cp's OEIS Frontend

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

%I A220218 #19 Dec 14 2024 05:26:22
%S A220218 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 A220218 29,30,31,33,34,35,36,37,38,39,41,42,43,44,45,46,47,48,49,50,51,52,53,
%U A220218 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 A220218 Numbers where all exponents in its prime factorization are one less than a prime.
%C A220218 Sequence has positive density, between 0.83 and 0.89; probably about 0.87951.
%C A220218 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
%C A220218 Numbers whose sets of unitary divisors (A077610) and modified exponential divisors (A379027) coincide. - _Amiram Eldar_, Dec 14 2024
%H A220218 Reinhard Zumkeller, <a href="/A220218/b220218.txt">Table of n, a(n) for n = 1..10000</a>
%H A220218 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 A220218 Select[Range[100],AllTrue[Transpose[FactorInteger[#]][[2]]+1,PrimeQ]&] (* _Harvey P. Dale_, Sep 29 2014 *)
%o A220218 (PARI) is(n)=vecmin(apply(n->isprime(n+1),factor(max(n,2))[,2])) \\ _Charles R Greathouse IV_, Dec 07 2012
%o A220218 (Haskell)
%o A220218 a220218 n = a220218_list !! (n-1)
%o A220218 a220218_list = 1 : filter
%o A220218                (all (== 1) . map (a010051' . (+ 1)) . a124010_row) [1..]
%o A220218 -- _Reinhard Zumkeller_, Nov 30 2015
%Y A220218 Apart from the first term, a subsequence of A096432.
%Y A220218 Cf. A010051, A124010, A077610, A379027.
%K A220218 nonn
%O A220218 1,2
%A A220218 _Charles R Greathouse IV_, Dec 07 2012