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

Author

Yasutoshi Kohmoto

Keywords

• nonn
• nice
• easy
• mult

Comments

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

Examples

			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.
		

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

Crossrefs

Cf. A006047, A037445, A038182, A074848.

Programs

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 *)

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

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

Extensions

More terms from Naohiro Nomoto, Jun 21 2001

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