cp's OEIS Frontend

A341508 a(n) = 0 if n is nonabundant, otherwise a(n) is the number of abundant divisors of the last abundant number in the iteration x -> A003961(x) (starting from x=n) before a nonabundant number is reached.

%I A341508 #22 Feb 20 2021 23:11:29
%S A341508 0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,2,0,0,0,0,0,1,0,0,0,0,
%T A341508 0,3,0,0,0,2,0,1,0,0,0,0,0,3,0,0,0,0,0,2,0,1,0,0,0,4,0,0,0,0,0,1,0,0,
%U A341508 0,1,0,5,0,0,0,0,0,1,0,3,0,0,0,3,0,0,0,1,0,3,0,0,0,0,0,4,0,0,0,2,0,1,0,1,0,0,0,5,0,0,0,2,0,1,0,0,0,0,0,1
%N A341508 a(n) = 0 if n is nonabundant, otherwise a(n) is the number of abundant divisors of the last abundant number in the iteration x -> A003961(x) (starting from x=n) before a nonabundant number is reached.
%C A341508 Question: Is a(A336389(n)) = 1 for all n >= 2? Note that all the terms of A047802 are obviously primitively abundant (in A091191).
%H A341508 Antti Karttunen, <a href="/A341508/b341508.txt">Table of n, a(n) for n = 1..65537</a>
%H A341508 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%H A341508 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%e A341508 Starting from n = 120 = 2^3 * 3 * 5, the number of its abundant divisors is A080224(120) = 7. Then we apply a prime shift (A003961) to obtain the next number, 3^3 * 5 * 7 = 945, which has A080224(945) = 1 abundant divisors (as 945 is a term of A091191). The next prime shift gives 5^3 * 7* 11 = 9625, which has zero abundant divisors (as it is nonabundant, in A263837), so A080224(9625) = 0, and a(120) = 1, the last nonzero value encountered.
%o A341508 (PARI)
%o A341508 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A341508 A080224(n) = sumdiv(n, d, sigma(d)>2*d);
%o A341508 A341508(n) = { my(t, u=0); while((t=A080224(n))>0, u=t; n = A003961(n)); (u); };
%Y A341508 Cf. A003961, A047802, A091191, A336389, A336835, A337468.
%Y A341508 Cf. A263837 (positions of zeros), A005101 (and of nonzeros).
%Y A341508 Differs from A080224 for the first time at n=120, with a(120) = 1, while A080224(120) = 7.
%K A341508 nonn
%O A341508 1,24
%A A341508 _Antti Karttunen_, Feb 20 2021