|
%I #14 May 20 2022 05:02:00
%S 1,1,2,2,3,3,3,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,
%T 7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,
%U 9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10
%N Number of times the term 2 has occurred so far in the range 1..n of A073751.
%C Number of prime factors (with multiplicity) in the primorial deflation of the n-th colossally abundant number [A342012(n) = A319626(A004490(n))], provided that the quotient A004490(1+n)/A004490(n) is always a prime.
%H Amiram Eldar, <a href="/A342010/b342010.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = Sum_{k=1..n} [2==A073751(k)], where [ ] is the Iverson bracket.
%F a(n) = A001222(A342012(n)) = A000120(A342013(n)).
%t Block[{a = {2}, b, c, f, k, m, n, q = {1}, lim = 105}, f[w_] := Block[{p = w[[1]], i = w[[2]]}, ((Log[(p^(i + 2) - 1)/(p^(i + 1) - 1)])/Log[p]) - 1]; m = {{2, 1}, {3, 0}}; c = 1; b = Array[f[m[[#]]] &, c + 1]; For[n = 2, n <= lim, n++, k = Position[b, Max[b]][[1, 1]]; AppendTo[a, m[[k, 1]]]; AppendTo[q, Boole[m[[k, 1]] == 2]]; m[[k, 2]]++; If[k > c, c++; AppendTo[m, {Prime[k + 1], 0}]; AppendTo[b, f[m[[-1]]]]]; b[[k]] = f[m[[k]]]]; Accumulate@ q] (* _Michael De Vlieger_, Mar 12 2021, after _T. D. Noe_ at A073751 *)
%o (PARI)
%o v073751 = readvec("b073751_to.txt"); \\ Prepared with gawk '{ print $2 }' < b073751.txt > b073751_to.txt
%o A073751(n) = v073751[n];
%o A342010(n) = sum(k=1,n,(2==A073751(k)));
%Y Cf. A000120, A001222, A004490, A073751, A319626, A342012, A342013.
%K nonn
%O 1,3
%A _Antti Karttunen_, Mar 08 2021
|
