A352518 Numbers > 1 that are not a prime power and whose prime indices and exponents are all themselves prime numbers.

225, 675, 1089, 1125, 2601, 3025, 3267, 3375, 6075, 7225, 7803, 8649, 11979, 15125, 15129, 24025, 25947, 27225, 28125, 29403, 30375, 31329, 33275, 34969, 35937, 36125, 40401, 42025, 44217, 45387, 54675, 62001, 65025, 70227, 81675, 84375, 87025, 93987

Offset: 1

Author

Gus Wiseman, Mar 24 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 terms together with their prime indices (not factors) begin:
     225: {2,2,3,3}
     675: {2,2,2,3,3}
    1089: {2,2,5,5}
    1125: {2,2,3,3,3}
    2601: {2,2,7,7}
    3025: {3,3,5,5}
    3267: {2,2,2,5,5}
    3375: {2,2,2,3,3,3}
    6075: {2,2,2,2,2,3,3}
    7225: {3,3,7,7}
    7803: {2,2,2,7,7}
    8649: {2,2,11,11}
   11979: {2,2,5,5,5}
   15125: {3,3,3,5,5}
   15129: {2,2,13,13}
   24025: {3,3,11,11}
   25947: {2,2,2,11,11}
   27225: {2,2,3,3,5,5}
   28125: {2,2,3,3,3,3,3}
For example, 7803 = prime(1)^3 prime(4)^2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

These partitions are counted by A352493.

This is the restriction of A346068 to numbers that are not a prime power.

The prime-power version is A352519, counted by A230595.

A000040 lists the primes.

A000961 lists prime powers.

A001694 lists powerful numbers, counted by A007690.

A038499 counts partitions of prime length.

A053810 lists all numbers p^q for p and q prime, counted by A001221.

A056166 = prime exponents are all prime, counted by A055923.

A076610 = prime indices are all prime, counted by A000607, powerful A339218.

A109297 = same indices as exponents, counted by A114640.

A112798 lists prime indices, reverse A296150, sum A056239.

A124010 gives prime signature, sorted A118914, sum A001222.

A257994 counts prime indices that are themselves prime, nonprime A330944.

A325131 = disjoint indices from exponents, counted by A114639.

Cf. A000720, A001597, A002035, A007821, A007916, A101436, A117958, A164336, A268335, A330945.

Programs

Mathematica

Select[Range[10000],!PrimePowerQ[#]&& And@@PrimeQ/@PrimePi/@First/@FactorInteger[#]&& And@@PrimeQ/@Last/@FactorInteger[#]&]

Formula

Sum_{n>=1} 1/a(n) = (Product_{p prime-indexed prime} (1 + Sum_{q prime} 1/p^q)) - (Sum_{p prime-indexed prime} Sum_{q prime} 1/p^q) - 1 = 0.0106862606... . - Amiram Eldar, Aug 04 2024