|
%I #13 Jan 09 2021 00:51:48
%S 21,39,57,63,65,87,91,111,115,117,129,133,147,159,171,183,185,189,203,
%T 213,235,237,247,259,261,267,273,299,301,303,305,319,321,325,333,339,
%U 351,365,371,377,387,393,399,417,427,441,445,453,477,481,489,497,507
%N Non-powers of primes whose prime indices have a common divisor > 1.
%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.
%C Also Heinz numbers of non-constant, non-relatively prime partitions. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
%F Equals A024619 /\ A318978.
%F Complement of A000961 \/ A289509.
%e The sequence of terms together with their prime indices begins:
%e 21: {2,4} 183: {2,18} 305: {3,18}
%e 39: {2,6} 185: {3,12} 319: {5,10}
%e 57: {2,8} 189: {2,2,2,4} 321: {2,28}
%e 63: {2,2,4} 203: {4,10} 325: {3,3,6}
%e 65: {3,6} 213: {2,20} 333: {2,2,12}
%e 87: {2,10} 235: {3,15} 339: {2,30}
%e 91: {4,6} 237: {2,22} 351: {2,2,2,6}
%e 111: {2,12} 247: {6,8} 365: {3,21}
%e 115: {3,9} 259: {4,12} 371: {4,16}
%e 117: {2,2,6} 261: {2,2,10} 377: {6,10}
%e 129: {2,14} 267: {2,24} 387: {2,2,14}
%e 133: {4,8} 273: {2,4,6} 393: {2,32}
%e 147: {2,4,4} 299: {6,9} 399: {2,4,8}
%e 159: {2,16} 301: {4,14} 417: {2,34}
%e 171: {2,2,8} 303: {2,26} 427: {4,18}
%t Select[Range[100],!(#==1||PrimePowerQ[#]||GCD@@PrimePi/@First/@FactorInteger[#]==1)&]
%Y A318978 allows prime powers, counted by A018783, with complement A289509.
%Y A327685 allows nonprime prime powers.
%Y A338330 is the coprime instead of relatively prime version.
%Y A338554 counts the partitions with these Heinz numbers.
%Y A338555 is the complement.
%Y A000740 counts relatively prime compositions.
%Y A000961 lists powers of primes, with complement A024619.
%Y A051424 counts pairwise coprime or singleton partitions.
%Y A108572 counts nontrivial periodic partitions, with Heinz numbers A001597.
%Y A291166 ranks relatively prime compositions, with complement A291165.
%Y A302696 gives the Heinz numbers of pairwise coprime partitions.
%Y A327516 counts pairwise coprime partitions, with Heinz numbers A302696.
%Y Cf. A000005, A000837, A007916, A056239, A112798, A289508, A302569, A302796, A318716, A327658, A328867, A328677, A338331, A338553.
%K nonn
%O 1,1
%A _Gus Wiseman_, Nov 03 2020
|
