A318182 - OEIS

%I #30 Aug 28 2018 04:50:13

%S 1,2,8,9,10,15,18,24,30,60,720,1020,4080,8925,14688,14976,16728,17850,

%T 35700,36720,37440,66912,71400,285600,308448,381888,428400,602208,

%U 636480,763776,856800,1321920,1505520,3011040,3084480,21679488,22276800,30844800

%N Numbers m such that A049417(A049417(m)) = k*m for some k where A049417 is the infinitary sigma function.

%C a(86) > 3*10^11. All the prime factors of the first 85 terms belong to the set {2, 3, 5, 7, 11, 13, 17, 41, 43, 257}. - _Giovanni Resta_, Aug 25 2018

%C Like in A019278, here there are many instances where if x is a term, then A049417(x) is also a term.

%C Additionally, there exist longer chains of 3 or 4 elements like:

%C - 8 (3), 15 (4), 24 (5), 60 (6);

%C - 9 (2), 10 (3), 18 (4), 30 (5);

%C - 31615920 (4), 50585472 (5), 126463680 (6), 252927360 (12);

%C - 963407296051200 (16), 3134896756992000 (17), 15414516736819200 (18);

%C - 3541951043592192 (5), 8854877608980480 (6), 17709755217960960 (12), 53129265653882880 (20);

%C - 4829933241262080 (11), 17709755217960960 (12), 53129265653882880 (20);

%C   7871002319093760 (9), 26564632826941440 (10), 70839020871843840 (13), 265646328269414400 (14).

%H Giovanni Resta, <a href="/A318182/b318182.txt">Table of n, a(n) for n = 1..85</a> (first 58 terms from Tomohiro Yamada)

%H Tomohiro Yamada, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_47_from211to218.pdf">Infinitary superperfect numbers</a>, Annales Mathematicae et Informaticae, 47 (2017) pp. 211-218. See Table 1.

%H Michel Marcus, <a href="/A318182/a318182.log.txt">Unexhaustive list of terms</a>

%o (PARI) a049417(n) = {my(b, f=factorint(n)); prod(k=1, #f[, 2], b = binary(f[k, 2]); prod(j=1, #b, if(b[j], 1+f[k, 1]^(2^(#b-j)), 1)));}

%o isok(n) = frac(a049417(a049417(n))/n) == 0;

%Y Cf. A049417 (infinitary sigma).

%Y Cf. A019278 (analog for sigma), A318175 (analog for bi-unitary sigma).

%K nonn

%O 1,2

%A _Michel Marcus_, Aug 20 2018

%E More terms from _Giovanni Resta_, Aug 25 2018