A291686 Numbers whose prime indices other than 1 are distinct prime numbers.
1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 15, 16, 17, 20, 22, 24, 30, 31, 32, 33, 34, 40, 41, 44, 48, 51, 55, 59, 60, 62, 64, 66, 67, 68, 80, 82, 83, 85, 88, 93, 96, 102, 109, 110, 118, 120, 123, 124, 127, 128, 132, 134, 136, 155, 157, 160, 164, 165, 166, 170, 176, 177
Offset: 1
Keywords
Examples
9 is not in the sequence because the prime indices of 9 = prime(2)*prime(2) are {2,2} which are prime numbers but not distinct.
15 is in the sequence because the prime indices of 15 = prime(2)*prime(3) are {2,3} which are distinct prime numbers.
21 is not in the sequence because the prime indices of 21 = prime(2)*prime(4) are {2,4} which are distinct but not all prime numbers.
24 is in the sequence because the prime indices of 24 = prime(1)*prime(1)*prime(1)*prime(2) are {1,1,1,2} which without the 1s are distinct prime numbers.
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1000
Crossrefs
Programs
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Mathematica
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]; Select[Range[100],Or[#===1,UnsameQ@@DeleteCases[primeMS[#],1]&&And@@(PrimeQ/@DeleteCases[primeMS[#],1])]&] -
PARI
ok(n)={my(t=n>>valuation(n,2)); issquarefree(t) && !#select(p->!isprime(primepi(p)), factor(t)[,1])} \\ Andrew Howroyd, Aug 26 2018
Formula
Sum_{n>=1} 1/a(n) = 2 * Product_{p in A006450} (1 + 1/p) converges since the sum of the reciprocals of A006450 converges. - Amiram Eldar, Feb 02 2021
Comments