A352492 - OEIS

%I #13 May 04 2022 03:34:23

%S 1,9,25,27,81,121,125,225,243,289,625,675,729,961,1089,1125,1331,1681,

%T 2025,2187,2601,3025,3125,3267,3375,3481,4489,4913,5625,6075,6561,

%U 6889,7225,7803,8649,9801,10125,11881,11979,14641,15125,15129,15625,16129,16875

%N Powerful numbers whose prime indices are all prime numbers.

%C 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.

%H Amiram Eldar, <a href="/A352492/b352492.txt">Table of n, a(n) for n = 1..10000</a>

%F Intersection of A001694 and A076610.

%F Sum_{n>=1} 1/a(n) = Product_{p in A006450} (1 + 1/(p*(p-1))) = 1.24410463... - _Amiram Eldar_, May 04 2022

%e The terms together with their prime indices (not prime factors) begin:

%e     1: {}

%e     9: {2,2}

%e    25: {3,3}

%e    27: {2,2,2}

%e    81: {2,2,2,2}

%e   121: {5,5}

%e   125: {3,3,3}

%e   225: {2,2,3,3}

%e   243: {2,2,2,2,2}

%e   289: {7,7}

%e   625: {3,3,3,3}

%e   675: {2,2,2,3,3}

%e   729: {2,2,2,2,2,2}

%e   961: {11,11}

%e For example, 675 = prime(2)^3 prime(3)^2 = 3^3 * 5^2.

%t Select[Range[1000],#==1||And@@PrimeQ/@PrimePi/@First/@FactorInteger[#]&&Min@@Last/@FactorInteger[#]>1&]

%Y Powerful numbers are A001694, counted by A007690.

%Y The version for prime exponents instead of indices is A056166, counted by A055923.

%Y This is the powerful case of A076610 (products of A006450), counted by A000607.

%Y The partitions with these Heinz numbers are counted by A339218.

%Y A000040 lists primes.

%Y A031368 lists primes of odd index, products A066208.

%Y A101436 counts exponents in prime factorization that are themselves prime.

%Y A112798 lists prime indices, reverse A296150, sum A056239.

%Y A124010 gives prime signature, sorted A118914, length A001221, sum A001222.

%Y A053810 lists all numbers p^q with p and q prime, counted by A230595.

%Y A257994 counts prime indices that are themselves prime, complement A330944.

%Y Cf. A000720, A000961, A001597, A007821, A007916, A052485, A109297, A140319, A164336, A181819, A325131, A330945, A346068.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 24 2022