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A325241
Numbers > 1 whose maximum prime exponent is one greater than their minimum.
11
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
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 15 2019
STATUS
approved