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%I #24 Sep 09 2020 03:40:14
%S 1,3,4,7,6,12,8,15,13,18,12,28,14,24,24,31,18,39,20,42,32,36,24,60,31,
%T 42,40,56,30,72,32,33,48,54,48,91,38,60,56,90,42,96,44,84,78,72,48,
%U 124,57,93,72,98,54,120,72,120,80,90,60,168,62,96,104,99,84,144,68,126,96,144,72,195,74,114,124
%N Sum of 5-infinitary divisors of n.
%C If n=Product p_i^r_i and d=Product p_i^s_i, each s_i has a digit a<=b in its 5-ary expansion everywhere that the corresponding r_i has a digit b, then d is a 5-infinitary-divisor of n.
%H Reinhard Zumkeller, <a href="/A097863/b097863.txt">Table of n, a(n) for n = 1..10000</a>
%F Denote by P_5={p^5^k} the two-parameter set when k=0,1,... and p runs prime values. Then every n has a unique representation of the form n=prod q_i prod (r_j)^2 prod (s_k)^3 prod (t_m)^4, where q_i, r_j, s_k, t_m are distinct elements of P_5. Using this representation, we have a(n)=prod (q_i+1)prod ((r_j)^2+r_j+1)prod ((s_k)^3+(s_k)^2+s_k+1) prod ((t_m)^4+(t_m)^3+(t_m)^2+t_m+1). - _Vladimir Shevelev_, May 08 2013
%e a(32) = a(2^10) = 2^10 + 2^0 = 32 + 1 = 33, in 5-ary expansion. This is the first term which is different from sigma(n).
%p A097863 := proc(n) option remember; local ifa, a, p, e, d, k ; ifa := ifactors(n)[2] ; a := 1 ; if nops(ifa) = 1 then p := op(1, op(1, ifa)) ; e := op(2, op(1, ifa)) ; d := convert(e, base, 5) ; for k from 0 to nops(d)-1 do a := a*(p^((1+op(k+1, d))*5^k)-1)/(p^(5^k)-1) ; end do: else for d in ifa do a := a*procname( op(1, d)^op(2, d)) ; end do: return a; end if; end proc:
%t f[p_, e_] := Module[{d = IntegerDigits[e, 5]}, m = Length[d]; Product[(p^((d[[j]] + 1)*5^(m - j)) - 1)/(p^(5^(m - j)) - 1), {j, 1, m}]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 09 2020 *)
%o (Haskell) following Bower and Harris, cf. A049418:
%o a097863 1 = 1
%o a097863 n = product $ zipWith f (a027748_row n) (a124010_row n) where
%o f p e = product $ zipWith div
%o (map (subtract 1 . (p ^)) $
%o zipWith (*) a000351_list $ map (+ 1) $ a031235_row e)
%o (map (subtract 1 . (p ^)) a000351_list)
%o -- _Reinhard Zumkeller_, Sep 18 2015
%Y Cf. A049417, A049418, A074847, A097464.
%Y Cf. A000351, A031235, A027748, A124010.
%K nonn,mult
%O 1,2
%A _Yasutoshi Kohmoto_
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