A353502 Numbers with all prime indices and exponents > 2.

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

Offset: 1

Author

Gus Wiseman, May 16 2022

Keywords

• nonn

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.

Examples

			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}
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000 (first 3000 terms from Amiram Eldar)

Crossrefs

The version for only parts is A007310, counted by A008483.

The version for <= 2 instead of > 2 is A018256, # of compositions A137200.

The version for only multiplicities is A036966, counted by A100405.

The version for indices and exponents prime (instead of > 2) is:

- listed by A346068

- counted by A351982

- only exponents: A056166, counted by A055923

- only parts: A076610, counted by A000607

The version for > 1 instead of > 2 is A062739, counted by A339222.

The version for compositions is counted by A353428, see A078012, A353400.

The partitions with these Heinz numbers are counted by A353501.

A000726 counts partitions with multiplicities <= 2, compositions A128695.

A001222 counts prime factors with multiplicity, distinct A001221.

A004250 counts partitions with some part > 2, compositions A008466.

A056239 adds up prime indices, row sums of A112798 and A296150.

A124010 gives prime signature, sorted A118914.

A295341 counts partitions with some multiplicity > 2, compositions A335464.

Cf. A000720, A003963, A065483, A114901, A181819, A353503, A353508.

Programs

Mathematica

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

Formula

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