cp's OEIS Frontend

A353502 Numbers with all prime indices and exponents > 2.

%I A353502 #27 Sep 06 2022 05:26:14
%S A353502 1,125,343,625,1331,2197,2401,3125,4913,6859,12167,14641,15625,16807,
%T A353502 24389,28561,29791,42875,50653,68921,78125,79507,83521,103823,117649,
%U A353502 130321,148877,161051,166375,205379,214375,226981,274625,279841,300125,300763,357911
%N A353502 Numbers with all prime indices and exponents > 2.
%C A353502 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 A353502 David A. Corneth, <a href="/A353502/b353502.txt">Table of n, a(n) for n = 1..10000</a> (first 3000 terms from Amiram Eldar)
%F A353502 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
%e A353502 The initial terms together with their prime indices:
%e A353502        1: {}
%e A353502      125: {3,3,3}
%e A353502      343: {4,4,4}
%e A353502      625: {3,3,3,3}
%e A353502     1331: {5,5,5}
%e A353502     2197: {6,6,6}
%e A353502     2401: {4,4,4,4}
%e A353502     3125: {3,3,3,3,3}
%e A353502     4913: {7,7,7}
%e A353502     6859: {8,8,8}
%e A353502    12167: {9,9,9}
%e A353502    14641: {5,5,5,5}
%e A353502    15625: {3,3,3,3,3,3}
%e A353502    16807: {4,4,4,4,4}
%e A353502    24389: {10,10,10}
%e A353502    28561: {6,6,6,6}
%e A353502    29791: {11,11,11}
%e A353502    42875: {3,3,3,4,4,4}
%t A353502 Select[Range[10000],#==1||!MemberQ[FactorInteger[#],{_?(#<5&),_}|{_,_?(#<3&)}]&]
%Y A353502 The version for only parts is A007310, counted by A008483.
%Y A353502 The version for <= 2 instead of > 2 is A018256, # of compositions A137200.
%Y A353502 The version for only multiplicities is A036966, counted by A100405.
%Y A353502 The version for indices and exponents prime (instead of > 2) is:
%Y A353502 - listed by A346068
%Y A353502 - counted by A351982
%Y A353502 - only exponents: A056166, counted by A055923
%Y A353502 - only parts: A076610, counted by A000607
%Y A353502 The version for > 1 instead of > 2 is A062739, counted by A339222.
%Y A353502 The version for compositions is counted by A353428, see A078012, A353400.
%Y A353502 The partitions with these Heinz numbers are counted by A353501.
%Y A353502 A000726 counts partitions with multiplicities <= 2, compositions A128695.
%Y A353502 A001222 counts prime factors with multiplicity, distinct A001221.
%Y A353502 A004250 counts partitions with some part > 2, compositions A008466.
%Y A353502 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A353502 A124010 gives prime signature, sorted A118914.
%Y A353502 A295341 counts partitions with some multiplicity > 2, compositions A335464.
%Y A353502 Cf. A000720, A003963, A065483, A114901, A181819, A353503, A353508.
%K A353502 nonn
%O A353502 1,2
%A A353502 _Gus Wiseman_, May 16 2022