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

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

Offset

1,1

Comments

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.

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 k such that A289508(k) = 1.

It is also the list of those k such that A289506(k) = A289507(k).

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

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.
From Gus Wiseman, Apr 13 2018: (Start)
Sequence of integer partitions with relatively prime parts begins:
02 : (1)
04 : (11)
06 : (21)
08 : (111)
10 : (31)
12 : (211)
14 : (41)
15 : (32)
16 : (1111)
18 : (221)
20 : (311)
22 : (51)
24 : (2111)
26 : (61)
28 : (411)
30 : (321)
32 : (11111)
33 : (52)
34 : (71)
35 : (43)
36 : (2211)
38 : (81)
40 : (3111)
(End)

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;

Mathematica

Select[Range[200], GCD@@PrimePi/@FactorInteger[#][[All, 1]]===1&] (* Gus Wiseman, Apr 13 2018 *)

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([primepi(p) for p in primefactors(n)]) == 1
print([n for n in range(1, 151) if ok(n)]) # Indranil Ghosh, Aug 06 2017

Crossrefs

Cf. A001222, A007359, A051424, A056239, A289506, A289507, A289508, A296150, A302696, A302697, A302698, A302796.

Sequence in context: A055956 A161207 A280877 * A336735 A304711 A350803

Adjacent sequences: A289506 A289507 A289508 * A289510 A289511 A289512

Keyword

nonn

Author

Christopher J. Smyth, Jul 11 2017

Status

approved