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Numbers where all exponents in its prime factorization are one less than a prime.
7

%I #14 Mar 20 2021 10:57:11

%S 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,

%T 29,30,31,33,34,35,36,37,38,39,41,42,43,44,45,46,47,48,49,50,51,52,53,

%U 55,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,73,74,75,76,77,78,79,80

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

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

%C 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

%H Reinhard Zumkeller, <a href="/A220218/b220218.txt">Table of n, a(n) for n = 1..10000</a>

%H Paul Erdős and Leon Mirsky, <a href="https://doi.org/10.1112/plms/s3-2.1.257">The distribution of values of the divisor function d(n)</a>, Proc. London Math. Soc., Vol. s3-2, No. 1 (1952), pp. 257-271; <a href="http://www.renyi.hu/~p_erdos/1952-12.pdf">alternative link</a>.

%t Select[Range[100],AllTrue[Transpose[FactorInteger[#]][[2]]+1,PrimeQ]&] (* _Harvey P. Dale_, Sep 29 2014 *)

%o (PARI) is(n)=vecmin(apply(n->isprime(n+1),factor(max(n,2))[,2])) \\ _Charles R Greathouse IV_, Dec 07 2012

%o (Haskell)

%o a220218 n = a220218_list !! (n-1)

%o a220218_list = 1 : filter

%o (all (== 1) . map (a010051' . (+ 1)) . a124010_row) [1..]

%o -- _Reinhard Zumkeller_, Nov 30 2015

%Y Apart from the first term, a subsequence of A096432.

%Y Cf. A010051, A124010.

%K nonn

%O 1,2

%A _Charles R Greathouse IV_, Dec 07 2012