cp's OEIS Frontend

A302696 Numbers whose prime indices (with repetition) are pairwise coprime. Nonprime Heinz numbers of integer partitions with pairwise coprime parts.

%I A302696 #27 Dec 07 2021 11:09:06
%S A302696 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 A302696 48,51,52,55,56,58,60,62,64,66,68,69,70,74,76,77,80,82,85,86,88,92,93,
%U A302696 94,95,96,102,104,106,110,112,116,118,119,120,122,123,124,128,132
%N A302696 Numbers whose prime indices (with repetition) are pairwise coprime. Nonprime Heinz numbers of integer partitions with pairwise coprime parts.
%C A302696 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 A302696 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A302696 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 A302696 Robert Israel, <a href="/A302696/b302696.txt">Table of n, a(n) for n = 1..10000</a>
%H A302696 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%e A302696 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 A302696 Missing from this list are: (2), (3), (4), (22), (5), (6), (7), (221), (8), (42), (9), (33), (222).
%p A302696 filter:= proc(n) local F;
%p A302696    F:= ifactors(n)[2];
%p A302696    if nops(F)=1 then if F[1][1] = 2 then return true else return false fi fi;
%p A302696    if ormap(t -> t[2]>1 and t[1] <> 2, F) then return false fi;
%p A302696    F:= map(t -> numtheory:-pi(t[1]), F);
%p A302696    ilcm(op(F))=convert(F,`*`)
%p A302696 end proc:
%p A302696 select(filter, [$1..200]); # _Robert Israel_, Sep 10 2020
%t A302696 primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A302696 Select[Range[200],Or[#===1,CoprimeQ@@primeMS[#]]&]
%o A302696 (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 A302696 Cf. A000837, A000961, A001222, A005117, A007359, A051424, A275024, A289508, A289509, A298748, A302568, A302569, A302697, A302698, A327512, A327513.
%K A302696 nonn
%O A302696 1,2
%A A302696 _Gus Wiseman_, Apr 11 2018
%E A302696 Clarification (with repetition) added to the definition by _Antti Karttunen_, Dec 06 2021