A001616 Number of parabolic vertices of Gamma_0(n). (Formerly M0247 N0087)

1, 2, 2, 3, 2, 4, 2, 4, 4, 4, 2, 6, 2, 4, 4, 6, 2, 8, 2, 6, 4, 4, 2, 8, 6, 4, 6, 6, 2, 8, 2, 8, 4, 4, 4, 12, 2, 4, 4, 8, 2, 8, 2, 6, 8, 4, 2, 12, 8, 12, 4, 6, 2, 12, 4, 8, 4, 4, 2, 12, 2, 4, 8, 12, 4, 8, 2, 6, 4, 8, 2, 16, 2, 4, 12, 6, 4, 8, 2, 12, 12, 4, 2, 12, 4, 4, 4, 8, 2, 16, 4, 6, 4, 4, 4, 16

Offset

1,2

Comments

Number of inequivalent cusps of Gamma_0(n). - Michael Somos, May 08 2015

References

B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 102.

Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (4).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

N. J. A. Sloane, Table of n, a(n) for n = 1..1000

Harriet Fell, Morris Newman, and Edward Ordman, Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B (1963), 61-68.

Steven R. Finch, Modular forms on SL_2(Z), December 28, 2005. [Cached copy, with permission of the author]

Steven R. Finch, Primitive Cusp Forms, April 27, 2009. [Cached copy, with permission of the author]

László Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv preprint arXiv:1310.7053 [math.NT], 2013-2014.

Formula

a(n) = Sum_{d|n} phi(gcd(d,n/d)), where phi() is Euler's totient function. - Joerg Arndt, Jul 17 2011

Multiplicative with a(p^e) = p^[e/2] + p^[(e-1)/2]. - David W. Wilson, Sep 01 2001

Example

G.f. = x + 2*x^2 + 2*x^3 + 3*x^4 + 2*x^5 + 4*x^6 + 2*x^7 + 4*x^8 + 4*x^9 + ...

Maple

with(numtheory); nupara := proc (n) local b, d; b := 0; for d to n do if modp(n, d) = 0 then b := b+eval(phi(gcd(d, n/d))) fi od; b end: # Gene Ward Smith, May 22 2006

Mathematica

Table[ Plus@@Map[ EulerPhi[ GCD[ #1, n/#1 ] ]&, Select[ Range[ n ], (Mod[ n, #1 ]==0)& ] ], {n, 1, 100} ] (* Olivier Gérard, Aug 15 1997 *)
a[ n_] := If[ n < 1, 0, Sum[ EulerPhi[ GCD[ d, n/d]], {d, Divisors@n}]]; (* Michael Somos, May 08 2015 *)
f[p_, e_] := p^Floor[e/2] + p^Floor[(e-1)/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 28 2023 *)

Prog

(PARI)  a(n)=sumdiv(n, d, eulerphi(gcd(d, n/d))); \\ Joerg Arndt, Jul 17 2011
(Haskell)
a001616 n = sum $ map a000010 $ zipWith gcd ds $ reverse ds
            where ds = a027750_row n
-- Reinhard Zumkeller, Jun 23 2013

Crossrefs

Cf. A027750, A000010, A027748, A124010.

Sequence in context: A144372 A182861 A049238 * A257599 A366620 A218702

Adjacent sequences: A001613 A001614 A001615 * A001617 A001618 A001619

Keyword

nonn,easy,nice,mult

Author

N. J. A. Sloane

Extensions

More terms from Olivier Gérard, Aug 15 1997

Status

approved