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
Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from N. J. A. Sloane)
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
(Python)
from math import prod
from sympy import factorint
def A001616(n): return prod(p**(e>>1)+p**(e-1>>1) for p, e in factorint(n).items()) # Chai Wah Wu, Jul 05 2024
CROSSREFS
KEYWORD
nonn,easy,nice,mult
AUTHOR
EXTENSIONS
More terms from Olivier Gérard, Aug 15 1997
STATUS
approved