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Numbers whose prime indices (with repetition) are pairwise coprime. Nonprime Heinz numbers of integer partitions with pairwise coprime parts.
78

%I #27 Dec 07 2021 11:09:06

%S 1,2,4,6,8,10,12,14,15,16,20,22,24,26,28,30,32,33,34,35,38,40,44,46,

%T 48,51,52,55,56,58,60,62,64,66,68,69,70,74,76,77,80,82,85,86,88,92,93,

%U 94,95,96,102,104,106,110,112,116,118,119,120,122,123,124,128,132

%N Numbers whose prime indices (with repetition) are pairwise coprime. Nonprime Heinz numbers of integer partitions with pairwise coprime parts.

%C A prime index of n is a number m such that prime(m) divides n. Two or more numbers are coprime if no pair has a common divisor other than 1. A single number is not considered coprime unless it is equal to 1.

%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

%C Number 36 = prime(1)*prime(1)*prime(2)*prime(2) is not included in the sequence, because the pair of prime indices {2,2} is not coprime. - _Gus Wiseman_, Dec 06 2021

%H Robert Israel, <a href="/A302696/b302696.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%e Sequence of integer partitions with pairwise coprime parts begins: (), (1), (11), (21), (111), (31), (211), (41), (32), (1111), (311), (51), (2111), (61), (411), (321).

%e Missing from this list are: (2), (3), (4), (22), (5), (6), (7), (221), (8), (42), (9), (33), (222).

%p filter:= proc(n) local F;

%p F:= ifactors(n)[2];

%p if nops(F)=1 then if F[1][1] = 2 then return true else return false fi fi;

%p if ormap(t -> t[2]>1 and t[1] <> 2, F) then return false fi;

%p F:= map(t -> numtheory:-pi(t[1]), F);

%p ilcm(op(F))=convert(F,`*`)

%p end proc:

%p select(filter, [$1..200]); # _Robert Israel_, Sep 10 2020

%t primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[200],Or[#===1,CoprimeQ@@primeMS[#]]&]

%o (PARI) isA302696(n) = if(isprimepower(n),!(n%2), if(!issquarefree(n>>valuation(n,2)), 0, my(pis=apply(primepi,factor(n)[,1])); (lcm(pis)==factorback(pis)))); \\ _Antti Karttunen_, Dec 06 2021

%Y Cf. A000837, A000961, A001222, A005117, A007359, A051424, A275024, A289508, A289509, A298748, A302568, A302569, A302697, A302698, A327512, A327513.

%K nonn

%O 1,2

%A _Gus Wiseman_, Apr 11 2018

%E Clarification (with repetition) added to the definition by _Antti Karttunen_, Dec 06 2021