A352518 - OEIS

%I #21 Aug 04 2024 03:02:36

%S 225,675,1089,1125,2601,3025,3267,3375,6075,7225,7803,8649,11979,

%T 15125,15129,24025,25947,27225,28125,29403,30375,31329,33275,34969,

%U 35937,36125,40401,42025,44217,45387,54675,62001,65025,70227,81675,84375,87025,93987

%N Numbers > 1 that are not a prime power and whose prime indices and exponents are all themselves 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="/A352518/b352518.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

%e     1089: {2,2,5,5}

%e     1125: {2,2,3,3,3}

%e     2601: {2,2,7,7}

%e     3025: {3,3,5,5}

%e     3267: {2,2,2,5,5}

%e     3375: {2,2,2,3,3,3}

%e     6075: {2,2,2,2,2,3,3}

%e     7225: {3,3,7,7}

%e     7803: {2,2,2,7,7}

%e     8649: {2,2,11,11}

%e    11979: {2,2,5,5,5}

%e    15125: {3,3,3,5,5}

%e    15129: {2,2,13,13}

%e    24025: {3,3,11,11}

%e    25947: {2,2,2,11,11}

%e    27225: {2,2,3,3,5,5}

%e    28125: {2,2,3,3,3,3,3}

%e For example, 7803 = prime(1)^3 prime(4)^2.

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

%Y These partitions are counted by A352493.

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

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

%Y A000040 lists the primes.

%Y A000961 lists prime powers.

%Y A001694 lists powerful numbers, counted by A007690.

%Y A038499 counts partitions of prime length.

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

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

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

%Y A109297 = same indices as exponents, counted by A114640.

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

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

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

%Y A325131 = disjoint indices from exponents, counted by A114639.

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

%K nonn

%O 1,1

%A _Gus Wiseman_, Mar 24 2022