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

Author

Gus Wiseman, Apr 15 2019

Keywords

• nonn

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

Examples

			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}
		

Links

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

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.

Programs

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 *)

PARI

is(n)={my(e=factor(n)[,2]); n>1 && vecmin(e) + 1 == vecmax(e); } \\ Amiram Eldar, Jan 30 2023

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

Formula

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