A323526 -id:A323526 - cp's OEIS frontend

A323520 Numbers of the form p^(k^2) where p is prime and k >= 0.

1, 2, 3, 5, 7, 11, 13, 16, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263

Offset: 1

Gus Wiseman, Jan 17 2019

nonn

Cf. A000290, A000961, A001248, A001597, A026478, A050376, A277562, A323519, A323526, A323528.

Select[Range[100],#==1||PrimePowerQ[#]&&IntegerQ[Sqrt[FactorInteger[#][[1,2]]]]&]

A323525 Number of ways to arrange the parts of a multiset whose multiplicities are the prime indices of n into a square matrix.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 0, 6, 4, 0, 12, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 36, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 84, 0, 0, 72, 0, 0, 0, 0, 0, 0, 0, 0, 126, 252, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jan 17 2019

nonn

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			The a(9) = 6 matrices:
  [1 1] [1 2] [1 2] [2 1] [2 1] [2 2]
  [2 2] [1 2] [2 1] [1 2] [2 1] [1 1]
The a(38) = 9 matrices:
  [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 2] [1 2 1] [2 1 1]
  [1 1 1] [1 1 1] [1 1 1] [1 1 2] [1 2 1] [2 1 1] [1 1 1] [1 1 1] [1 1 1]
  [1 1 2] [1 2 1] [2 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1]

The positions of 0's are numbers whose sum of prime indices is not a perfect square (A323527).
The positions of 1's are primes indexed by squares (A323526).
Cf. A008480, A026478, A056239, A103198, A112798, A120732, A318284, A318762, A323519, A323528, A323531.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,Reverse[primeMS[n]]];
Table[If[IntegerQ[Sqrt[Total[primeMS[n]]]],Length[Permutations[nrmptn[n]]],0],{n,100}]

If A056239(n) is a perfect square, a(n) = A318762(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

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

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A323520 Numbers of the form p^(k^2) where p is prime and k >= 0.

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A323525 Number of ways to arrange the parts of a multiset whose multiplicities are the prime indices of n into a square matrix.

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

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

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