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A291686 Numbers whose prime indices other than 1 are distinct prime numbers. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 26 23:42 EDT 2021. Contains 347673 sequences. (Running on oeis4.)