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A146564 a(n) is the number of solutions of the equation k*n/(k-n) = c. k,c integers.

%I A146564 #37 Oct 04 2024 00:23:57
%S A146564 1,4,4,7,4,13,4,10,7,13,4,22,4,13,13,13,4,22,4,22,13,13,4,31,7,13,10,
%T A146564 22,4,40,4,16,13,13,13,37,4,13,13,31,4,40,4,22,22,13,4,40,7,22,13,22,
%U A146564 4,31,13,31,13,13,4,67,4,13,22,19,13,40,4,22,13,40,4,52
%N A146564 a(n) is the number of solutions of the equation k*n/(k-n) = c. k,c integers.
%C A146564 In general, if n is a prime p then a(p)=4, and k is from {p-1, p+1, 2*p, p^2+p}.
%C A146564 In general, if n is a squared prime p^2 then a(p^2)=7, and k is from {p^2-p, p^2-1, p^2+1, p^2+p, p^3-p^2, p^3+p^2, p^4+p^2}.
%C A146564 The sequence counts solutions with k>0 and any sign of c, or, alternatively, solutions with c>0 and any sign of k. If solutions were constrained to k>0 and c>0, A048691 would result. - _R. J. Mathar_, Nov 21 2008
%H A146564 Nathaniel Johnston, <a href="/A146564/b146564.txt">Table of n, a(n) for n = 1..10000</a>
%H A146564 Umberto Cerruti, <a href="/A146564/a146564.pdf">Percorsi tra i numeri</a> (in Italian), pages 2-4.
%F A146564 Conjecture: a(n) = A048691(n)+A063647(n). - _R. J. Mathar_, Nov 21 2008 (See Corollary 4 in Cerruti's paper.)
%F A146564 a(n) = Sum_{d|n} psi(2^omega(d)), where psi is A001615 and omega is A001221. - _Enrique Pérez Herrero_, Apr 13 2012
%e A146564 For n=7 we search the number of integer solutions of the equation 7*k/(k-7). This holds for k from {6,8,14,56}. Then a(7)=4. For n=10 we search the number of integer solutions of the equation 10*k/(k-10). This holds for k from {5,6,8,9,11,12,14,15,20,30,35,60,110}. Then a(10)=13.
%p A146564 A146564 := proc(n) local b,d,k,c ; b := numtheory[divisors](n^2) ; kbag := {} ; for d in b do k := d+n ; if k > 0 then kbag := kbag union {k} ; fi ; k := -d+n ; if k > 0 then kbag := kbag union {k} ; fi; end do; RETURN(nops(kbag)) ; end: for n from 1 to 800 do printf("%d,",A146564(n)) ; od: # _R. J. Mathar_, Nov 21 2008
%t A146564 psi[n_] := Module[{pp, ee}, {pp, ee} = Transpose[FactorInteger[n]]; If[Max[pp] == 3, n, Times@@(pp+1) * Times@@(pp^(ee-1))]];
%t A146564 a[n_] := Sum[psi[2^PrimeNu[d]], {d, Divisors[n]}]-1;
%t A146564 a /@ Range[72] (* _Jean-François Alcover_, Jan 18 2020 *)
%o A146564 (PARI)
%o A146564 jordantot(n,k)=sumdiv(n,d,d^k*moebius(n/d));
%o A146564 dedekindpsi(n)=jordantot(n,2)/eulerphi(n);
%o A146564 A146564(n)=sumdiv(n, d, dedekindpsi(2^omega(d)));
%o A146564 for(n=1, 200, print(n" "A146564(n))) \\ _Enrique Pérez Herrero_, Apr 14 2012
%o A146564 (Magma) [# [k:k in {1..n^2+n} diff {n}| IsIntegral(k*n/(k-n))]:n in [1..75]]; // _Marius A. Burtea_, Oct 18 2019
%Y A146564 Cf. A191973.
%K A146564 nonn,easy
%O A146564 1,2
%A A146564 _Ctibor O. Zizka_, Nov 01 2008
%E A146564 Extended beyond a(11) by _R. J. Mathar_, Nov 21 2008