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

1,1

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).

Links

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

Example

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: {{},{},{},{},{}}

Mathematica

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

Prog

(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

Crossrefs

Subsequence of A122132.

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

Sequence in context: A020662 A306202 A328335 * A235034 A331682 A107037

Adjacent sequences: A302566 A302567 A302568 * A302570 A302571 A302572

Keyword

nonn

Author

Gus Wiseman, Apr 10 2018

Status

approved