A146564 - OEIS

A146564

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

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, 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, 4, 31, 13, 31, 13, 13, 4, 67, 4, 13, 22, 19, 13, 40, 4, 22, 13, 40, 4, 52

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OFFSET

1,2

COMMENTS

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}.

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}.

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

LINKS

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Umberto Cerruti, Percorsi tra i numeri (in Italian), pages 2-4.

FORMULA

Conjecture: a(n) = A048691(n)+A063647(n). - R. J. Mathar, Nov 21 2008 (See Corollary 4 in Cerruti's paper.)

a(n) = Sum_{d|n} psi(2^omega(d)), where psi is A001615 and omega is A001221. - Enrique Pérez Herrero, Apr 13 2012

EXAMPLE

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.

MAPLE

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

MATHEMATICA

psi[n_] := Module[{pp, ee}, {pp, ee} = Transpose[FactorInteger[n]]; If[Max[pp] == 3, n, Times@@(pp+1) * Times@@(pp^(ee-1))]];

a[n_] := Sum[psi[2^PrimeNu[d]], {d, Divisors[n]}]-1;

a /@ Range[72] (* Jean-François Alcover, Jan 18 2020 *)

PROG

(PARI)

jordantot(n, k)=sumdiv(n, d, d^k*moebius(n/d));

dedekindpsi(n)=jordantot(n, 2)/eulerphi(n);

A146564(n)=sumdiv(n, d, dedekindpsi(2^omega(d)));

for(n=1, 200, print(n" "A146564(n))) \\ Enrique Pérez Herrero, Apr 14 2012

(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

CROSSREFS