cp's OEIS Frontend

A342669 Even numbers which are either primitively nondeficient (A006039), or become such after applying prime shift A003961 some number of times to them.

%I A342669 #19 Mar 29 2021 15:05:08
%S A342669 6,20,28,70,88,104,120,180,272,300,304,368,420,464,496,504,550,572,
%T A342669 630,650,660,748,780,836,924,990,1020,1050,1092,1140,1170,1184,1312,
%U A342669 1376,1380,1430,1470,1504,1650,1696,1740,1860,1870,1888,1952,2002,2090,2210,2220,2310,2460,2470,2530,2580,2584,2730,2820,2856,2990
%N A342669 Even numbers which are either primitively nondeficient (A006039), or become such after applying prime shift A003961 some number of times to them.
%C A342669 Even numbers k for which A341624(k) = 1.
%C A342669 Even numbers whose closure under map x -> A003961(x) contains a primitive non-deficient number (one of the terms of A006039). Shifting each term k exactly A336835(k)-1 times with A003961 towards larger primes gives those numbers, but not in monotonic order, producing instead a permutation of A006039.
%C A342669 Sequence 2*A246277(A006039(.)), sorted into ascending order.
%C A342669 If there are any two terms, x and y, such that the other is a multiple of the other, then A336835(x) != A336835(y), and furthermore, for any term k present here, for all its proper divisors (d|k, d<k) it holds that A336835(d) < A336835(k), in other words, they reach the deficiency earlier (by prime shifting) than k itself.
%H A342669 Antti Karttunen, <a href="/A342669/b342669.txt">Table of n, a(n) for n = 1..10000</a>
%H A342669 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%H A342669 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%e A342669 For n = 120 = 2^3 * 3 * 5, A341620(120) = 8, so it is not primitive nondeficient. However, prime-shifting it once gives A003961(120) = 945 = 3^3 * 5 * 7, which is one of the terms of A006039 as A341620(945) = 1. Therefore 120 is included in the sequence.
%o A342669 (PARI)
%o A342669 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A342669 A341620(n) = sumdiv(n,d,(sigma(d)>=(2*d)));
%o A342669 A341624(n) = { my(t, u=0); while((t=A341620(n))>0, u=t; n = A003961(n)); (u); };
%o A342669 isA342669(n) = (!(n%2)&&(1==A341624(n)));
%Y A342669 Cf. A000396, A006039 (even terms of these form a subsequence).
%Y A342669 Cf. A003961, A246277, A336835, A341605/A341606, A341620, A341624.
%K A342669 nonn
%O A342669 1,1
%A A342669 _Antti Karttunen_, Mar 20 2021