login
Numbers k for which A003973(k) >= 2*A003961(k).
16

%I #33 Sep 05 2020 18:55:30

%S 120,180,240,300,360,420,480,504,540,600,630,660,720,780,840,900,924,

%T 960,990,1008,1020,1050,1080,1092,1140,1170,1200,1260,1320,1380,1440,

%U 1470,1500,1512,1560,1620,1650,1680,1740,1800,1848,1860,1890,1920,1980,2016,2040,2100,2160,2184,2220,2280,2310,2340,2400,2460

%N Numbers k for which A003973(k) >= 2*A003961(k).

%C Provided that there are no odd perfect numbers, then these are equal to numbers k for which A003961(k) is in A005231, i.e., numbers that become odd abundant numbers when prime-shifted once.

%C Not all terms are even. The first odd term is a(8313165) = 334639305 = A064989(A115414(1)). (See A337385). For any odd term x present, A064989(x) is also present, for example, A064989(334639305) = 19399380 = a(482324).

%H Antti Karttunen, <a href="/A337386/b337386.txt">Table of n, a(n) for n = 1..25993; all terms < 2^20</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%t Select[Range[2500], If[# == 1, 1, DivisorSigma[1, # ]] >= 2# &@ Apply[Times, FactorInteger[#] /. {p_, e_} /; e > 0 :> Prime[PrimePi@ p + 1]^e] &] (* _Michael De Vlieger_, Aug 27 2020 *)

%o (PARI)

%o A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

%o isA337386(n) = (sigma(A003961(n))>=2*A003961(n));

%Y Cf. A000203, A003961, A003973, A005231, A115414, A336389, A337468.

%Y Subsequence of A005101, of A337381, and of A246282.

%Y Subsequences: A337385 (odd terms), A337479 (primitive elements).

%K nonn

%O 1,1

%A _Antti Karttunen_, Aug 27 2020