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A146564 a(n) is the number of solutions of the equation k*n/(k-n) = c. k,c integers. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

In general if n is a prime then a(p)=4, k is from {p-1,p+1,2*p,p^2+p}, if n =p^2 then a(p^2)=7, 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. [From 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

PROG

(PARI)

\\ Enrique Pérez Herrero.- Apr 14 2012

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)))

CROSSREFS

Cf. A191973.

Sequence in context: A204156 A163106 A258972 * A048785 A271781 A243454

Adjacent sequences:  A146561 A146562 A146563 * A146565 A146566 A146567

KEYWORD

nonn,easy

AUTHOR

Ctibor O. Zizka, Nov 01 2008

EXTENSIONS

Extended beyond a(11) by R. J. Mathar, Nov 21 2008

STATUS

approved

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Last modified January 23 01:55 EST 2018. Contains 298093 sequences.