A001616 -id:A001616 - cp's OEIS frontend

A054728 a(n) is the smallest level N such that genus of modular curve X_0(N) is n (or -1 if no such curve exists).

1, 11, 22, 30, 38, 42, 58, 60, 74, 66, 86, 78, 106, 105, 118, 102, 134, 114, 223, 132, 166, 138, 188, 156, 202, 168, 214, 174, 236, 186, 359, 204, 262, 230, 278, 222, 298, 240, 314, 246, 326, 210, 346, 270, 358, 282, 557, 306, 394, 312, 412, 318

Offset: 0

Janos A. Csirik, Apr 21 2000

sign

a(150) = -1, a(n) > 0 for 0<=n<=149.

a(9999988) = 119999861 is the largest value in the first 1+10^7 terms of the sequence. - Gheorghe Coserea, May 24 2016

J. A. Csirik, The genus of X_0(N) is not 150, preprint, 2000.

Gheorghe Coserea, Table of n, a(n) for n = 0..200010
János A. Csirik, Joseph L. Wetherell, Michael E. Zieve, On the genera of X_0(N), arXiv:math/0006096 [math.NT], 2000.

Cf. A001617, A054727, A054729.

a1617[n_] := If[n<1, 0, 1+Sum[MoebiusMu[d]^2 n/d/12 - EulerPhi[GCD[d, n/d]]/2, {d, Divisors[n]}] - Count[(#^2 - # + 1)/n& /@ Range[n], ?IntegerQ]/3 - Count[(#^2+1)/n& /@ Range[n], ?IntegerQ]/4];
seq[n_] := Module[{inv, bnd}, inv = Table[-1, {n+1}]; bnd = 12n + 18 Floor[Sqrt[n]] + 100; For[k = 1, k <= bnd, k++, g = a1617[k]; If[g <= n && inv[[g+1]] == -1, inv[[g+1]] = k]]; inv];
seq[51] (* Jean-François Alcover, Nov 20 2018, after Gheorghe Coserea and Michael Somos in A001617 *)

A000089(n) = {
  if (n%4 == 0 || n%4 == 3, return(0));
  if (n%2 == 0, n \= 2);
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, if (f[k,1] % 4 == 3, 0, 2));
};
A000086(n) = {
  if (n%9 == 0 || n%3 == 2, return(0));
  if (n%3 == 0, n \= 3);
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, if (f[k,1] % 3 == 2, 0, 2));
};
A001615(n) = {
  my(f = factor(n), fsz = matsize(f)[1],
     g = prod(k=1, fsz, (f[k,1]+1)),
     h = prod(k=1, fsz, f[k,1]));
  return((n*g)\h);
};
A001616(n) = {
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, f[k,1]^(f[k,2]\2) + f[k,1]^((f[k,2]-1)\2));
};
A001617(n) = 1 + A001615(n)/12 - A000089(n)/4 - A000086(n)/3 - A001616(n)/2;
seq(n) = {
  my(inv = vector(n+1,g,-1), bnd = 12*n + 18*sqrtint(n) + 100, g);
  for (k = 1, bnd, g = A001617(k);
       if (g <= n && inv[g+1] == -1, inv[g+1] = k));
  return(inv);
};
seq(51)  \\ Gheorghe Coserea, May 21 2016

A001617(a(A001617(n))) = A001617(n) and a(A054729(n)) = -1 for all n>=1. - Gheorghe Coserea, May 22 2016

A091404 Numbers n such that genus of group Gamma_0(n) is 2.

Original entry on oeis.org

22, 23, 26, 28, 29, 31, 37, 50

Offset: 1

N. J. A. Sloane, Mar 02 2004

I assume it is known that there are no further terms? A reference for this would be nice.

B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 103.
G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see Prop. 1.40 and 1.43.

Cf. A001617, A001615, A000089, A000086, A001616, A091401, A091403.

a89[n_] := a89[n] = Product[{p, e} = pe; Which[p < 3 && e == 1, 1, p == 2 && e > 1, 0, Mod[p, 4] == 1, 2, Mod[p, 4] == 3, 0, True, a89[p^e]], {pe, FactorInteger[n]}];
a86[n_] := a86[n] = Product[{p, e} = pe; Which[p == 1 || p == 3 && e == 1, 1, p == 3 && e > 1, 0, Mod[p, 3] == 1, 2, Mod[p, 3] == 2, 0, True, a86[p^e]], {pe, FactorInteger[n]}];
a1615[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}];
a1616[n_] := Sum[EulerPhi[GCD[d, n/d]], {d, Divisors[n]}];
a1617[n_] := 1 + a1615[n]/12 - a89[n]/4 - a86[n]/3 - a1616[n]/2;
Position[Array[a1617, 100], 2] // Flatten (* Jean-François Alcover, Oct 19 2018 *)

Numbers n such that A001617(n) = 2.

A332713 a(n) = Sum_{d|n} phi(d/gcd(d, n/d)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 7, 8, 10, 11, 12, 13, 14, 15, 13, 17, 16, 19, 20, 21, 22, 23, 21, 22, 26, 22, 28, 29, 30, 31, 24, 33, 34, 35, 32, 37, 38, 39, 35, 41, 42, 43, 44, 40, 46, 47, 39, 44, 44, 51, 52, 53, 44, 55, 49, 57, 58, 59, 60, 61, 62, 56, 46, 65, 66, 67, 68, 69, 70

Offset: 1

Ilya Gutkovskiy, Feb 20 2020

Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

Cf. A000010, A001616, A010052, A046790 (numbers n such that a(n) < n), A055653, A061884, A078779 (fixed points), A332619, A332686, A332712.

Table[Sum[EulerPhi[d/GCD[d, n/d]], {d, Divisors[n]}], {n, 1, 70}]
A055653[n_] := Sum[Boole[GCD[d, n/d] == 1] EulerPhi[d], {d, Divisors[n]}]; a[n_] := Sum[Boole[IntegerQ[(n/d)^(1/2)]] A055653[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]

a(n) = sumdiv(n, d, eulerphi(d/gcd(d, n/d))); \\ Michel Marcus, Feb 20 2020

Dirichlet g.f.: zeta(s) * zeta(2*s) * zeta(s - 1) * Product_{p prime} (1 - p^(-s) + p^(-2*s) - p^(1 - 2*s)).
a(n) = Sum_{d|n} phi(lcm(d, n/d)/d).
a(n) = Sum_{d|n} A010052(n/d) * A055653(d).
Sum_{k=1..n} a(k) ~ c * Pi^6 * n^2 / 1080, where c = A330523 = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.5358961538283379998085... - Vaclav Kotesovec, Feb 22 2020
From Richard L. Ollerton, May 10 2021: (Start)
a(n) = Sum_{k=1..n} phi(gcd(n,k)/gcd(gcd(n,k),n/gcd(n,k)))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} phi(n/gcd(n,k)/gcd(gcd(n,k),n/gcd(n,k)))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} phi(lcm(gcd(n,k),n/gcd(n,k))/gcd(n,k))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} phi(lcm(gcd(n,k),n/gcd(n,k))*gcd(n,k)/n)/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} A010052(gcd(n,k))*A055653(n/gcd(n,k))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} A010052(n/gcd(n,k))*A055653(gcd(n,k))/phi(n/gcd(n,k)). (End)

A124316 a(n) = Sum_{d|n} sigma(gcd(d,n/d)), where sigma is the sum of divisors function, A000203.

Original entry on oeis.org

1, 2, 2, 5, 2, 4, 2, 8, 6, 4, 2, 10, 2, 4, 4, 15, 2, 12, 2, 10, 4, 4, 2, 16, 8, 4, 10, 10, 2, 8, 2, 22, 4, 4, 4, 30, 2, 4, 4, 16, 2, 8, 2, 10, 12, 4, 2, 30, 10, 16, 4, 10, 2, 20, 4, 16, 4, 4, 2, 20, 2, 4, 12, 37, 4, 8, 2, 10, 4, 8, 2, 48, 2, 4, 16, 10, 4, 8, 2, 30, 23, 4, 2, 20, 4, 4, 4, 16, 2, 24

Offset: 1

Robert G. Wilson v, Sep 30 2006

Apparently multiplicative and the inverse Mobius transform of A069290. - R. J. Mathar, Feb 07 2011

Antti Karttunen, Table of n, a(n) for n = 1..10000
Ekkehard Krätzel, Werner Georg Nowak, and László Tóth, On certain arithmetic functions involving the greatest common divisor, Cent. Eur. J. Math., 10 (2012), 761-774.
Manfred Kühleitner and Werner Georg Nowak, On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions, Cent. Eur. J. Math., Vol. 11, No. 3 (2013), pp. 477-486; arXiv preprint, arXiv:1204.1146 [math.NT], 2012.
László Tóth, Multiplicative arithmetic functions of several variables: a survey, in: T. Rassias, P. Pardalos (eds.) Mathematics Without Boundaries, Springer, New York, NY, 2024; arXiv preprint, arXiv:1310.7053 [math.NT], 2013-2014.

Cf. A000203, A001616, A001620, A069290, A061884, A082633, A124315.

A124316 := proc(n) local a,d;
  a := 0 ;
  for d in numtheory[divisors](n) do
     igcd(d,n/d) ;
     a := a+numtheory[sigma](%) ;
   end do:
   a;
end proc: # R. J. Mathar, Apr 14 2011

Table[Plus @@ Map[DivisorSigma[1, GCD[ #, n/# ]] &, Divisors@n], {n, 90}]
f[p_, e_] := (If[OddQ[e], 2*p^((e+3)/2), p^(e/2 + 1)*(p+1)] - (e+3)*p + e + 1)/(p-1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Mar 28 2024 *)

a(n) = sumdiv(n, d, sigma(gcd(d, n/d))); \\ Michel Marcus, Feb 13 2016

from sympy import divisors, divisor_sigma, gcd
def a(n): return sum([divisor_sigma(gcd(d, n/d)) for d in divisors(n)]) # Indranil Ghosh, May 25 2017

From Amiram Eldar, Mar 28 2024: (Start)
Multiplicative with a(p^e) = (p^(e/2 + 1)*(p+1) - (e+3)*p + e + 1)/(p-1)^2, if e is even, and (2*p^((e+3)/2) - (e+3)*p + e + 1)/(p-1)^2 if e is odd.
Dirichlet g.f.: zeta(s)^2 * zeta(2*s-1).
Sum_{k=1..n} a(k) = (log(n)^2/4 + (2*gamma - 1/2)*log(n) + 5*gamma^2/2 - 2*gamma - 3*gamma_1 + 1/2) * n + O(n^(2/3)*log(n)^(16/9)), where gamma is Euler's constant (A001620) and gamma_1 is the first Stieltjes constant (A082633) (Krätzel et al., 2012). (End)

A054730 Odd n such that genus of modular curve X_0(N) is never equal to n.

Original entry on oeis.org

49267, 74135, 94091, 96463, 102727, 107643, 118639, 138483, 145125, 181703, 182675, 208523, 221943, 237387, 240735, 245263, 255783, 267765, 269627, 272583, 277943, 280647, 283887, 286815, 309663, 313447, 322435, 326355, 336675, 347823, 352719

Offset: 1

Janos A. Csirik, Apr 21 2000

nonn

There are 4329 odd integers in the sequence less than 10^7. - Gheorghe Coserea, May 23 2016

J. A. Csirik, The genus of X_0(N) is not 150, preprint, 2000.

Gheorghe Coserea, Table of n, a(n) for n = 1..4329
J. A. Csirik, M. Zieve, and J. Wetherell, On the genera of X0(N), arXiv:math/0006096 [math.NT], 2000.

Cf. A054726, A054727, A054729.

A000089(n) = {
  if (n%4 == 0 || n%4 == 3, return(0));
  if (n%2 == 0, n \= 2);
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, if (f[k, 1] % 4 == 3, 0, 2));
};
A000086(n) = {
  if (n%9 == 0 || n%3 == 2, return(0));
  if (n%3 == 0, n \= 3);
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, if (f[k, 1] % 3 == 2, 0, 2));
};
A001615(n) = {
  my(f = factor(n), fsz = matsize(f)[1],
     g = prod(k=1, fsz, (f[k, 1]+1)),
     h = prod(k=1, fsz, f[k, 1]));
  return((n*g)\h);
};
A001616(n) = {
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, f[k, 1]^(f[k, 2]\2) + f[k, 1]^((f[k, 2]-1)\2));
};
A001617(n) = 1 + A001615(n)/12 - A000089(n)/4 - A000086(n)/3 - A001616(n)/2;
scan(n) = {
  my(inv = vector(n+1, g, -1), bnd = 12*n + 18*sqrtint(n) + 100, g);
  for (k = 1, bnd, g = A001617(k);
       if (g <= n && inv[g+1] == -1, inv[g+1] = k));
  select(x->(x%2==1), apply(x->(x-1), Vec(select(x->x==-1, inv, 1))));
};
scan(400*1000)

More terms from Gheorghe Coserea, May 23 2016

Offset corrected by Gheorghe Coserea, May 23 2016

A111248 Dimension of the space of weight 2 modular forms for Gamma_0(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 3, 3, 2, 5, 1, 4, 4, 5, 2, 7, 2, 6, 4, 5, 3, 8, 5, 5, 6, 7, 3, 10, 3, 8, 6, 6, 6, 12, 3, 7, 6, 10, 4, 12, 4, 9, 10, 8, 5, 14, 8, 13, 8, 10, 5, 15, 8, 12, 8, 9, 6, 18, 5, 10, 12, 14, 8, 16, 6, 12, 10, 16, 7, 20, 6, 11, 16, 13, 10, 18, 7, 18, 15, 12, 8, 22, 10, 13, 12, 16, 8, 26

Offset: 1

Steven Finch, Oct 31 2005

nonn

Equivalently, this is the dimension of the space of level n, weight 2 modular forms. - Steven Finch, Apr 03 2009

Klaus Brockhaus, Table of n, a(n) for n = 1..10000
S. R. Finch, Modular forms on SL_2(Z), December 28, 2005. [Cached copy, with permission of the author]

Cf. A001616, A001617. [Steven Finch, Apr 03 2009]

Cf. A006571.

[ Dimension(ModularForms(Gamma0(n), 2)) : n in [1..100] ]; // Klaus Brockhaus, Mar 10 2011

a(n) = A001617(n) + A001616(n) - 1. - Steven Finch, Apr 03 2009

More terms from Steven Finch, Apr 03 2009

A273445 a(n) is the number of solutions of the equation n = A001617(k).

Original entry on oeis.org

15, 12, 8, 11, 7, 14, 4, 13, 7, 12, 4, 15, 4, 9, 6, 10, 5, 16, 2, 20, 3, 14, 7, 11, 2, 13, 5, 11, 3, 14, 3, 9, 6, 13, 3, 17, 3, 14, 4, 10, 4, 20, 3, 15, 3, 12, 1, 15, 2, 20, 4, 11, 3, 13, 3, 16, 3, 12, 3, 15, 3, 12, 5, 9, 4, 15, 2, 14, 5, 17, 3, 13

Offset: 0

Gheorghe Coserea, May 22 2016

nonn

The zeros of the sequence are given by A054729. The first five zeros of the sequence have indexes 150, 180, 210, 286, 304.

			For n = 0 the a(0) = 15 solutions are:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25 (A091401).
For n = 1 the a(1) = 12 solutions are:
11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, 49 (A091403).
For n = 2 the a(2) = 8 solutions are:
22, 23, 26, 28, 29, 31, 37, 50 (A091404).

Gheorghe Coserea, Table of n, a(n) for n = 0..100001
J. A. Csirik, M. Zieve, and J. Wetherell, On the genera of X0(N), arXiv:math/0006096 [math.NT], 2000.

Cf. A001617, A054729, A091401, A091403, A091404.

(* b = A001617 *) nmax = 71;
b[n_] := b[n] = If[n < 1, 0, 1 + Sum[ MoebiusMu[ d]^2 n/d / 12 - EulerPhi[ GCD[ d, n/d]] / 2, {d, Divisors[n]}] - Count[(#^2 - # + 1)/n& /@ Range[n], ?IntegerQ]/3 -Count[(#^2 + 1)/n& /@ Range[n], ?IntegerQ]/4];
Clear[f];
f[m_] := f[m] = Module[{}, A001617 = Array[b, m]; a[n_] := Count[A001617, n]; Table[a[n], {n, 0, nmax}]];
f[m = nmax]; f[m = m + nmax];
While[Print["m = ", m]; f[m] != f[m - nmax], m = m + nmax];
A273445 = f[m] (* Jean-François Alcover, Dec 16 2018, using Michael Somos' code for A001617 *)

A000089(n) = {
  if (n%4 == 0 || n%4 == 3, return(0));
  if (n%2 == 0, n \= 2);
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, if (f[k,1] % 4 == 3, 0, 2));
};
A000086(n) = {
  if (n%9 == 0 || n%3 == 2, return(0));
  if (n%3 == 0, n \= 3);
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, if (f[k,1] % 3 == 2, 0, 2));
};
A001615(n) = {
  my(f = factor(n), fsz = matsize(f)[1],
     g = prod(k=1, fsz, (f[k,1]+1)),
     h = prod(k=1, fsz, f[k,1]));
  return((n*g)\h);
};
A001616(n) = {
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, f[k,1]^(f[k,2]\2) + f[k,1]^((f[k,2]-1)\2));
};
A001617(n) = 1 + A001615(n)/12 - A000089(n)/4 - A000086(n)/3 - A001616(n)/2;
seq(n) = {
  my(a = vector(n+1,g,0), bnd = 12*n + 18*sqrtint(n) + 100, g);
  for (k = 1, bnd, g = A001617(k);
       if (g <= n, a[g+1]++));
  return(a);
};
seq(72)

a(n) = card {k, n = A001617(k)}.

A372866 a(n) is the sum, over all positive integers x, y such that x*y <= n, of phi(gcd(x,y)).

Original entry on oeis.org

1, 3, 5, 8, 10, 14, 16, 20, 24, 28, 30, 36, 38, 42, 46, 52, 54, 62, 64, 70, 74, 78, 80, 88, 94, 98, 104, 110, 112, 120, 122, 130, 134, 138, 142, 154, 156, 160, 164, 172, 174, 182, 184, 190, 198, 202, 204, 216, 224, 236, 240, 246, 248, 260, 264, 272, 276, 280

Offset: 1

Jeffrey Shallit, Jul 04 2024

nonn

A number-theoretic sum involving the Euler phi function and gcd's.

Theorem 1.1 of Kiuchi and Tsuruta (2024) gives an estimate for a(n).

			For n = 3, the (a,b) pairs that appear in the sum are (1,1), (1,2), (1,3), (2,1), (3,1); the gcd of all is 1, and the sum of the phi-function at these 5 values of 1 is 5.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
I. Kiuchi and Y. Tsuruta, On sums involving the Euler totient function, Bull. Aust. Math. Soc. 109 (2024), 486-497.

Partial sums of A001616.

Cf. A000010.

a:= proc(n) option remember; uses numtheory; `if`(n<1, 0,
      a(n-1)+add(phi(igcd(d, n/d)), d=divisors(n)))
    end:
seq(a(n), n=1..58);  # Alois P. Heinz, Jul 04 2024

a[n_] := a[n] = DivisorSum[n, EulerPhi[GCD[#, n/#]] &] + a[n - 1]; a[1] = 1; Array[a, 120] (* Michael De Vlieger, Jul 04 2024 *)

a(n) = sum(k=1, n, sumdiv(k, d, eulerphi(gcd(d, k/d)))); \\ Michel Marcus, Jul 04 2024

from math import gcd
from functools import cache
from sympy import divisors, totient as phi
@cache
def a(n):
    if n == 1: return 1
    return a(n-1) + sum(phi(gcd(a, (b:=n//a))) for a in divisors(n))
print([a(n) for n in range(1, 61)]) # Michael S. Branicky, Jul 04 2024

from math import prod
from itertools import count, islice
from sympy import factorint
def A372866_gen(): # generator of terms
    c = 0
    for n in count(1):
        c += prod(p**(e>>1)+p**(e-1>>1) for p, e in factorint(n).items())
        yield c
A372866_list = list(islice(A372866_gen(),20)) # Chai Wah Wu, Jul 05 2024

A130107 Möbius transform of A063659.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 6, 3, 5, 4, 10, 2, 12, 6, 8, 6, 16, 5, 18, 4, 12, 10, 22, 6, 19, 12, 16, 6, 28, 8, 30, 12, 20, 16, 24, 5, 36, 18, 24, 12, 40, 12, 42, 10, 20, 22, 46, 12, 41, 19, 32, 12, 52, 16, 40, 18, 36, 28, 58, 8, 60, 30, 30, 24, 48, 20, 66, 16, 44, 24

Offset: 1

Gary W. Adamson, May 07 2007

Double inverse Möbius transform of A130107 = A001615: (1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, ...).

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

Enrique Pérez Herrero, Table of n, a(n) for n = 1..2000
Steven R. Finch, Primitive Cusp Forms, April 27, 2009. [Cached copy, with permission of the author]

Cf. A130107, A054525, A063659, A001615.

with(numtheory): A130107 := proc(n) local dp, mtdp, d, p;
dp := n -> n*mul((1+1/p), p=factorset(n));
mtdp := n -> add(mobius(n/d)*dp(d), d=divisors(n));
add(mobius(n/d)*mtdp(d), d=divisors(n)) end:
seq(A130107(n), n=1..76); # Peter Luschny, Apr 06 2014

JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/#]&]/;(n>0)&&IntegerQ[n];
DedekindPsi[n_]:=JordanTotient[n,2]/EulerPhi[n];
A063659[n_]:=DivisorSum[n,MoebiusMu[n/#]*DedekindPsi[#]&];
A130107[n_]:=DivisorSum[n,MoebiusMu[n/#]*A063659[#]&]:
Table[A130107[n],{n,1,30}]
(* Enrique Pérez Herrero, Apr 03 2014 *)
a[ n_] := If[ n < 2, Boole[n == 1], Times @@ (Which[ #2 == 1, # - 1, #2 == 2, #^2 - # - 1, True, #^(#2 - 3) (#^2 - 1) (# - 1)] &) @@@ FactorInteger[n]]; (* Michael Somos, Jun 17 2015 *)

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k,]; if( e==1, p - 1, e==2, p^2 - p - 1, p^(e-3) * (p^2 - 1) * (p-1))))}; /* Michael Somos, Jun 17 2015 */

A054525 * A063659, (1, 2, 3, 3, 5, 6, 7, 6, 8, 10, ...).
Multiplicative with a(p^e) = p-1 if e=1,  a(p^e) = p^2-p-1 if e=2, a(p^e) = p^(e-3)*(p+1)*(p-1)^2. - Enrique Pérez Herrero, Apr 03 2014
Dirichlet g.f.: zeta(s-1) / (zeta(s) * zeta(2s)). - Álvar Ibeas, Mar 07 2015
Sum_{k=1..n} a(k) ~ 270*n^2 / Pi^6. - Vaclav Kotesovec, Jan 11 2019

More terms from Enrique Pérez Herrero, Apr 03 2014

A159633 Dimension of Eisenstein subspace of the space of modular forms of weight k/2, level 4*n and trivial character, where k>=5 is odd.

Original entry on oeis.org

2, 3, 4, 6, 4, 6, 4, 8, 8, 6, 4, 12, 4, 6, 8, 12, 4, 12, 4, 12, 8, 6, 4, 16, 12, 6, 12, 12, 4, 12, 4, 16, 8, 6, 8, 24, 4, 6, 8, 16, 4, 12, 4, 12, 16, 6, 4, 24, 16, 18, 8, 12, 4, 18, 8, 16, 8, 6, 4, 24, 4, 6, 16, 24, 8, 12, 4, 12, 8, 12, 4, 32, 4, 6, 24, 12, 8, 12, 4, 24, 24, 6, 4, 24, 8, 6, 8, 16, 4

Offset: 1

Steven Finch, Apr 17 2009

nonn

Denote dim{M_k(Gamma_0(N))} by m(k,N) and dim{S_k(Gamma_0(N))} by s(k,N).
We have:
m(3/2,N)-s(3/2,N)+m(1/2,N)-s(1/2,N) =
m(5/2,N)-s(5/2,N) = m(7/2,N)-s(7/2,N) =
m(9/2,N)-s(9/2,N) = m(11/2,N)-s(11/2,N) = ...
m(k/2,N)-s(k/2,N) = ...
where N is any positive multiple of 4 and k>=5 is odd.
a(n) = A159635(n) - A159636(n). - Steven Finch, Apr 22 2009
Conjecture: a(n) = 2*chi(n) - if(mod(n+2,4)=0, chi(n)/2, 0) with chi(n) = A001616(n) = Sum_{d|n} phi(gcd(d,n/d)); checked up to n=1024. - Wouter Meeussen, Apr 02 2014

K. Ono, The Web of Modularity: Arithmetic of Coefficients of Modular Forms and q-series. American Mathematical Society, 2004 (p. 16, theorem 1.56).

H. Cohen and J. Oesterle, Dimensions des espaces de formes modulaires, Modular Functions of One Variable. VI, Proc. 1976 Bonn conf., Lect. Notes in Math. 627, Springer-Verlag, 1977, pp. 69-78.
S. R. Finch, Primitive Cusp Forms, April 27, 2009. [Cached copy, with permission of the author]
Peter Humphries, Answer to: "A conjecture related to the Cohen-Oesterlé dimension formula", MathOverflow, 2014.
MAGMA Calculator.

Cf. A159630, A159631, A159632, A159634, A159635, A159636. - Steven Finch, Apr 22 2009

Cf. A001616.

[[4*n,Dimension(HalfIntegralWeightForms(4*n,5/2))-Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n,5/2)))] : n in [1..100]]; [[4*n,Dimension(HalfIntegralWeightForms(4*n,7/2))-Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n,7/2)))] : n in [1..100]]; [[4*n,Dimension(HalfIntegralWeightForms(4*n,3/2))-Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n,3/2)))+Dimension(HalfIntegralWeightForms(4*n,1/2))-Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n,1/2)))] : n in [1..100]];

(* see link, conjecture proved by P. Humphries *)
chi[n_Integer]:=Sum[EulerPhi[GCD[d,n/d]],{d,Divisors[n]}];
2 chi[#] - If[Mod[# + 2, 4] == 0, chi[#]/2, 0] & /@ Range[89]
(* Wouter Meeussen, Apr 06 2014 *)

cp's OEIS Frontend

A054728 a(n) is the smallest level N such that genus of modular curve X_0(N) is n (or -1 if no such curve exists).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A091404 Numbers n such that genus of group Gamma_0(n) is 2.

Views

Author

Keywords

Comments

References

Crossrefs

Programs

Mathematica

Formula

A332713 a(n) = Sum_{d|n} phi(d/gcd(d, n/d)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A124316 a(n) = Sum_{d|n} sigma(gcd(d,n/d)), where sigma is the sum of divisors function, A000203.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A054730 Odd n such that genus of modular curve X_0(N) is never equal to n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

PARI

Extensions

A111248 Dimension of the space of weight 2 modular forms for Gamma_0(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Formula

Extensions

A273445 a(n) is the number of solutions of the equation n = A001617(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula