A038148 Number of 3-infinitary divisors of n: if n = Product p(i)^r(i) and d = Product p(i)^s(i), each s(i) has a digit a <= b in its ternary expansion everywhere that the corresponding r(i) has a digit b, then d is a 3-infinitary-divisor of n.

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

Offset

1,2

Comments

Multiplicative: If e = sum d_k 3^k, then a(p^e) = prod (d_k+1). - Christian G. Bower, May 19 2005

Links

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

Frédéric Chyzak, Ivan Gutman, and Peter Paule, Predicting the number of hexagonal systems with 24 and 25 hexagons, Communications in Mathematical and Computer Chemistry (1999) No. 40, 139-151. See p. 141.

J. O. M. Pedersen, Tables of Aliquot Cycles [Broken link]

J. O. M. Pedersen, Tables of Aliquot Cycles [Via Internet Archive Wayback-Machine]

J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]

Index entries for sequences computed from exponents in factorization of n

Formula

a(1) = 1; for n > 1, a(n) = A006047(A067029(n)) * a(A028234(n)). [After Christian G. Bower's 2005 comment.] - Antti Karttunen, May 28 2017

Example

2^3*3 is a 3-infinitary-divisor of 2^5*3^2 because 2^3*3 = 2^10*3^1 and 2^5*3^2 = 2^12*3^2 in ternary expanded power. All corresponding digits satisfy the condition. 1 <= 1, 0 <= 2, 1 <= 2.

Maple

A038148 := 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, 3) ; a := a*mul(d+1, d=e) ; end do: end if; end proc:
seq(A038148(n), n=1..50) ; # R. J. Mathar, Feb 08 2011

Mathematica

a[1] = 1; a[n_] := (k = 1; Do[k = k * Times @@ (IntegerDigits[f, 3] + 1), {f, FactorInteger[n][[All, 2]]}]; k); Table[a[n], {n, 1, 102}](* Jean-François Alcover, Feb 03 2012, after R. J. Mathar *)

Prog

(PARI)
A006047(n) = { my(m=1, d); while(n, d = (n%3); m *= (1+d); n = (n-d)/3); m; };
A038148(n) = factorback(apply(e -> A006047(e), factorint(n)[, 2])); \\ (After A037445) - Antti Karttunen, May 28 2017
(Scheme) (define (A038148 n) (if (= 1 n) n (* (A006047 (A067029 n)) (A038148 (A028234 n))))) ;; Antti Karttunen, May 28 2017

Crossrefs

Cf. A006047, A037445, A038182, A074848.

Sequence in context: A366147 A367170 A369306 * A366742 A141829 A111336

Adjacent sequences: A038145 A038146 A038147 * A038149 A038150 A038151

Keyword

nonn,nice,easy,mult

Author

Yasutoshi Kohmoto

Extensions

More terms from Naohiro Nomoto, Jun 21 2001

Data section further extended to 105 terms by Antti Karttunen, May 28 2017

Status

approved