

A353502


Numbers with all prime indices and exponents > 2.


5



1, 125, 343, 625, 1331, 2197, 2401, 3125, 4913, 6859, 12167, 14641, 15625, 16807, 24389, 28561, 29791, 42875, 50653, 68921, 78125, 79507, 83521, 103823, 117649, 130321, 148877, 161051, 166375, 205379, 214375, 226981, 274625, 279841, 300125, 300763, 357911
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OFFSET

1,2


COMMENTS

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.


LINKS



FORMULA

Sum_{n>=1} 1/a(n) = Product_{p prime > 3} (1 + 1/(p^2*(p1))) = (72/95)*A065483 = 1.0154153584... .  Amiram Eldar, May 28 2022


EXAMPLE

The initial terms together with their prime indices:
1: {}
125: {3,3,3}
343: {4,4,4}
625: {3,3,3,3}
1331: {5,5,5}
2197: {6,6,6}
2401: {4,4,4,4}
3125: {3,3,3,3,3}
4913: {7,7,7}
6859: {8,8,8}
12167: {9,9,9}
14641: {5,5,5,5}
15625: {3,3,3,3,3,3}
16807: {4,4,4,4,4}
24389: {10,10,10}
28561: {6,6,6,6}
29791: {11,11,11}
42875: {3,3,3,4,4,4}


MATHEMATICA

Select[Range[10000], #==1!MemberQ[FactorInteger[#], {_?(#<5&), _}{_, _?(#<3&)}]&]


CROSSREFS

The version for <= 2 instead of > 2 is A018256, # of compositions A137200.
The version for indices and exponents prime (instead of > 2) is:
The partitions with these Heinz numbers are counted by A353501.
A000726 counts partitions with multiplicities <= 2, compositions A128695.
A295341 counts partitions with some multiplicity > 2, compositions A335464.


KEYWORD

nonn


AUTHOR



STATUS

approved



