cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A001615 Dedekind psi function: n * Product_{p|n, p prime} (1 + 1/p).

Original entry on oeis.org

1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, 14, 24, 24, 24, 18, 36, 20, 36, 32, 36, 24, 48, 30, 42, 36, 48, 30, 72, 32, 48, 48, 54, 48, 72, 38, 60, 56, 72, 42, 96, 44, 72, 72, 72, 48, 96, 56, 90, 72, 84, 54, 108, 72, 96, 80, 90, 60, 144, 62, 96, 96, 96, 84, 144, 68, 108, 96
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of primitive sublattices of index n in generic 2-dimensional lattice; also index of Gamma_0(n) in SL_2(Z).
A generic 2-dimensional lattice L = consists of all vectors of the form mV + nW, (m,n integers). A sublattice S = has index |ad-bc| and is primitive if gcd(a,b,c,d) = 1. The generic lattice L has precisely a(2) = 3 sublattices of index 2, namely <2V,W>, and (which = ) and so on for other indices.
The sublattices of index n are in 1-to-1 correspondence with matrices [a b; 0 d] with a>0, ad=n, b in [0..d-1]. The number of these is Sum_{d|n} = sigma(n), which is A000203. A sublattice is primitive if gcd(a,b,d) = 1; the number of these is n * product_{p|n} (1+1/p), which is the present sequence.
SL_2(Z) = Gamma is the group of all 2 X 2 matrices [a b; c d] where a,b,c,d are integers with ad-bc = 1 and Gamma_0(N) is usually defined as the subgroup of this for which N|c. But conceptually Gamma is best thought of as the group of (positive) automorphisms of a lattice , its typical element taking V -> aV + bW, W -> cV + dW and then Gamma_0(N) can be defined as the subgroup consisting of the automorphisms that fix the sublattice of index N. - J. H. Conway, May 05 2001
Dedekind proved that if n = k_i*j_i for i in I represents all the ways to write n as a product, and e_i=gcd(k_i,j_i), then a(n)= sum(k_i / (e_i * phi(e_i)), i in I ) [cf. Dickson, History of the Theory of Numbers, Vol. 1, p. 123].
Also a(n)= number of cyclic subgroups of order n in an Abelian group of order n^2 and type (1,1) (Fricke). - Len Smiley, Dec 04 2001
The polynomial degree of the classical modular equation of degree n relating j(z) and j(nz) is psi(n) (Fricke). - Michael Somos, Nov 10 2006; clarified by Katherine E. Stange, Mar 11 2022
The Mobius transform of this sequence is A063659. - Gary W. Adamson, May 23 2008
The inverse Mobius transform of this sequence is A060648. - Vladeta Jovovic, Apr 05 2009
The Dirichlet inverse of this sequence is A008836(n) * A048250(n). - Álvar Ibeas, Mar 18 2015
The Riemann Hypothesis is true if and only if a(n)/n - e^gamma*log(log(n)) < 0 for any n > 30. - Enrique Pérez Herrero, Jul 12 2011
The Riemann Hypothesis is also equivalent to another inequality, see the Sole and Planat link. - Thomas Ordowski, May 28 2017
An infinitary analog of this sequence is the sum of the infinitary divisors of n (see A049417). - Vladimir Shevelev, Apr 01 2014
Problem: are there composite numbers n such that n+1 divides psi(n)? - Thomas Ordowski, May 21 2017
The sum of divisors d of n such that n/d is squarefree. - Amiram Eldar, Jan 11 2019
Psi(n)/n is a new maximum for each primorial (A002110) [proof in link: Patrick Sole and Michel Planat, Proposition 1 page 2]. - Bernard Schott, May 21 2020
From Jianing Song, Nov 05 2022: (Start)
a(n) is the number of subgroups of C_n X C_n that are isomorphic to C_n, where C_n is the cyclic group of order n. Proof: the number of elements of order n in C_n X C_n is A007434(n) (they are the elements of the form (a,b) in C_n X C_n where gcd(a,b,n) = 1), and each subgroup isomorphic to C_n contains phi(n) generators, so the number of such subgroups is A007434(n)/phi(n) = a(n).
The total number of order-n subgroups of C_n X C_n is A000203(n). (End)

Examples

			Let L = <V,W> be a 2-dimensional lattice. The 6 primitive sublattices of index 4 are generated by <4V,W>, <V,4W>, <4V,W+-V>, <2V+W,2W>, <2V,2W+V>. Compare A000203.
G.f. = x + 3*x^2 + 4*x^3 + 6*x^4 + 6*x^5 + 12*x^6 + 8*x^7 + 12*x^8 + 12*x^9 + ...

References

  • Tom Apostol, Intro. to Analyt. Number Theory, page 71, Problem 11, where this is called phi_1(n).
  • David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, p. 228.
  • R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 220.
  • Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 147.
  • B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 79.
  • G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (1).
  • 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

Crossrefs

Other sequences that count lattices/sublattices: A000203 (with primitive condition removed), A003050 (hexagonal lattice instead), A003051, A054345, A160889, A160891.
Cf. A002110, A027748, A124010, A019269.
Cf. A301594.
Cf. A063659 (Möbius transform), A082020 (average order), A156303 (Euler transform), A173290 (partial sums), A175836 (partial products), A203444 (range).
Cf. A210523 (record values).
Algebraic combinations with other core sequences: A000082, A033196, A175732, A291784, A344695.
Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10: A034444 (k=0), this sequence (k=1), A065958 (k=2), A065959 (k=3), A065960 (k=4), A351300 (k=5), A351301 (k=6), A351302 (k=7), A351303 (k=8), A351304 (k=9), A351305 (k=10).
Cf. A082695 (Dgf at s=3), A339925 (Dgf at s=4).

Programs

  • Haskell
    import Data.Ratio (numerator)
a001615 n = numerator (fromIntegral n * (product $
            map ((+ 1) . recip . fromIntegral) $ a027748_row n))
-- Reinhard Zumkeller, Jun 03 2013, Apr 12 2012
  • Magma
    m:=75; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[MoebiusMu(k)^2*x^k/(1-x^k)^2: k in [1..2*m]]) )); // G. C. Greubel, Nov 23 2018
  • Maple
    A001615 := proc(n) n*mul((1+1/i[1]),i=ifactors(n)[2]) end; # Mark van Hoeij, Apr 18 2012
  • Mathematica
    Join[{1}, Table[n Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]), {n, 2, 100}]] (* T. D. Noe, Jun 11 2006 *)
Table[DirichletConvolve[j, MoebiusMu[j]^2, j, n], {n, 100}] (* Jan Mangaldan, Aug 22 2013 *)
a[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}]; (* Michael Somos, Jan 10 2015 *)
Table[n Product[1 + 1/p, {p, Select[Divisors[n], PrimeQ]}], {n, 1, 100}] (* Vaclav Kotesovec, May 08 2021 *)
Table[n DivisorSum[n, MoebiusMu[#]^2/# &], {n, 20}] (* Eric W. Weisstein, Mar 09 2025 *)
  • PARI
    {a(n) = if( n<1, 0, direuler( p=2, n, (1 + X) / (1 - p*X)) [n])};
  • PARI
    {a(n) = if( n<1, 0, n * sumdiv( n, d, moebius(d)^2 / d))}; /* Michael Somos, Nov 10 2006 */
  • PARI
    a(n)=my(f=factor(n)); prod(i=1,#f~, f[i,1]^f[i,2] + f[i,1]^(f[i,2]-1)) \\ Charles R Greathouse IV, Aug 22 2013
  • PARI
    a(n) = n * sumdivmult(n, d, issquarefree(d)/d) \\ Charles R Greathouse IV, Sep 09 2014
  • Python
    from math import prod
from sympy import primefactors
def A001615(n):
    plist = primefactors(n)
    return n*prod(p+1 for p in plist)//prod(plist) # Chai Wah Wu, Jun 03 2021
  • Sage
    def A001615(n) : return n*mul(1+1/p for p in prime_divisors(n))
[A001615(n) for n in (1..69)] # Peter Luschny, Jun 10 2012

Formula

Dirichlet g.f.: zeta(s) * zeta(s-1) / zeta(2*s). - Michael Somos, May 19 2000
Multiplicative with a(p^e) = (p+1)*p^(e-1). - David W. Wilson, Aug 01 2001
a(n) = A003557(n)*A048250(n) = n*A000203(A007947(n))/A007947(n). - Labos Elemer, Dec 04 2001
a(n) = n*Sum_{d|n} mu(d)^2/d, Dirichlet convolution of A008966 and A000027. - Benoit Cloitre, Apr 07 2002
a(n) = Sum_{d|n} mu(n/d)^2 * d. - Joerg Arndt, Jul 06 2011
From Enrique Pérez Herrero, Aug 22 2010: (Start)
a(n) = J_2(n)/J_1(n) = J_2(n)/phi(n) = A007434(n)/A000010(n), where J_k is the k-th Jordan Totient Function.
a(n) = (1/phi(n))*Sum_{d|n} mu(n/d)*d^(b-1), for b=3. (End)
a(n) = n / Sum_{d|n} mu(d)/a(d). - Enrique Pérez Herrero, Jun 06 2012
a(n^k)= n^(k-1) * a(n). - Enrique Pérez Herrero, Jan 05 2013
If n is squarefree, then a(n) = A049417(n) = A000203(n). - Vladimir Shevelev, Apr 01 2014
a(n) = Sum_{d^2 | n} mu(d) * A000203(n/d^2). - Álvar Ibeas, Dec 20 2014
The average order of a(n) is 15*n/Pi^2. - Enrique Pérez Herrero, Jan 14 2012. See Apostol. - N. J. A. Sloane, Sep 04 2017
G.f.: Sum_{k>=1} mu(k)^2*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Oct 25 2018
a(n) = Sum_{d|n} 2^omega(d) * phi(n/d), Dirichlet convolution of A034444 and A000010. - Daniel Suteu, Mar 09 2019
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} 2^omega(gcd(n,k)).
a(n) = Sum_{k=1..n} 2^omega(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
a(n) = abs(A158523(n)) = A158523(n) * A008836(n). - Enrique Pérez Herrero, Nov 07 2022
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^2). - Ridouane Oudra, Mar 26 2025

Extensions

More terms from Olivier Gérard, Aug 15 1997

A061019 Negate primes in factorization of n.

Original entry on oeis.org

1, -2, -3, 4, -5, 6, -7, -8, 9, 10, -11, -12, -13, 14, 15, 16, -17, -18, -19, -20, 21, 22, -23, 24, 25, 26, -27, -28, -29, -30, -31, -32, 33, 34, 35, 36, -37, 38, 39, 40, -41, -42, -43, -44, -45, 46, -47, -48, 49, -50, 51, -52, -53, 54, 55, 56, 57, 58, -59, 60, -61, 62, -63, 64, 65, -66, -67, -68, 69, -70
Offset: 1

Views

Author

Marc LeBrun, Apr 13 2001

Keywords

Comments

Inverse Moebius transform of A158523. - Corrected by Antti Karttunen, Nov 26 2024

Examples

			a(6) = (-2)(-3) = +6, while a(8) = (-2)^3 = -8.

Links

Crossrefs

Cf. A000027, A001222, A061020, A001615, A158523 (Möbius transform).
Cf. A027746.
Cf. A239122 (partial sums).

Programs

  • Haskell
    a061019 1 = 1
a061019 n = product $ map negate $ a027746_row n
-- Reinhard Zumkeller, Feb 08 2012
  • Mathematica
    Table[n (-1)^PrimeOmega[n],{n,70}] (* Harvey P. Dale, Oct 05 2011 *)
  • PARI
    a(n) = if( bitand(bigomega(n),1), - n, n ); /* Joerg Arndt, Sep 19 2012 */
  • Python
    from functools import reduce
from operator import ixor
from sympy import factorint
def A061019(n): return -n if reduce(ixor, factorint(n).values(),0)&1 else n # Chai Wah Wu, Dec 20 2022

Formula

a(n) = n*lambda(n), where lambda is Liouville's function: A008836.
a(n) = (-1)^(number of primes dividing n)*n = n * (-1)^A001222(n) = n*A008836(n).
Totally multiplicative with a(p) = -p for prime p. [Jaroslav Krizek, Nov 01 2009]
Dirichlet g.f.: zeta(2*s-2)/zeta(s-1). Dirichlet inverse of A055615, all terms turned positive there. - R. J. Mathar, Apr 16 2011
a(n) = Sum_{d|n} lambda(d)*psi(d) = sum_{d|n} A008836(d)* A001615(d) = n/lambda(n). - Enrique Pérez Herrero, Sep 18 2012

A347236 a(n) = Sum_{d|n} A061019(d) * A003961(n/d), where A061019 negates the primes in the prime factorization, while A003961 shifts the factorization one step towards larger primes.

Original entry on oeis.org

1, 1, 2, 7, 2, 2, 4, 13, 19, 2, 2, 14, 4, 4, 4, 55, 2, 19, 4, 14, 8, 2, 6, 26, 39, 4, 68, 28, 2, 4, 6, 133, 4, 2, 8, 133, 4, 4, 8, 26, 2, 8, 4, 14, 38, 6, 6, 110, 93, 39, 4, 28, 6, 68, 4, 52, 8, 2, 2, 28, 6, 6, 76, 463, 8, 4, 4, 14, 12, 8, 2, 247, 6, 4, 78, 28, 8, 8, 4, 110, 421, 2, 6, 56, 4, 4, 4, 26, 8, 38, 16
Offset: 1

Views

Author

Antti Karttunen, Aug 24 2021

Keywords

Comments

Dirichlet convolution of A003961 and A061019.
Dirichlet convolution of A003973 and A158523.
Multiplicative because A003961 and A061019 are.
All terms are positive because all terms of A347237 are nonnegative and A347237(1) = 1.
Union of sequences A001359 and A108605 (= 2*A001359) seems to give the positions of 2's in this sequence.

Links

Crossrefs

Cf. A000040, A001223, A001359, A003961, A003973, A061019, A108605, A158523, A347237 (Möbius transform), A347238 (Dirichlet inverse), A347239.
Cf. also A347136.
Cf. A151800.

Programs

  • Mathematica
    f[p_, e_] := ((np = NextPrime[p])^(e + 1) - (-p)^(e + 1))/(np + p); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 02 2021 *)
  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A061019(n) = (((-1)^bigomega(n))*n);
A347236(n) = sumdiv(n,d,A061019(d)*A003961(n/d));

Formula

a(n) = Sum_{d|n} A003961(n/d) * A061019(d).
a(n) = Sum_{d|n} A003973(n/d) * A158523(d).
a(n) = Sum_{d|n} A347237(d).
a(n) = A347239(n) - A347238(n).
For all n >= 1, a(A000040(n)) = A001223(n).
Multiplicative with a(p^e) = (A151800(p)^(e+1)-(-p)^(e+1))/(A151800(p)+p). - Sebastian Karlsson, Sep 02 2021

A378434 Arithmetic mean between the Dirichlet inverses of {sum of unitary divisors} and {sum of squarefree divisors}.

Original entry on oeis.org

1, -3, -4, 5, -6, 12, -8, -9, 9, 18, -12, -20, -14, 24, 24, 16, -18, -27, -20, -30, 32, 36, -24, 36, 20, 42, -24, -40, -30, -72, -32, -30, 48, 54, 48, 48, -38, 60, 56, 54, -42, -96, -44, -60, -54, 72, -48, -64, 35, -60, 72, -70, -54, 72, 72, 72, 80, 90, -60, 120, -62, 96, -72, 56, 84, -144, -68, -90, 96, -144, -72, -90
Offset: 1

Views

Author

Antti Karttunen, Nov 26 2024

Keywords

Comments

Arithmetic mean between A158523 and A178450.
Apparently differs from A378433 at positions given by A048111: 16, 32, 36, 48, 64, 72, 80, 81, 96, ...

Links

Crossrefs

Cf. A034448, A048111, A048250, A158523, A178450, A325973, A378433, A378435 (Dirichlet inverse).

Programs

  • PARI
    A158523(n) = { my(f = factor(n)); prod(i = 1, #f~, (-1)^f[i, 2]*(f[i, 1]+1)*f[i, 1]^(f[i, 2]-1)); }; \\ From A158523
A178450(n) = { my(f=factor(n)); prod(i=1, #f~, if(!(f[i,2]%2), 2*(f[i, 1]^(f[i, 2]/2)), -(1+f[i,1])*(f[i, 1]^((f[i, 2]-1)/2)))); };
A378434(n) = ((A158523(n)+A178450(n))/2);

Formula

a(n) = (1/2) * (A158523(n)+A178450(n)).

A347237 Möbius transform of A347236.

Original entry on oeis.org

1, 0, 1, 6, 1, 0, 3, 6, 17, 0, 1, 6, 3, 0, 1, 42, 1, 0, 3, 6, 3, 0, 5, 6, 37, 0, 49, 18, 1, 0, 5, 78, 1, 0, 3, 102, 3, 0, 3, 6, 1, 0, 3, 6, 17, 0, 5, 42, 89, 0, 1, 18, 5, 0, 1, 18, 3, 0, 1, 6, 5, 0, 51, 330, 3, 0, 3, 6, 5, 0, 1, 102, 5, 0, 37, 18, 3, 0, 3, 42, 353, 0, 5, 18, 1, 0, 1, 6, 7, 0, 9, 30, 5, 0, 3, 78, 3, 0, 17
Offset: 1

Views

Author

Antti Karttunen, Aug 24 2021

Keywords

Comments

Dirichlet convolution of A003972 (prime shifted phi) with A061019.
Dirichlet convolution of A003961 with A158523.
Multiplicative because A003972 and A061019 are, and also because A347236 is.
From Antti Karttunen, Aug 25 2021: (Start)
All terms are nonnegative because sequence is multiplicative and a(p^k) >= 0 for all primes p and k >= 0.
Proof: For any prime p, sequence a(p^k), k>=0, is obtained as an ordinary convolution of sequences (-p)^k and the first differences of q^k, where q = A151800(p). (E.g., for powers of 2, the sequences convolved are A122803 and A025192, giving A102901.) This convolution is an alternating sum, with the terms 1*(q-1)*q^(k-1), -(p)*(q-1)*q^(k-2), (p^2)*(q-1)*q^(k-3), -(p^3)*(q-1)*q^(k-4), ..., (p^(k-1))*(q-1), -(p^k), for odd k, with sum of each consecutive pair being nonnegative because q >= p+1, while with an even exponent k, the leftover term p^k at the end is also positive, thus the whole sum is nonnegative also in that case.
(End)

Links

Crossrefs

Cf. A000040, A001223, A003961, A003972, A008683, A008836, A016825 (positions of zeros), A061019, A102901, A120612, A158523, A347236.
Cf. also A347137.
Cf. A151800.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A061019(n) = (((-1)^bigomega(n))*n);
A347237(n) = sumdiv(n,d,A061019(d)*eulerphi(A003961(n/d)));
\\ Or alternatively as:
A158523(n) = { my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); ((-1)^e)*(p+1)*(p^(e-1))); };
A347237(n) = sumdiv(n,d,A003961(n/d)*A158523(d));

Formula

a(n) = Sum_{d|n} A008683(n/d) * A347236(d).
a(n) = Sum_{d|n} A003972(n/d) * A061019(d).
a(n) = Sum_{d|n} A003961(n/d) * A158523(d).
For all n >= 1, a(A000040(n)) = A001223(n) - 1.
For all n >= 0, a(2^n) = A102901(n).
For all n >= 0, a(3^n) = A120612(n).
Multiplicative with a(p^e) = (-p)^e + (A151800(p)-1)*(A151800(p)^e-(-p)^e)/(A151800(p)+p). - Sebastian Karlsson, Sep 02 2021

A378435 Dirichlet inverse of the arithmetic mean between the Dirichlet inverses of {sum of unitary divisors} and {sum of squarefree divisors}.

Original entry on oeis.org

1, 3, 4, 4, 6, 12, 8, 6, 7, 18, 12, 16, 14, 24, 24, 9, 18, 21, 20, 24, 32, 36, 24, 24, 16, 42, 16, 32, 30, 72, 32, 15, 48, 54, 48, 25, 38, 60, 56, 36, 42, 96, 44, 48, 42, 72, 48, 36, 29, 48, 72, 56, 54, 48, 72, 48, 80, 90, 60, 96, 62, 96, 56, 24, 84, 144, 68, 72, 96, 144, 72, 33, 74, 114, 64, 80, 96, 168, 80, 54, 34
Offset: 1

Views

Author

Antti Karttunen, Nov 26 2024

Keywords

Comments

The first negative term is a(2592) = -48.
Apparently differs from A325973 at positions given by A048111: 16, 32, 36, 48, 64, 72, 80, 81, 96, ...

Links

Crossrefs

Dirichlet inverse of A378434.
Cf. A034448, A048111, A048250, A158523, A178450, A325973, A378433.

Programs

  • PARI
    A158523(n) = { my(f = factor(n)); prod(i = 1, #f~, (-1)^f[i, 2]*(f[i, 1]+1)*f[i, 1]^(f[i, 2]-1)); }; \\ From A158523
A178450(n) = { my(f=factor(n)); prod(i=1, #f~, if(!(f[i,2]%2), 2*(f[i, 1]^(f[i, 2]/2)), -(1+f[i,1])*(f[i, 1]^((f[i, 2]-1)/2)))); };
A378434(n) = ((A158523(n)+A178450(n))/2);
memoA378435 = Map();
A378435(n) = if(1==n,1,my(v); if(mapisdefined(memoA378435,n,&v), v, v = -sumdiv(n,d,if(dA378434(n/d)*A378435(d),0)); mapput(memoA378435,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA378434(n/d) * a(d).
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.