cp's OEIS Frontend

A289509 Numbers k such that the gcd of the indices j for which the j-th prime prime(j) divides k is 1.

%I A289509 #38 Jun 18 2021 02:37:36
%S A289509 2,4,6,8,10,12,14,15,16,18,20,22,24,26,28,30,32,33,34,35,36,38,40,42,
%T A289509 44,45,46,48,50,51,52,54,55,56,58,60,62,64,66,68,69,70,72,74,75,76,77,
%U A289509 78,80,82,84,85,86,88,90,92,93,94,95,96,98,99,100,102,104
%N A289509 Numbers k such that the gcd of the indices j for which the j-th prime prime(j) divides k is 1.
%C A289509 Any integer k in the sequence encodes (by 'Heinz encoding' cf. A056239) a multiset of integers whose gcd is 1, namely the multiset containing r_j copies of j if k factors as Product_j prime(j)^{r_j} with gcd_j j = 1.
%C A289509 Clearly the sequence contains all even numbers and no odd primes or odd prime powers. It also clearly contains all numbers that are divisible by consecutive primes.
%C A289509 The sequence is the list of those k such that A289508(k) = 1.
%C A289509 It is also the list of those k such that A289506(k) = A289507(k).
%C A289509 Heinz numbers of integer partitions with relatively prime parts, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - _Gus Wiseman_, Apr 13 2018
%H A289509 Alois P. Heinz, <a href="/A289509/b289509.txt">Table of n, a(n) for n = 1..20000</a>
%e A289509 6 is a term because 6 = p_1*p_2 and gcd(1,2) = 1.
%e A289509 From _Gus Wiseman_, Apr 13 2018: (Start)
%e A289509 Sequence of integer partitions with relatively prime parts begins:
%e A289509 02 : (1)
%e A289509 04 : (11)
%e A289509 06 : (21)
%e A289509 08 : (111)
%e A289509 10 : (31)
%e A289509 12 : (211)
%e A289509 14 : (41)
%e A289509 15 : (32)
%e A289509 16 : (1111)
%e A289509 18 : (221)
%e A289509 20 : (311)
%e A289509 22 : (51)
%e A289509 24 : (2111)
%e A289509 26 : (61)
%e A289509 28 : (411)
%e A289509 30 : (321)
%e A289509 32 : (11111)
%e A289509 33 : (52)
%e A289509 34 : (71)
%e A289509 35 : (43)
%e A289509 36 : (2211)
%e A289509 38 : (81)
%e A289509 40 : (3111)
%e A289509 (End)
%p A289509 p:=1:for ind to 10000 do p:=nextprime(p);primeindex[p]:=ind;od:
%p A289509 out:=[]:for n from 2 to 100 do m:=[];f:=ifactors(n)[2];g:=0;
%p A289509 for k to nops(f) do mk:=primeindex[f[k][1]];m:=[op(m),mk];
%p A289509 g:=gcd(g,mk);od; if g=1 then out:=[op(out),n];fi;od:out;
%t A289509 Select[Range[200],GCD@@PrimePi/@FactorInteger[#][[All,1]]===1&] (* _Gus Wiseman_, Apr 13 2018 *)
%o A289509 (PARI) isok(n) = my(f=factor(n)); gcd(apply(x->primepi(x), f[,1])) == 1; \\ _Michel Marcus_, Jul 19 2017
%o A289509 (Python)
%o A289509 from sympy import gcd, primepi, primefactors
%o A289509 def ok(n): return gcd([primepi(p) for p in primefactors(n)]) == 1
%o A289509 print([n for n in range(1, 151) if ok(n)]) # _Indranil Ghosh_, Aug 06 2017
%Y A289509 Cf. A001222, A007359, A051424, A056239, A289506, A289507, A289508, A296150, A302696, A302697, A302698, A302796.
%K A289509 nonn
%O A289509 1,1
%A A289509 _Christopher J. Smyth_, Jul 11 2017