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

1,2

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

Prog

(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;

Crossrefs

Cf. A049417 (infinitary sigma).

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

Sequence in context: A050569 A352698 A318175 * A047469 A283774 A037456

Adjacent sequences: A318179 A318180 A318181 * A318183 A318184 A318185

Keyword

nonn

Author

Michel Marcus, Aug 20 2018

Extensions

More terms from Giovanni Resta, Aug 25 2018

Status

approved