A261256 - OEIS

%I #39 Feb 19 2022 03:57:22

%S 4,24,72,160,432,896,2592,5632,12800,26624,61440,124416,278528,622592,

%T 1376256,2949120,5971968,12058624,25690112,60817408,130023424,

%U 262144000,528482304,1107296256,2264924160,4586471424,9395240960,19864223744,40265318400,83751862272

%N Let S_k denote the sequence of numbers j such that A001222(j) - A001221(j) = k. Then a(n) is the n-th term of S_n.

%C S_0 would correspond to the squarefree numbers (A005117), that is, numbers j such that A001222(j) = A001221(j). Note that S_0 is excluded from the scheme. - _Michel Marcus_, Sep 21 2015

%H Charlie Neder, <a href="/A261256/b261256.txt">Table of n, a(n) for n = 1..500</a>

%F a(n+1) > 2*a(n).

%F a(n) >= 2^prime(n) for n < 5.

%F a(n) = A257851(n,n-1). - _Reinhard Zumkeller_, Nov 29 2015

%F a(n) = b(n)*2^(n+1), where b(n) consists of the values of k/2^excess(k) over odd k, sorted in ascending order. In particular, a(n) <= prime(n)*2^(n+1), with equality only when n = 2. - _Charlie Neder_, Jan 31 2019

%e For n = 1, S_1 = {4, 9, 12, 18, 20, 25, ...}, so a(1) = S_1(1) = 4.

%e For n = 2, S_2 = {8, 24, 27, 36, 40, 54, ...}, so a(2) = S_2(2) = 24.

%e For n = 3, S_3 = {16, 48, 72, 80, 81, 108, ...}, so a(3) = S_3(3) = 72.

%e For n = 4, S_4 = {32, 96, 144, 160, 216, 224, ...}, so a(4) = S_4(4) = 160.

%e For n = 5, S_5 = {64, 192, 288, 320, 432, 448, ...}, so a(5) = S_5(5) = 432.

%t OutSeq = {}; For[i = 1, i <= 16, i++, l = Select[Range[10^2*2^i], PrimeOmega[#] - PrimeNu[#] == i &]; AppendTo[OutSeq, l[[i]]]]; OutSeq

%o (PARI) a(n) = {ik = 1; nbk = 0; while (nbk != n, ik++; if (bigomega(ik) == omega(ik) + n, nbk++);); ik;} \\ _Michel Marcus_, Oct 06 2015

%o (Haskell)

%o a261256 n = a257851 n (n - 1)  -- _Reinhard Zumkeller_, Nov 29 2015

%Y Cf. A001221, A001222.

%Y Cf. A060687, A195086, A195087, A195088, A195089, A195090, A195091, A195092, A195093.

%Y Cf. A046660, A257851, A264959.

%K nonn

%O 1,1

%A _Carlos Eduardo Olivieri_, Aug 12 2015

%E a(17)-a(21) from _Jon E. Schoenfield_, Sep 12 2015

%E More terms from _Charlie Neder_, Jan 31 2019