cp's OEIS Frontend

A342013 Position of the n-th colossally abundant number in A329886, the primorial inflation of Doudna-tree.

%I A342013 #17 Mar 14 2021 20:42:04
%S A342013 1,2,5,9,19,21,37,75,139,267,535,539,555,1067,2091,4139,8279,16471,
%T A342013 32855,32919,32923,65691,131227,262299,524599,1048887,2097463,4194615,
%U A342013 4194647,8388951,16777559,33554775,67109207,67109463,134218327,268436655,536872111,536872119,1073743031,2147484855,2147485879,4294969527,8589936823
%N A342013 Position of the n-th colossally abundant number in A329886, the primorial inflation of Doudna-tree.
%C A342013 a(n) is the unique integer k such that A329886(k) = A004490(n).
%C A342013 Like A342012, also this sequence is monotonic. Proof: the doubling step corresponds here to step *2 + 1, and "bumping up" some of the prime factors likewise results a larger A156552-code, thus both steps keep the result growing.
%C A342013 The binary length of these numbers (A070939, = 1+A000523) grows by 0 or 1 at each step, thus the next colossally abundant number is always found on either on the same row (right of the current CA-number), or the next row of A329886, the row immediately below. The next CA-number will be on the same row only when its factorization contains neither a new prime nor yet another instance of prime 2.
%H A342013 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A342013 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A342013 a(n) = A156552(A342012(n)) = A156552(A319626(A004490(n))).
%o A342013 (PARI)
%o A342013 A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res};
%o A342013 A342013(n) = A156552(A342012(n)); \\ Uses also code from A342012.
%Y A342013 Cf. A004490, A005940, A073751, A108951, A156552, A319626, A329886, A329900, A342000, A342010 (the binary weight), A342011, A342012.
%K A342013 nonn
%O A342013 1,2
%A A342013 _Antti Karttunen_ and _Michael De Vlieger_, Mar 08 2021