A302596 -id:A302596 - cp's OEIS frontend

A346068 Numbers that are the product of distinct primes with prime subscripts raised to prime powers.

1, 9, 25, 27, 121, 125, 225, 243, 289, 675, 961, 1089, 1125, 1331, 1681, 2187, 2601, 3025, 3125, 3267, 3375, 3481, 4489, 4913, 6075, 6889, 7225, 7803, 8649, 11881, 11979, 15125, 15129, 16129, 24025, 24649, 25947, 27225, 28125, 29403, 29791, 30375, 31329, 32041, 33275, 34969

Offset: 1

Ilya Gutkovskiy, Jul 30 2021

nonn

			675 = 3^3 * 5^2 = prime(prime(1))^prime(2) * prime(prime(2))^prime(1), therefore 675 is a term.

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

Intersection of A056166 and A076610.

Cf. A006450, A302590, A302596, A321874.

Join[{1}, Select[Range[35000], AllTrue[Join[PrimePi[(t = Transpose @ FactorInteger[#])[[1]]], t[[2]]], PrimeQ] &]] (* Amiram Eldar, Jul 30 2021 *)

from sympy import factorint, isprime, primepi
def ok(n):
    f = factorint(n)
    if not all(isprime(e) for e in f.values()): return False
    return all(isprime(primepi(p)) for p in f)
print(list(filter(ok, range(35000)))) # Michael S. Branicky, Jul 30 2021

Sum_{n>=1} 1/a(n) = Product_{p in A006450} (1 + Sum_{q prime} 1/p^q) = 1.2271874... - Amiram Eldar, Jul 31 2021

A371127 Powers of 2 times powers > 1 of a prime-indexed prime number.

Original entry on oeis.org

3, 5, 6, 9, 10, 11, 12, 17, 18, 20, 22, 24, 25, 27, 31, 34, 36, 40, 41, 44, 48, 50, 54, 59, 62, 67, 68, 72, 80, 81, 82, 83, 88, 96, 100, 108, 109, 118, 121, 124, 125, 127, 134, 136, 144, 157, 160, 162, 164, 166, 176, 179, 191, 192, 200, 211, 216, 218, 236, 241

Offset: 1

Gus Wiseman, Mar 18 2024

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 terms together with their prime indices begin:
      3: {2}
      5: {3}
      6: {1,2}
      9: {2,2}
     10: {1,3}
     11: {5}
     12: {1,1,2}
     17: {7}
     18: {1,2,2}
     20: {1,1,3}
     22: {1,5}
     24: {1,1,1,2}
     25: {3,3}
     27: {2,2,2}
     31: {11}
     34: {1,7}
     36: {1,1,2,2}

Subset of A302540.
Subset of A336101 = powers of 2 times powers of primes.
Positions of 2's in A370820.
Counting prime factors instead of divisors gives A371287.
A000005 counts divisors.
A000961 lists powers of primes, A302596 of prime index.
A001221 counts distinct prime factors.
A003963 gives product of prime indices.
A027746 lists prime factors, indices A112798, length A001222.
A076610 lists products of primes of prime index.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 counts choices of a prime factor of each prime index.
Cf. A000079, A005179, A007416, A303975, A319899, A355739, A370348, A370802, A371165-A371170, A371178.

Select[Range[100],Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]==2&]

A371287 Numbers whose product of prime indices has exactly two distinct prime factors.

Original entry on oeis.org

13, 15, 26, 29, 30, 33, 35, 37, 39, 43, 45, 47, 51, 52, 55, 58, 60, 61, 65, 66, 69, 70, 71, 73, 74, 75, 77, 78, 79, 85, 86, 87, 89, 90, 91, 93, 94, 95, 99, 101, 102, 104, 105, 107, 110, 111, 116, 117, 119, 120, 122, 123, 129, 130, 132, 135, 137, 138, 139, 140

Offset: 1

Gus Wiseman, Mar 21 2024

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 terms together with their prime indices begin:
  13: {6}
  15: {2,3}
  26: {1,6}
  29: {10}
  30: {1,2,3}
  33: {2,5}
  35: {3,4}
  37: {12}
  39: {2,6}
  43: {14}
  45: {2,2,3}
  47: {15}
  51: {2,7}
  52: {1,1,6}
  55: {3,5}
  58: {1,10}
  60: {1,1,2,3}

Positions of 2's in A303975 (positions of 1's are A320698).
Counting divisors (not factors) gives A371127, positions of 2's in A370820.
A000005 counts divisors.
A000961 lists powers of primes, of prime index A302596.
A001221 counts distinct prime factors.
A001358 lists semiprimes, squarefree A006881.
A003963 gives product of prime indices.
A027746 lists prime factors, indices A112798, length A001222.
A076610 lists products of primes of prime index.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 counts choices of a prime factor of each prime index.
Cf. A000079, A007416, A056239, A302540, A319899, A336101, A355739, A370802.

Select[Range[100],2==PrimeNu[Times @@ PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]

A001221(A003963(a(n))) = A303975(a(n)) = 2.

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A346068 Numbers that are the product of distinct primes with prime subscripts raised to prime powers.

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A371127 Powers of 2 times powers > 1 of a prime-indexed prime number.

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A371287 Numbers whose product of prime indices has exactly two distinct prime factors.

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