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Odd numbers whose product of prime indices is a prime power.
1

%I #7 Dec 16 2018 17:58:55

%S 1,3,5,7,9,11,17,19,21,23,25,27,31,41,49,53,57,59,63,67,81,83,97,103,

%T 109,115,121,125,127,131,133,147,157,159,171,179,189,191,211,227,241,

%U 243,277,283,289,311,331,343,353,361,367,371,393,399,401,419,431,441

%N Odd numbers whose product of prime indices is a prime power.

%C Differs from A322400 in having 1 and lacking 377, the MM-number of {{1,2},{1,3}}.

%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

%e The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. The sequence of multiset partitions whose MM-numbers belong to this sequence begins:

%e 1: {}

%e 3: {{1}}

%e 5: {{2}}

%e 7: {{1,1}}

%e 9: {{1},{1}}

%e 11: {{3}}

%e 17: {{4}}

%e 19: {{1,1,1}}

%e 21: {{1},{1,1}}

%e 23: {{2,2}}

%e 25: {{2},{2}}

%e 27: {{1},{1},{1}}

%e 31: {{5}}

%e 41: {{6}}

%e 49: {{1,1},{1,1}}

%e 53: {{1,1,1,1}}

%e 57: {{1},{1,1,1}}

%e 59: {{7}}

%e 63: {{1},{1},{1,1}}

%e 67: {{8}}

%e 81: {{1},{1},{1},{1}}

%e 83: {{9}}

%e 97: {{3,3}}

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

%t Select[Range[1,100,2],#==1||PrimePowerQ[Times@@primeMS[#]]&]

%o (PARI) isok(n) = {if (n % 2, my(f = factor(n), pk = prod(k=1, #f~, primepi(f[k,1]))); (pk == 1) || isprimepower(pk););} \\ _Michel Marcus_, Dec 16 2018

%Y Cf. A003963, A056239, A112798, A290103, A302242, A320325, A320698.

%K nonn

%O 1,2

%A _Gus Wiseman_, Dec 15 2018