A325241 Numbers > 1 whose maximum prime exponent is one greater than their minimum.

12, 18, 20, 28, 44, 45, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 98, 99, 108, 116, 117, 124, 126, 132, 140, 147, 148, 150, 153, 156, 164, 171, 172, 175, 180, 188, 198, 200, 204, 207, 212, 220, 228, 234, 236, 242, 244, 245, 252, 260, 261, 268, 275, 276, 279

Offset

1,1

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose maximum multiplicity is one greater than their minimum (counted by A325279).

The asymptotic density of this sequence is 1/zeta(3) - 1/zeta(2) = A088453 - A059956 = 0.22398... . - Amiram Eldar, Jan 30 2023

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Formula

A051903(a(n)) - A051904(a(n)) = 1.

Example

The sequence of terms together with their prime indices begins:
  12: {1,1,2}
  18: {1,2,2}
  20: {1,1,3}
  28: {1,1,4}
  44: {1,1,5}
  45: {2,2,3}
  50: {1,3,3}
  52: {1,1,6}
  60: {1,1,2,3}
  63: {2,2,4}
  68: {1,1,7}
  72: {1,1,1,2,2}
  75: {2,3,3}
  76: {1,1,8}
  84: {1,1,2,4}
  90: {1,2,2,3}
  92: {1,1,9}
  98: {1,4,4}
  99: {2,2,5}

Mathematica

Select[Range[100], Max@@FactorInteger[#][[All, 2]]-Min@@FactorInteger[#][[All, 2]]==1&]
Select[Range[300],  Min[e = FactorInteger[#][[;; , 2]]] +1 == Max[e] &] (* Amiram Eldar, Jan 30 2023 *)

Prog

(Python)
from sympy import factorint
def ok(n):
    e = sorted(factorint(n).values())
    return n > 1 and max(e) == 1 + min(e)
print([k for k in range(280) if ok(k)]) # Michael S. Branicky, Dec 18 2021
(PARI) is(n)={my(e=factor(n)[, 2]); n>1 && vecmin(e) + 1 == vecmax(e); } \\ Amiram Eldar, Jan 30 2023

Crossrefs

Positions of 1's in A062977. Supersequence of A054753, A096156.

Cf. A001221, A001222, A001694, A051903, A051904, A052485, A056239, A112798, A118914, A325240, A325259, A325270, A325279.

Cf. A059956, A088453.

Sequence in context: A267117 A187039 A360554 * A072357 A340780 A054753

Adjacent sequences: A325238 A325239 A325240 * A325242 A325243 A325244

Keyword

nonn

Author

Gus Wiseman, Apr 15 2019

Status

approved