A001616 Number of parabolic vertices of Gamma_0(n).
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
Examples
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 + ...
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.
Programs
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Haskell
a001616 n = sum $ map a000010 $ zipWith gcd ds $ reverse ds where ds = a027750_row n -- Reinhard Zumkeller, Jun 23 2013 -
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
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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 *) -
PARI
a(n)=sumdiv(n,d,eulerphi(gcd(d,n/d))); \\ Joerg Arndt, Jul 17 2011
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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
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
Extensions
More terms from Olivier Gérard, Aug 15 1997
Comments