A341123 -id:A341123 - cp's OEIS frontend

A341122 Number of partitions of n into 4 prime powers (including 1).

1, 1, 2, 3, 5, 5, 8, 9, 12, 13, 17, 17, 22, 22, 26, 27, 33, 31, 39, 38, 44, 43, 51, 47, 58, 54, 63, 60, 71, 64, 79, 74, 88, 82, 99, 88, 108, 97, 116, 105, 126, 110, 134, 119, 141, 126, 153, 133, 164, 143, 172, 149, 184, 155, 194, 168, 204, 173, 215, 180, 227, 192, 238

Offset: 4

Ilya Gutkovskiy, Feb 05 2021

nonn

Cf. A000961, A010055, A071330, A341112, A341123, A341124, A341125, A341126, A341127.

q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(q(i), b(n-i, min(n-i, i), t-1), 0)))
    end:
a:= n-> b(n$2, 4):
seq(a(n), n=4..66);  # Alois P. Heinz, Feb 05 2021

q[n_] := q[n] = Length[FactorInteger[n]] < 2;
b[n_, i_, t_] := b[n, i, t] = If[n == 0,
     If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
     If[q[i], b[n - i, Min[n - i, i], t - 1], 0]]];
a[n_] := b[n, n, 4];
Table[a[n], {n, 4, 66}] (* Jean-François Alcover, Feb 22 2022, after Alois P. Heinz *)

A341124 Number of partitions of n into 6 prime powers (including 1).

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 12, 17, 20, 27, 31, 41, 45, 56, 63, 77, 83, 101, 108, 128, 136, 160, 168, 196, 204, 236, 245, 281, 288, 331, 340, 387, 395, 450, 457, 519, 525, 594, 598, 677, 678, 763, 764, 855, 851, 957, 949, 1062, 1053, 1177, 1161, 1300, 1276, 1425, 1403, 1564

Offset: 6

Ilya Gutkovskiy, Feb 05 2021

nonn

Cf. A000961, A010055, A071330, A341112, A341122, A341123, A341125, A341126, A341127.

q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(q(i), b(n-i, min(n-i, i), t-1), 0)))
    end:
a:= n-> b(n$2, 6):
seq(a(n), n=6..62);  # Alois P. Heinz, Feb 05 2021

q[n_] := q[n] = Length[FactorInteger[n]] < 2;
b[n_, i_, t_] := b[n, i, t] = If[n == 0,
     If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
     If[q[i], b[n - i, Min[n - i, i], t - 1], 0]]];
a[n_] := b[n, n, 6];
Table[a[n], {n, 6, 62}] (* Jean-François Alcover, Feb 22 2022, after Alois P. Heinz *)

A341112 Number of partitions of n into 3 prime powers (including 1).

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 6, 6, 8, 8, 10, 9, 12, 10, 13, 12, 15, 13, 17, 15, 18, 15, 19, 16, 21, 17, 23, 18, 24, 19, 27, 23, 30, 24, 32, 25, 32, 26, 34, 26, 36, 26, 36, 28, 38, 28, 40, 30, 42, 32, 43, 30, 45, 32, 47, 35, 49, 30, 50, 35, 51, 36, 53, 35, 55, 37, 54, 40, 57, 36, 61, 40, 61

Offset: 3

Ilya Gutkovskiy, Feb 05 2021

nonn

Cf. A000961, A010055, A071330, A282064, A307727, A307825, A341122, A341123, A341124, A341125, A341126, A341127.

q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(q(i), b(n-i, min(n-i, i), t-1), 0)))
    end:
a:= n-> b(n$2, 3):
seq(a(n), n=3..75);  # Alois P. Heinz, Feb 05 2021

q[n_] := q[n] = Length[FactorInteger[n]] < 2;
b[n_, i_, t_] := b[n, i, t] = If[n == 0,
     If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
     If[q[i], b[n - i, Min[n - i, i], t - 1], 0]]];
a[n_] := b[n, n, 3];
Table[a[n], {n, 3, 75}] (* Jean-François Alcover, Feb 22 2022, after Alois P. Heinz *)

A341125 Number of partitions of n into 7 prime powers (including 1).

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 13, 18, 22, 30, 35, 47, 54, 68, 78, 97, 107, 132, 146, 173, 190, 225, 242, 285, 305, 352, 377, 434, 456, 525, 553, 627, 659, 748, 778, 881, 916, 1028, 1068, 1197, 1232, 1381, 1421, 1578, 1619, 1801, 1837, 2041, 2079, 2296, 2337, 2583, 2613

Offset: 7

Ilya Gutkovskiy, Feb 05 2021

nonn

Cf. A000961, A010055, A071330, A341112, A341122, A341123, A341124, A341126, A341127.

q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(q(i), b(n-i, min(n-i, i), t-1), 0)))
    end:
a:= n-> b(n$2, 7):
seq(a(n), n=7..60);  # Alois P. Heinz, Feb 05 2021

q[n_] := q[n] = Length[FactorInteger[n]] < 2;
b[n_, i_, t_] := b[n, i, t] = If[n == 0,
     If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
     If[q[i], b[n - i, Min[n - i, i], t - 1], 0]]];
a[n_] := b[n, n, 7];
Table[a[n], {n, 7, 60}] (* Jean-François Alcover, Feb 22 2022, after Alois P. Heinz *)

A341126 Number of partitions of n into 8 prime powers (including 1).

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 13, 19, 23, 32, 38, 51, 60, 77, 90, 113, 128, 158, 179, 215, 240, 287, 316, 373, 409, 475, 517, 599, 645, 741, 799, 908, 971, 1104, 1173, 1326, 1408, 1580, 1670, 1874, 1967, 2198, 2310, 2563, 2680, 2976, 3097, 3426, 3566, 3926, 4070, 4485

Offset: 8

Ilya Gutkovskiy, Feb 05 2021

nonn

Cf. A000961, A010055, A071330, A341112, A341122, A341123, A341124, A341125, A341127.

q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(q(i), b(n-i, min(n-i, i), t-1), 0)))
    end:
a:= n-> b(n$2, 8):
seq(a(n), n=8..60);  # Alois P. Heinz, Feb 05 2021

q[n_] := q[n] = Length[FactorInteger[n]] < 2;
b[n_, i_, t_] := b[n, i, t] = If[n == 0,
     If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
     If[q[i], b[n - i, Min[n - i, i], t - 1], 0]]];
a[n_] := b[n, n, 8];
Table[a[n], {n, 8, 60}] (* Jean-François Alcover, Feb 22 2022, after _Alois P. Heinz *)

A341127 Number of partitions of n into 9 prime powers (including 1).

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 13, 19, 24, 33, 40, 54, 64, 83, 99, 125, 144, 180, 206, 250, 284, 341, 383, 455, 506, 593, 656, 762, 835, 965, 1054, 1206, 1309, 1491, 1610, 1825, 1964, 2213, 2374, 2664, 2843, 3179, 3387, 3769, 3998, 4440, 4695, 5194, 5480, 6043, 6357

Offset: 9

Ilya Gutkovskiy, Feb 05 2021

nonn

Cf. A000961, A010055, A071330, A341112, A341122, A341123, A341124, A341125, A341126.

q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(q(i), b(n-i, min(n-i, i), t-1), 0)))
    end:
a:= n-> b(n$2, 9):
seq(a(n), n=9..60);  # Alois P. Heinz, Feb 05 2021

q[n_] := q[n] = Length[FactorInteger[n]] < 2;
b[n_, i_, t_] := b[n, i, t] = If[n == 0,
     If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
     If[q[i], b[n - i, Min[n - i, i], t - 1], 0]]];
a[n_] := b[n, n, 9];
Table[a[n], {n, 9, 60}] (* Jean-François Alcover, Feb 22 2022, after Alois P. Heinz *)

A341142 Number of partitions of n into 5 distinct prime powers (including 1).

Original entry on oeis.org

1, 0, 1, 2, 3, 3, 5, 5, 7, 8, 10, 12, 15, 15, 18, 21, 23, 26, 31, 33, 36, 41, 43, 48, 52, 58, 62, 72, 72, 82, 85, 95, 97, 112, 112, 125, 127, 142, 142, 161, 159, 181, 180, 200, 196, 222, 217, 243, 239, 269, 261, 291, 284, 316, 308, 341, 332, 370, 358, 394, 381, 427, 414, 456

Offset: 15

Ilya Gutkovskiy, Feb 05 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 15..5000

Cf. A000961, A010055, A341123, A341132, A341134, A341140, A341141, A341143, A341144, A341145.

q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(q(i), b(n-i, min(n-i, i-1), t-1), 0)))
    end:
a:= n-> b(n$2, 5):
seq(a(n), n=15..78);  # Alois P. Heinz, Feb 05 2021

q[n_] := q[n] = PrimeNu[n] < 2;
b[n_, i_, t_] := b[n, i, t] = If[n == 0,
     If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
     If[q[i], b[n - i, Min[n - i, i - 1], t - 1], 0]]];
a[n_] := b[n, n, 5];
Table[a[n], {n, 15, 78}] (* Jean-François Alcover, Jul 13 2021, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n,{5}],?(Max[PrimeNu[#]]<2&&Length[#]==Length[Union[#]]&)],{n,15,80}] (* _Harvey P. Dale, Dec 22 2024 *)

A341134 Number of ways to write n as an ordered sum of 5 prime powers (including 1).

Original entry on oeis.org

1, 5, 15, 35, 70, 121, 190, 280, 395, 535, 711, 920, 1160, 1425, 1725, 2041, 2395, 2775, 3200, 3645, 4146, 4640, 5190, 5730, 6325, 6915, 7625, 8270, 9030, 9745, 10576, 11320, 12320, 13185, 14305, 15281, 16510, 17480, 18855, 19835, 21306, 22435, 24010, 25025, 26810, 27790, 29590

Offset: 5

Ilya Gutkovskiy, Feb 05 2021

nonn

Cf. A000961, A010055, A282062, A282064, A341123, A341133, A341135, A341136, A341137, A341138, A341139.

q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(q(j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 5):
seq(a(n), n=5..51);  # Alois P. Heinz, Feb 05 2021

nmax = 51; CoefficientList[Series[Sum[Boole[PrimePowerQ[k] || k == 1] x^k, {k, 1, nmax}]^5, {x, 0, nmax}], x] // Drop[#, 5] &

A365877 a(n) is the number of quadratic equations ux^2 + vx + w = 0 with distinct solution sets L != {} and integer coefficients u, v, w, where n >= abs(u) + abs(v) + abs(w) and the sum of the solutions equals the product of the solutions.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 11, 15, 17, 23, 25, 32, 35, 40, 44, 53, 56, 66, 71, 78, 83, 95, 100, 111, 117, 127, 134, 150, 154, 171, 180, 191, 199, 213, 220, 240, 250, 263, 272, 294, 301, 324, 335, 348, 360, 386, 395, 419, 430, 448, 461, 490, 500, 522, 536, 556, 571, 603

Offset: 1

Felix Huber, Sep 22 2023

nonn

According to Vieta's formulas, x_1 + x_2 = -v/u and x_1*x_2 = w/u. So x_1 + x_2 = x_1*x_2 when v = -w. Furthermore, the discriminant must not be negative, i.e., v^2 - 4*u*w = v^2 + 4*u*v >= 0.

			For n = 11 the a(11) = 23 equations are given by (u, v, w) = (1, 0, 0), (1, 1, -1), (2, 1, -1), (1, 2, -2), (3, 1, -1), (4, 1, -1), (5, 1, -1), (3, 2, -2), (1, 3, -3), (6, 1, -1), (2, 3, -3), (7, 1, -1), (5, 2, -2), (1, 4, -4), (-1, 4, -4), (8, 1, -1), (4, 3, -3), (9, 1, -1), (7, 2, -2), (5, 3, -3), (3, 4, -4), (1, 5, -5), (-1, 5, -5).
Equations multiplied by -1 do not have a different solution set; for example, (- 1, -1, 1) has the same solution set as (1, 1, -1).
Equations with GCD(u, v, w) != 1 are excluded, because their solution set are assigned to equations with lower n. For example, (2, 0, 0) is not included here, because its solution set is already assigned to (1, 0, 0).
Equations with a double solution are considered to have two equal solutions. For example, (-1, 4, -4) has the two solutions x_1 = x_2 = 2.

Wikipedia, Quadratic equation.
Wikipedia, Vieta's formulas.

Cf. A364384, A364385, A341123, A365892.

Partial sums of A365876.

A365876:= proc(n) local u, v, a, min; u := n; v := 0; a := 0; min := true; while min = true do if u <> 0 and gcd(u, v) = 1 then a := a + 1; end if; u := u - 2; v:=(n-abs(u))/2; if u < -1/9*n then min := false; end if; end do; return a; end proc;
A365877:= proc(n) local s; option remember; if n = 1 then A365876(1); else procname(n - 1) + A365876(n); end if; end proc; seq(A365877(n), n = 1 .. 59);

from math import gcd
def A365877(n):
    if n == 1: return 1
    c = 1
    for m in range(2,n+1):
        for v in range(1,m+1>>1):
            u = m-(v<<1)
            if gcd(u,v)==1:
                v2, u2 = v*v, v*(u<<2)
                if v2+u2 >= 0:
                    c +=1
                if v2-u2 >= 0:
                    c +=1
    return c # Chai Wah Wu, Oct 05 2023

a(n) = Sum_{k=1..n} A365876(k).

a(n) = A341123(n) for 1 <= n <= 13.

A347646 Number of partitions of n into at most 5 prime powers (including 1).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 9, 12, 16, 20, 24, 30, 35, 42, 47, 55, 62, 71, 77, 88, 96, 107, 114, 127, 135, 149, 156, 173, 180, 198, 205, 224, 233, 254, 262, 287, 297, 321, 330, 358, 366, 396, 403, 436, 441, 478, 478, 517, 521, 560, 562, 609, 607, 655, 655, 702, 699, 756, 746

Offset: 0

Ilya Gutkovskiy, Sep 09 2021

nonn

Cf. A000961, A023893, A341123, A347643, A347644, A347645, A347647.

cp's OEIS Frontend

A341122 Number of partitions of n into 4 prime powers (including 1).

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A341124 Number of partitions of n into 6 prime powers (including 1).

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A341112 Number of partitions of n into 3 prime powers (including 1).

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A341125 Number of partitions of n into 7 prime powers (including 1).

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A341126 Number of partitions of n into 8 prime powers (including 1).

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A341127 Number of partitions of n into 9 prime powers (including 1).

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A341142 Number of partitions of n into 5 distinct prime powers (including 1).

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A341134 Number of ways to write n as an ordered sum of 5 prime powers (including 1).

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Mathematica

A365877 a(n) is the number of quadratic equations u*x^2 + v*x + w = 0 with distinct solution sets L != {} and integer coefficients u, v, w, where n >= abs(u) + abs(v) + abs(w) and the sum of the solutions equals the product of the solutions.

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A347646 Number of partitions of n into at most 5 prime powers (including 1).

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A365877 a(n) is the number of quadratic equations ux^2 + vx + w = 0 with distinct solution sets L != {} and integer coefficients u, v, w, where n >= abs(u) + abs(v) + abs(w) and the sum of the solutions equals the product of the solutions.