cp's OEIS Frontend

A074848 Number of 4-infinitary divisors of n.

%I A074848 #31 Apr 25 2025 12:41:08
%S A074848 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,
%T A074848 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,
%U A074848 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
%N A074848 Number of 4-infinitary divisors of n.
%C A074848 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.
%H A074848 Antti Karttunen, <a href="/A074848/b074848.txt">Table of n, a(n) for n = 1..10000</a>
%H A074848 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A074848 Multiplicative: If e = sum d_k 4^k, then a(p^e) = prod (d_k+1). - _Christian G. Bower_, May 19 2005
%F A074848 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
%e A074848 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.
%p A074848 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:
%p A074848 seq(A074848(n),n=1..70) ; # _R. J. Mathar_, Feb 08 2011
%t A074848 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 *)
%o A074848 (PARI)
%o A074848 A268444(n) = { my(m=1, d); while(n, d = (n%4); m *= (1+d); n = (n-d)/4); m; };
%o A074848 A074848(n) = factorback(apply(e -> A268444(e), factorint(n)[, 2])) \\ (After A037445) - _Antti Karttunen_, May 28 2017
%o A074848 (Scheme) (definec (A074848 n) (if (= 1 n) n (* (A268444 (A067029 n)) (A074848 (A028234 n))))) ;; _Antti Karttunen_, May 28 2017
%o A074848 (Python)
%o A074848 from math import prod
%o A074848 from sympy import factorint
%o A074848 from gmpy2 import digits
%o A074848 def A268444(n):
%o A074848     s = digits(n,4)
%o A074848     return prod((int(d)+1)**s.count(d) for d in '123')
%o A074848 def A074848(n): return prod(A268444(e) for e in factorint(n).values()) # _Chai Wah Wu_, Apr 24 2025
%Y A074848 Cf. A037445, A038148, A074847, A268444.
%K A074848 nonn,mult
%O A074848 1,2
%A A074848 _Yasutoshi Kohmoto_, Sep 10 2002
%E A074848 More terms from _Antti Karttunen_, May 28 2017
%E A074848 Name shortened by _Amiram Eldar_, Sep 09 2020