

A001616


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


25



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
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OFFSET

1,2


COMMENTS

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


REFERENCES

B. Schoeneberg, Elliptic Modular Functions, SpringerVerlag, NY, 1974, p. 102.
G. 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
Fell, Harriet; Newman, Morris; Ordman, Edward; Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B 1963 6168.
S. R. Finch, Modular forms on SL_2(Z), December 28, 2005. [Cached copy, with permission of the author]
S. R. Finch, Primitive Cusp Forms, April 27, 2009. [Cached copy, with permission of the author]
L. Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv preprint arXiv:1310.7053, 2013


FORMULA

a(n) = sum(d divides 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^[(e1)/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 *)


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 A218702 A324181
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



