

A289509


Numbers n with the property that the gcd of the indices j for which the jth prime prime(j) divides n is 1.


3



2, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 58, 60, 62, 64, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 88, 90, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104
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OFFSET

1,1


COMMENTS

Any integer n 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 n factors as Product_j prime(j)^{r_j} with gcd_j j = 1.
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.
The sequence is the list of those n such that A289508(n) = 1.
It is also the list of those n such that A289506(n) = A289507(n).


LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..20000


EXAMPLE

6 is a term because 6 = p_1*p_2 and gcd(1,2) = 1.


MAPLE

p:=1:for ind to 10000 do p:=nextprime(p); primeindex[p]:=ind; od:
out:=[]:for n from 2 to 100 do m:=[]; f:=ifactors(n)[2]; g:=0;
for k to nops(f) do mk:=primeindex[f[k][1]]; m:=[op(m), mk];
g:=gcd(g, mk); od; if g=1 then out:=[op(out), n]; fi; od:out;


PROG

(PARI) isok(n) = my(f=factor(n)); gcd(apply(x>primepi(x), f[, 1])) == 1; \\ Michel Marcus, Jul 19 2017
(Python)
from sympy import gcd, primepi, primefactors
def ok(n): return gcd(map(primepi, primefactors(n)))==1
print [n for n in xrange(1, 151) if ok(n)] # Indranil Ghosh, Aug 06 2017


CROSSREFS

Cf. A289506, A289507, A289508.
Sequence in context: A055956 A161207 A280877 * A195125 A228295 A184592
Adjacent sequences: A289506 A289507 A289508 * A289510 A289511 A289512


KEYWORD

nonn


AUTHOR

Christopher J. Smyth, Jul 11 2017


STATUS

approved



