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
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 15 2019
STATUS
approved