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

4, 9, 12, 20, 44, 52, 60, 108, 124, 125, 132, 140, 156, 172, 188, 204, 236, 300, 308, 396, 412, 436, 476, 492, 612, 644, 700, 836, 876, 884, 891, 924, 972, 980, 1004, 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

Offset: 1

Author

Antti Karttunen, Feb 12 2020

Keywords

• nonn

Comments

Numbers k > 1 such that A332224(A156552(k)) = A087808(sigma(A156552(k))) is equal to 2*A048675(k) = A048675(k^2).

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).

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).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..150 (all terms < 10000, computed using Hans Havermann's factorization of A156552)

Crossrefs

Cf. A000203, A048675, A156552, A323243, A332223, A332224, A332229, A332445, A332446.

Cf. A324201 (a subsequence).

Programs

PARI

for(n=2,2048,if(A048675(A332223(n))==2*A048675(n),print1(n,", ")))

PARI

\\ To find all terms < 10000:
v156552sigs = readvec("a156552.txt"); \\ Use the factorization file for A156552 prepared by Hans Havermann, available at https://oeis.org/A156552/a156552.txt
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)));
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A087808(n) = if(n<1, 0, if(n%2==0, 2*A087808(n/2), A087808((n-1)/2)+1));
isA322225(n) = (A087808(A323243(n)) == 2*A048675(n));
for(n=2,10000,if(isA322225(n),print1(n,", ")));