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

Crossrefs

Cf. A037445, A038148, A074847, A268444.

Sequence in context: A299701 A286605 A035149 * A252505 A365173 A366991

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