A323520 -id:A323520 - cp's OEIS frontend

A384914 The number of unordered factorizations of n into numbers of the form p^(k^2) where p is prime and k >= 0 (A323520).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 12 2025

First differs from A203640, A295658 and A365333 at n = 64, from A043289 and A053164 at n = 81, and from A063775 at n = 512.

			a(16) = 2 since 4 has 2 factorizations: 2^1 * 2^1 * 2^1 * 2^1 and 2^4, with exponents 1 and 4 that are squares.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000290, A001156, A197680, A323520.
Cf. A043289, A053164, A063775, A203640, A295658, A365333.
Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384912, A384913, A384915, A384916.

s[n_] := s[n] = If[n == 0, 1, Sum[Sum[d * Boole[IntegerQ[Sqrt[d]]], {d, Divisors[j]}] * s[n-j], {j, 1, n}] / n];
f[p_, e_] := s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = if(n < 1, 1, sum(j = 1, n, sumdiv(j, d, d*issquare(d)) * s(n-j))/n);
a(n) = vecprod(apply(s, factor(n)[, 2]));

Multiplicative with a(p^e) = A001156(e).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.08451356983124311685..., where f(x) = (1-x) / Product_{k>=1} (1-x^(k^2)).

A323519 a(n) is the number of ways to fill a square matrix with the multiset of prime factors of n, if the number of prime factors (counted with multiplicity) is a perfect square, and a(n) = 0 otherwise.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 4, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 6, 1, 0, 0, 4, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 4, 0, 4, 0, 0, 1, 12, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 12, 0, 0

Offset: 1

Gus Wiseman, Jan 17 2019

nonn

			The a(60) = 12 matrices:
  [2 2] [2 2] [2 3] [2 3] [2 5] [2 5] [3 2] [3 2] [3 5] [5 2] [5 2] [5 3]
  [3 5] [5 3] [2 5] [5 2] [2 3] [3 2] [2 5] [5 2] [2 2] [2 3] [3 2] [2 2]

Positions of 0's are A323521.
Positions of 1's are A323520.
Cf. A000290, A008480, A026478, A056239, A089299, A103198, A112798, A120732, A323433, A323525, A323531.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[IntegerQ[Sqrt[PrimeOmega[n]]],Length[Permutations[primeMS[n]]],0],{n,100}]

If A001222(n) is a perfect square, then a(n) = A008480(n). Otherwise, a(n) = 0.

A323528 Numbers whose sum of prime indices is a perfect square.

Original entry on oeis.org

1, 2, 7, 9, 10, 12, 16, 23, 38, 51, 53, 65, 68, 77, 78, 94, 97, 98, 99, 104, 105, 110, 125, 126, 129, 132, 135, 140, 150, 151, 162, 168, 172, 176, 178, 180, 200, 205, 216, 224, 227, 240, 246, 249, 259, 288, 298, 311, 320, 328, 332, 333, 341, 361, 370, 377, 384

Offset: 1

Gus Wiseman, Jan 17 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The sum of prime indices of n is A056239(n).

Also Heinz numbers of integer partitions of perfect squares, where the Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			10 is in the sequence because 10 = 2*5 = prime(1)*prime(3) and 1 + 3 = 4 is a square.

Robert Israel, Table of n, a(n) for n = 1..10000

Complement of A323527.

Cf. A000290, A001248, A026478, A056239, A112798, A323520, A323521, A323525, A323526.

select(k-> issqr(add(numtheory[pi](i[1])*i[2], i=ifactors(k)[2])), [$1..400])[]; # Alois P. Heinz, Jan 22 2019

Select[Range[100],IntegerQ[Sqrt[Sum[PrimePi[f[[1]]]*f[[2]],{f,FactorInteger[#]}]]]&]

isok(n) = {my(f=factor(n)); issquare(sum(k=1, #f~, primepi(f[k, 1])*f[k,2]));} \\ Michel Marcus, Jan 18 2019

A323521 Numbers whose number of prime factors counted with multiplicity (A001222) is not a perfect square.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 18, 20, 21, 22, 25, 26, 27, 28, 30, 32, 33, 34, 35, 38, 39, 42, 44, 45, 46, 48, 49, 50, 51, 52, 55, 57, 58, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 85, 86, 87, 91, 92, 93, 94, 95, 96, 98, 99, 102, 105, 106

Offset: 1

Gus Wiseman, Jan 17 2019

nonn

Cf. A000037, A001222, A000290, A026478, A063989, A323519, A323520, A323527.

Select[Range[100],!IntegerQ[Sqrt[PrimeOmega[#]]]&]

A323526 One and prime numbers whose prime index is a perfect square.

Original entry on oeis.org

1, 2, 7, 23, 53, 97, 151, 227, 311, 419, 541, 661, 827, 1009, 1193, 1427, 1619, 1879, 2143, 2437, 2741, 3083, 3461, 3803, 4211, 4637, 5051, 5519, 6007, 6481, 6997, 7573, 8161, 8737, 9341, 9931, 10627, 11321, 12049, 12743, 13499, 14327, 15101, 15877, 16747, 17609

Offset: 1

Gus Wiseman, Jan 17 2019

nonn

Cf. A000290, A000720, A001248, A011757, A026478, A056239, A323520, A323521, A323525, A323527, A323528.

Mathematica
```
Array[Prime[#^2]&,20]
```

vector(50, n, if (n==1, 1, prime((n-1)^2))) \\ Michel Marcus, Feb 15 2019

a(n) = A011757(n-1) for n > 1. - Alois P. Heinz, Jan 17 2019

A323527 Numbers whose sum of prime indices is not a perfect square.

Original entry on oeis.org

3, 4, 5, 6, 8, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 79, 80, 81

Offset: 1

Gus Wiseman, Jan 17 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The sum of prime indices of n is A056239(n).

Complement of A323528.

Cf. A000037, A000720, A001248, A026478 (Omega is square), A056239, A112798, A323520, A323521 (Omega is not square), A323525, A323526.

remove(k-> issqr(add(numtheory[pi](i[1])*i[2], i=ifactors(k)[2])), [$1..99])[];  # Alois P. Heinz, Jan 22 2019

Select[Range[100],!IntegerQ[Sqrt[Sum[PrimePi[f[[1]]]*f[[2]],{f,FactorInteger[#]}]]]&]

isok(n) = {my(f=factor(n)); !issquare(sum(k=1, #f~, primepi(f[k, 1])*f[k,2]));} \\ Michel Marcus, Jan 18 2019

cp's OEIS Frontend

A384914 The number of unordered factorizations of n into numbers of the form p^(k^2) where p is prime and k >= 0 (A323520).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A323519 a(n) is the number of ways to fill a square matrix with the multiset of prime factors of n, if the number of prime factors (counted with multiplicity) is a perfect square, and a(n) = 0 otherwise.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A323528 Numbers whose sum of prime indices is a perfect square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A323521 Numbers whose number of prime factors counted with multiplicity (A001222) is not a perfect square.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A323526 One and prime numbers whose prime index is a perfect square.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A323527 Numbers whose sum of prime indices is not a perfect square.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

PARI