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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. 6

%I

%S 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,

%T 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,

%U 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

%N 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.

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

%H Antti Karttunen, <a href="/A038148/b038148.txt">Table of n, a(n) for n = 1..10000</a>

%H J. O. M. Pedersen, <a href="http://amicable.homepage.dk/tables.htm">Tables of Aliquot Cycles</a> [Broken link]

%H J. O. M. Pedersen, <a href="http://web.archive.org/web/20140502102524/http://amicable.homepage.dk/tables.htm">Tables of Aliquot Cycles</a> [Via Internet Archive Wayback-Machine]

%H J. O. M. Pedersen, <a href="/A063990/a063990.pdf">Tables of Aliquot Cycles</a> [Cached copy, pdf file only]

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F 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

%e 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.

%p A038148 := proc(n) if n= 1 then 1; else ifa := ifactors(n)[2] ;

%p 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:

%p seq(A038148(n),n=1..50) ; # _R. J. Mathar_, Feb 08 2011

%t 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-Fran├žois Alcover_, Feb 03 2012, after _R. J. Mathar_ *)

%o (PARI)

%o A006047(n) = { my(m=1, d); while(n, d = (n%3); m *= (1+d); n = (n-d)/3); m; };

%o A038148(n) = factorback(apply(e -> A006047(e), factorint(n)[, 2])); \\ (After A037445) - _Antti Karttunen_, May 28 2017

%o (Scheme) (define (A038148 n) (if (= 1 n) n (* (A006047 (A067029 n)) (A038148 (A028234 n))))) ;; _Antti Karttunen_, May 28 2017

%Y Cf. A006047, A037445, A038182, A074848.

%K nonn,nice,easy,mult

%O 1,2

%A _Yasutoshi Kohmoto_

%E More terms from _Naohiro Nomoto_, Jun 21 2001

%E Data section further extended to 105 terms by _Antti Karttunen_, May 28 2017

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Last modified October 17 12:33 EDT 2019. Contains 328112 sequences. (Running on oeis4.)