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A325241
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Numbers > 1 whose maximum prime exponent is one greater than their minimum.
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9
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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
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OFFSET
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1,1
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COMMENTS
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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).
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LINKS
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FORMULA
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EXAMPLE
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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}
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MATHEMATICA
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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 *)
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PROG
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(Python)
from sympy import factorint
def ok(n):
e = sorted(factorint(n).values())
return n > 1 and max(e) == 1 + min(e)
(PARI) is(n)={my(e=factor(n)[, 2]); n>1 && vecmin(e) + 1 == vecmax(e); } \\ Amiram Eldar, Jan 30 2023
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CROSSREFS
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Cf. A001221, A001222, A001694, A051903, A051904, A052485, A056239, A112798, A118914, A325240, A325259, A325270, A325279.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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