A220218 Numbers where all exponents in its prime factorization are one less than a prime.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Author

Charles R Greathouse IV, Dec 07 2012

Keywords

• nonn

Comments

Sequence has positive density, between 0.83 and 0.89; probably about 0.87951.

The numbers of terms not exceeding 10^k, for k=1,2,..., are 9, 90, 880, 8796, 87956, 879518, 8795126, 87951173, 879511794, ... The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^3 + Sum_{q prime >= 5} (p-1)/p^q) = 0.87951176583716527413... - Amiram Eldar, Mar 20 2021

Numbers whose sets of unitary divisors (A077610) and modified exponential divisors (A379027) coincide. - Amiram Eldar, Dec 14 2024

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Paul Erdős and Leon Mirsky, The distribution of values of the divisor function d(n), Proc. London Math. Soc., Vol. s3-2, No. 1 (1952), pp. 257-271; alternative link.

Crossrefs

Apart from the first term, a subsequence of A096432.

Cf. A010051, A124010, A077610, A379027.

Programs

Haskell

a220218 n = a220218_list !! (n-1)
a220218_list = 1 : filter
               (all (== 1) . map (a010051' . (+ 1)) . a124010_row) [1..]
-- Reinhard Zumkeller, Nov 30 2015

Mathematica

Select[Range[100],AllTrue[Transpose[FactorInteger[#]][[2]]+1,PrimeQ]&] (* Harvey P. Dale, Sep 29 2014 *)

PARI

is(n)=vecmin(apply(n->isprime(n+1),factor(max(n,2))[,2])) \\ Charles R Greathouse IV, Dec 07 2012