cp's OEIS Frontend

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

%I A352518 #21 Aug 04 2024 03:02:36
%S A352518 225,675,1089,1125,2601,3025,3267,3375,6075,7225,7803,8649,11979,
%T A352518 15125,15129,24025,25947,27225,28125,29403,30375,31329,33275,34969,
%U A352518 35937,36125,40401,42025,44217,45387,54675,62001,65025,70227,81675,84375,87025,93987
%N A352518 Numbers > 1 that are not a prime power and whose prime indices and exponents are all themselves prime numbers.
%C A352518 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 A352518 Amiram Eldar, <a href="/A352518/b352518.txt">Table of n, a(n) for n = 1..10000</a>
%F A352518 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
%e A352518 The terms together with their prime indices (not factors) begin:
%e A352518      225: {2,2,3,3}
%e A352518      675: {2,2,2,3,3}
%e A352518     1089: {2,2,5,5}
%e A352518     1125: {2,2,3,3,3}
%e A352518     2601: {2,2,7,7}
%e A352518     3025: {3,3,5,5}
%e A352518     3267: {2,2,2,5,5}
%e A352518     3375: {2,2,2,3,3,3}
%e A352518     6075: {2,2,2,2,2,3,3}
%e A352518     7225: {3,3,7,7}
%e A352518     7803: {2,2,2,7,7}
%e A352518     8649: {2,2,11,11}
%e A352518    11979: {2,2,5,5,5}
%e A352518    15125: {3,3,3,5,5}
%e A352518    15129: {2,2,13,13}
%e A352518    24025: {3,3,11,11}
%e A352518    25947: {2,2,2,11,11}
%e A352518    27225: {2,2,3,3,5,5}
%e A352518    28125: {2,2,3,3,3,3,3}
%e A352518 For example, 7803 = prime(1)^3 prime(4)^2.
%t A352518 Select[Range[10000],!PrimePowerQ[#]&& And@@PrimeQ/@PrimePi/@First/@FactorInteger[#]&& And@@PrimeQ/@Last/@FactorInteger[#]&]
%Y A352518 These partitions are counted by A352493.
%Y A352518 This is the restriction of A346068 to numbers that are not a prime power.
%Y A352518 The prime-power version is A352519, counted by A230595.
%Y A352518 A000040 lists the primes.
%Y A352518 A000961 lists prime powers.
%Y A352518 A001694 lists powerful numbers, counted by A007690.
%Y A352518 A038499 counts partitions of prime length.
%Y A352518 A053810 lists all numbers p^q for p and q prime, counted by A001221.
%Y A352518 A056166 = prime exponents are all prime, counted by A055923.
%Y A352518 A076610 = prime indices are all prime, counted by A000607, powerful A339218.
%Y A352518 A109297 = same indices as exponents, counted by A114640.
%Y A352518 A112798 lists prime indices, reverse A296150, sum A056239.
%Y A352518 A124010 gives prime signature, sorted A118914, sum A001222.
%Y A352518 A257994 counts prime indices that are themselves prime, nonprime A330944.
%Y A352518 A325131 = disjoint indices from exponents, counted by A114639.
%Y A352518 Cf. A000720, A001597, A002035, A007821, A007916, A101436, A117958, A164336, A268335, A330945.
%K A352518 nonn
%O A352518 1,1
%A A352518 _Gus Wiseman_, Mar 24 2022