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A378884 Numbers that are not powers of primes and whose two smallest prime divisors are consecutive primes.

%I A378884 #8 Dec 10 2024 09:58:55
%S A378884 6,12,15,18,24,30,35,36,42,45,48,54,60,66,72,75,77,78,84,90,96,102,
%T A378884 105,108,114,120,126,132,135,138,143,144,150,156,162,165,168,174,175,
%U A378884 180,186,192,195,198,204,210,216,221,222,225,228,234,240,245,246,252,255,258
%N A378884 Numbers that are not powers of primes and whose two smallest prime divisors are consecutive primes.
%C A378884 Subsequence of A104210 and first differs from at an n = 15: A104210(15) = 70 = 2 * 5 * 7 is not a term of this sequence.
%C A378884 All the positive multiples of 6 (A008588 \ {0}) are terms.
%C A378884 Numbers k such that nextprime(lpf(k)) = A151800(A020639(k)) | k.
%C A378884 The asymptotic density of this sequence is Sum_{k>=1} (Product_{j=1..k-1} (1-1/prime(j)))/(prime(k)*prime(k+1)) = 0.2178590011934... .
%H A378884 Amiram Eldar, <a href="/A378884/b378884.txt">Table of n, a(n) for n = 1..10000</a>
%e A378884 12 = 2^2 * 3 is a term since 2 and 3 are consecutive primes.
%e A378884 70 = 2 * 5 * 7 is not a term since 2 and 5 are not consecutive primes.
%e A378884 165 = 3 * 5 * 11 is a term since 3 and 5 are consecutive primes.
%t A378884 q[k_] := Module[{p = FactorInteger[k][[;; , 1]]}, Length[p] > 1 && p[[2]] == NextPrime[p[[1]]]]; Select[Range[300], q]
%o A378884 (PARI) is(k) = if(k == 1, 0, my(p = factor(k)[,1]); #p > 1 && p[2] == nextprime(p[1]+1));
%Y A378884 Subsequence of A024619, A104210 and A378885.
%Y A378884 Subsequences: A006094, A256617.
%Y A378884 Cf. A008588, A020639, A151800.
%K A378884 nonn,easy
%O A378884 1,1
%A A378884 _Amiram Eldar_, Dec 09 2024