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

1,2

Comments

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

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

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

Example

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.

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])]&]

Prog

(PARI) ok(n)={my(t=n>>valuation(n, 2)); issquarefree(t) && !#select(p->!isprime(primepi(p)), factor(t)[, 1])} \\ Andrew Howroyd, Aug 26 2018

Crossrefs

Cf. A000009, A000961, A001222, A003963, A005117, A006450, A007716, A056239, A076610, A275024, A279790, A281113, A294788, A294788, A301750, A302242, A302243.

Sequence in context: A333909 A088232 A070994 * A057197 A067936 A087721

Adjacent sequences: A291683 A291684 A291685 * A291687 A291688 A291689

Keyword

nonn

Author

Gus Wiseman, Apr 09 2018

Status

approved