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Charles R Greathouse IV, <a href="/A302569/b302569.txt">Table of n, a(n) for n = 1..10000</a>

(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

Subsequence of A122132.

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allocated for Gus WisemanNumbers 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

1,1

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).

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

primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Select[Range[200], Or[PrimeQ[#], CoprimeQ@@primeMS[#]]&]

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

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Gus Wiseman, Apr 10 2018

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allocated for Gus Wiseman

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