A378744 - OEIS

%I #21 Dec 10 2024 11:59:13

%S 2,2,3,2,5,3,7,4,3,11,4,7,15,8,6,7,21,8,12,25,9,15,10,9,18,36,19,38,

%T 20,6,23,16,48,10,52,18,19,29,20,21,63,13,67,34,69,12,13,39,80,41,28,

%U 84,18,30,31,31,94,32,49,98,20,15,54,109,55,17,39,24,121,61,42,32,43,44,67,45,35,70,36,29,146,30,150

%N The least k such that prime(1+n)^k > 2 * prime(n)^k.

%C A000040(n)^a(n) = A378745(n) is always term of A337372, i.e., is primitively prime-shift abundant.

%H Antti Karttunen, <a href="/A378744/b378744.txt">Table of n, a(n) for n = 1..20000</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.

%H <a href="/index/Pri#gaps">Index entries for primes, gaps between</a>.

%F a(n) = ceiling(log(2) / log(A000040(n+1)/A000040(n))).

%F For all n >= 1, A341609(A000040(n)^a(n)) = 1.

%e For n=1, prime(1)=2 and prime(2)=3, and 3^1 is not larger than 2*2^1, but 3^2 > 2*2^2, therefore a(1) = 2.

%e For n=3, prime(3)=5 and prime(4)=7, with 7 < 2*5, 7^2 = 49 < 2*25, and 7^3 = 343 > 2*125, therefore a(3) = 3.

%o (PARI) A378744(n) = { my(p=prime(n), q=prime(1+n)); for(k=-1+floor(log(2)/log(q/p)), oo, if(q^k > 2*(p^k), return(k))); };

%Y Cf. A000040, A001359, A003961, A337372, A341609, A378745 [= A000040(n)^a(n)].

%K nonn,new

%O 1,1

%A _Antti Karttunen_, Dec 08 2024