A318182 Numbers m such that A049417(A049417(m)) = k*m for some k where A049417 is the infinitary sigma function.

1, 2, 8, 9, 10, 15, 18, 24, 30, 60, 720, 1020, 4080, 8925, 14688, 14976, 16728, 17850, 35700, 36720, 37440, 66912, 71400, 285600, 308448, 381888, 428400, 602208, 636480, 763776, 856800, 1321920, 1505520, 3011040, 3084480, 21679488, 22276800, 30844800

Offset: 1

Author

Michel Marcus, Aug 20 2018

Keywords

• nonn

Comments

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

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

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

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

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

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

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

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

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

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

Links

• Giovanni Resta, Table of n, a(n) for n = 1..85 (first 58 terms from Tomohiro Yamada)
• Tomohiro Yamada, Infinitary superperfect numbers, Annales Mathematicae et Informaticae, 47 (2017) pp. 211-218. See Table 1.
• Michel Marcus, Unexhaustive list of terms

Crossrefs

Cf. A049417 (infinitary sigma).

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

Programs

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)));}
isok(n) = frac(a049417(a049417(n))/n) == 0;

Extensions

More terms from Giovanni Resta, Aug 25 2018