A302569 Numbers that are either prime or whose prime indices are pairwise coprime. Heinz numbers of integer partitions with pairwise coprime parts.

2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 22, 23, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 48, 51, 52, 53, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89

Offset: 1

Author

Gus Wiseman, Apr 10 2018

Keywords

• nonn

Comments

A prime index of n is a number m such that prime(m) divides n.

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

Examples

			Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of multiset systems.
02: {{}}
03: {{1}}
04: {{},{}}
05: {{2}}
06: {{},{1}}
07: {{1,1}}
08: {{},{},{}}
10: {{},{2}}
11: {{3}}
12: {{},{},{1}}
13: {{1,2}}
14: {{},{1,1}}
15: {{1},{2}}
16: {{},{},{},{}}
17: {{4}}
19: {{1,1,1}}
20: {{},{},{2}}
22: {{},{3}}
23: {{2,2}}
24: {{},{},{},{1}}
26: {{},{1,2}}
28: {{},{},{1,1}}
29: {{1,3}}
30: {{},{1},{2}}
31: {{5}}
32: {{},{},{},{},{}}
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A122132.

Cf. A000961, A001222, A005117, A007359, A007716, A051424, A056239, A076610, A101268, A275024, A302505, A302568.

Programs

Mathematica

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[200],Or[PrimeQ[#],CoprimeQ@@primeMS[#]]&]

PARI

is(n)=if(n<9, return(n>1)); n>>=valuation(n,2); if(n<9, return(1)); my(f=factor(n)); if(vecmax(f[,2])>1, return(0)); if(#f~==1, return(1)); my(v=apply(primepi, f[,1]),P=vecprod(v)); for(i=1,#v, if(gcd(v[i],P/v[i])>1, return(0))); 1 \\ Charles R Greathouse IV, Nov 11 2021