

A369765 Maximal coefficient of (1 + x) * (1 + x^2) * (1 + x^3) * (1 + x^5) * ... * (1 + x^prime(n-1)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 6, 7, 13, 19, 32, 53, 90, 156, 277, 494, 878, 1566, 2836, 5146, 9401, 17358, 32042, 59434, 110292, 204332, 380548, 713601, 1342448, 2538012, 4808578, 9043605, 17070234, 32268611, 61271738, 116123939, 220993892, 421000142, 802844420, 1534312896

Offset: 0

Ilya Gutkovskiy, Jan 31 2024

nonn

Cf. A008578, A025591, A036497, A350457, A350514, A369708.

Table[Max[CoefficientList[Product[(1 + x^If[k == 1, 1, Prime[k - 1]]), {k, 1, n}], x]], {n, 0, 40}]

from collections import Counter
from sympy import prime
def A369765(n):
    c = {0:1,1:1}
    for k in range(1,n):
        p, d = prime(k), Counter(c)
        for j in c:
            d[j+p] += c[j]
        c = d
    return max(c.values()) # Chai Wah Wu, Feb 01 2024

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

A379563 Smallest degree of x with the largest coefficient in Product_{k=1..n} (1 + x^prime(k)).

Original entry on oeis.org

0, 0, 0, 5, 5, 5, 18, 18, 35, 42, 58, 73, 89, 114, 137, 163, 190, 220, 249, 281, 318, 356, 393, 437, 480, 530, 580, 632, 685, 740, 796, 860, 925, 994, 1063, 1138, 1213, 1292, 1373, 1457, 1543, 1633, 1723, 1819, 1915, 2014, 2113, 2219, 2330, 2444, 2558, 2675, 2794, 2915, 3040, 3169

Offset: 0

Ilya Gutkovskiy, Dec 26 2024

nonn

Cf. A000586, A350393, A350457, A379564.

A379564 Largest degree of x with the largest coefficient in Product_{k=1..n} (1 + x^prime(k)).

Original entry on oeis.org

0, 2, 5, 5, 12, 23, 23, 40, 42, 58, 71, 87, 108, 124, 144, 165, 191, 220, 252, 287, 321, 356, 398, 437, 483, 530, 581, 632, 686, 740, 797, 860, 926, 994, 1064, 1138, 1214, 1292, 1374, 1457, 1544, 1633, 1724, 1819, 1916, 2014, 2114, 2219, 2331, 2444, 2559, 2675, 2795, 2915, 3041, 3169

Offset: 0

Ilya Gutkovskiy, Dec 26 2024

nonn

Cf. A000586, A350394, A350457, A379563.

A379976 Absolute value of the minimum coefficient of (1 - x^2) * (1 - x^3) * (1 - x^5) * ... * (1 - x^prime(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 4, 6, 8, 12, 18, 30, 46, 70, 113, 186, 314, 531, 894, 1561, 2705, 4817, 8502, 15030, 26502, 47200, 84698, 151809, 273961, 496807, 900596, 1643185, 2999067, 5498916, 10110030, 18596096, 34300223, 63585519, 118208807, 219235308, 405259618, 752027569, 1400505025

Offset: 0

Ilya Gutkovskiy, Jan 07 2025

nonn

Robert Israel, Table of n, a(n) for n = 0..1500

Cf. A046675, A086394, A350457, A350514, A367843.

P:= 1: R:= 1:
for i from 1 to 100 do
  P:= P * (1 - x^ithprime(i));
  R:= R, abs(min(coeffs(expand(P),x)))
od:
R; # Robert Israel, Feb 07 2025

Table[Min[CoefficientList[Product[(1 - x^Prime[k]), {k, 1, n}], x]], {n, 0, 50}] // Abs

a(n) = abs(vecmin(Vec(prod(k=1, n, 1-x^prime(k))))); \\ Michel Marcus, Jan 18 2025

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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A350514 Maximal coefficient of Product_{j=1..n} (1 - x^prime(j)).

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A367843 Maximum of the absolute value of the coefficients of (1 - x^2) * (1 - x^3) * (1 - x^5) * ... * (1 - x^prime(n)).

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A369708 Maximal coefficient of (1 + x^3) * (1 + x^5) * (1 + x^7) * ... * (1 + x^prime(n)).

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A350504 Maximal coefficient of (1 + x) * (1 + x^3) * (1 + x^5) * ... * (1 + x^(2*n-1)).

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A369765 Maximal coefficient of (1 + x) * (1 + x^2) * (1 + x^3) * (1 + x^5) * ... * (1 + x^prime(n-1)).

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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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A379563 Smallest degree of x with the largest coefficient in Product_{k=1..n} (1 + x^prime(k)).

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A379564 Largest degree of x with the largest coefficient in Product_{k=1..n} (1 + x^prime(k)).

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A379976 Absolute value of the minimum coefficient of (1 - x^2) * (1 - x^3) * (1 - x^5) * ... * (1 - x^prime(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)).