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A332225 Numbers k > 1 for which A048675(A332223(k)) is equal to 2*A048675(k).

%I A332225 #37 Feb 15 2020 21:09:08
%S A332225 4,9,12,20,44,52,60,108,124,125,132,140,156,172,188,204,236,300,308,
%T A332225 396,412,436,476,492,612,644,700,836,876,884,891,924,972,980,1004,
%U A332225 1044,1092,1100,1116,1148,1188,1196,1236,1260,1268,1292,1300,1308,1372,1380,1476,1620,1628,1724,1860,1900,2140,2244,2324,2356,2444,2460,2652,2660,2700
%N A332225 Numbers k > 1 for which A048675(A332223(k)) is equal to 2*A048675(k).
%C A332225 Numbers k > 1 such that A332224(A156552(k)) = A087808(sigma(A156552(k))) is equal to 2*A048675(k) = A048675(k^2).
%C A332225 Notably, of the first 150 terms (4 .. 9996), 156 = 2^2 * 3 * 13 is the only even term that does not map to a prime, as A156552(156) = 267 = 3*89 (and sigma(267) = 360 = 4*90).
%C A332225 Although sigma(A156552(k)) = A323243(k) is a multiple of 4 for most of the terms k present in this sequence, there are exceptions, for example 840350 = A005940(1+A332445(1)) = 2^1 * 5^2 * 7^5 is one, as A048675(A332223(840350)) = 98 = 2*A048675(840350) and A323243(840350) = 2394 == 2 (mod 4).
%H A332225 Antti Karttunen, <a href="/A332225/b332225.txt">Table of n, a(n) for n = 1..150</a> (all terms < 10000, computed using Hans Havermann's factorization of A156552)
%o A332225 (PARI) for(n=2,2048,if(A048675(A332223(n))==2*A048675(n),print1(n,", ")))
%o A332225 (PARI)
%o A332225 \\ To find all terms < 10000:
%o A332225 v156552sigs = readvec("a156552.txt"); \\ Use the factorization file for A156552 prepared by Hans Havermann, available at https://oeis.org/A156552/a156552.txt
%o A332225 A323243(n) = if(n<=2,n-1,my(prsig=v156552sigs[n],ps=prsig[1],es=prsig[2]); prod(i=1,#ps,((ps[i]^(1+es[i]))-1)/(ps[i]-1)));
%o A332225 A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
%o A332225 A087808(n) = if(n<1, 0, if(n%2==0, 2*A087808(n/2), A087808((n-1)/2)+1));
%o A332225 isA322225(n) = (A087808(A323243(n)) == 2*A048675(n));
%o A332225 for(n=2,10000,if(isA322225(n),print1(n,", ")));
%Y A332225 Cf. A000203, A048675, A156552, A323243, A332223, A332224, A332229, A332445, A332446.
%Y A332225 Cf. A324201 (a subsequence).
%K A332225 nonn
%O A332225 1,1
%A A332225 _Antti Karttunen_, Feb 12 2020