A074848 Number of 4-infinitary divisors of n.

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 2, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 4, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 4, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 6, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 4, 2, 4, 2, 12, 4, 4, 4, 8, 2, 12, 4, 6, 4, 4, 4, 8, 2, 6, 6, 9, 2, 8, 2, 8, 8

Offset

1,2

Comments

If n = Product p(i)^r(i) and d = Product p(i)^s(i), each s(i) has a digit a<=b in its 4-ary expansion everywhere that the corresponding r(i) has a digit b, then d is a 4-infinitary-divisor of n.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from exponents in factorization of n

Formula

Multiplicative: If e = sum d_k 4^k, then a(p^e) = prod (d_k+1). - Christian G. Bower, May 19 2005

a(1) = 1; for n > 1, a(n) = A268444(A067029(n)) * a(A028234(n)). [After _Christian G. Bower's 2005 formula.] - Antti Karttunen, May 28 2017

Example

2^4*3 is a 4-infinitary-divisor of 2^5*3^2 because 2^4*3 = 2^10*3^1 and 2^5*3^2 = 2^11*3^2 in 4-ary expanded power. All corresponding digits satisfy the condition. 1<=1, 0<=1, 1<=2.

Maple

A074848 := proc(n) if n= 1 then 1; else ifa := ifactors(n)[2] ; a := 1; for f in ifa do e := convert(op(2, f), base, 4) ; a := a*mul(d+1, d=e) ; end do: end if; end proc:
seq(A074848(n), n=1..70) ; # R. J. Mathar, Feb 08 2011

Mathematica

f[p_, e_] := Times @@ (IntegerDigits[e, 4] + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]  (* Amiram Eldar, Sep 09 2020 *)

Prog

(PARI)
A268444(n) = { my(m=1, d); while(n, d = (n%4); m *= (1+d); n = (n-d)/4); m; };
A074848(n) = factorback(apply(e -> A268444(e), factorint(n)[, 2])) \\ (After A037445) - Antti Karttunen, May 28 2017
(Scheme) (definec (A074848 n) (if (= 1 n) n (* (A268444 (A067029 n)) (A074848 (A028234 n))))) ;; Antti Karttunen, May 28 2017
(Python)
from math import prod
from sympy import factorint
from gmpy2 import digits
def A268444(n):
    s = digits(n, 4)
    return prod((int(d)+1)**s.count(d) for d in '123')
def A074848(n): return prod(A268444(e) for e in factorint(n).values()) # Chai Wah Wu, Apr 24 2025

Crossrefs

Cf. A037445, A038148, A074847, A268444.

Sequence in context: A286605 A035149 A391239 * A252505 A365173 A390555

Adjacent sequences: A074845 A074846 A074847 * A074849 A074850 A074851

Keyword

nonn,mult

Author

Yasutoshi Kohmoto, Sep 10 2002

Extensions

More terms from Antti Karttunen, May 28 2017

Name shortened by Amiram Eldar, Sep 09 2020

Status

approved