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 1-7 of 7 results.

A059379 Array of values of Jordan function J_k(n) read by antidiagonals (version 1).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 2, 8, 7, 1, 4, 12, 26, 15, 1, 2, 24, 56, 80, 31, 1, 6, 24, 124, 240, 242, 63, 1, 4, 48, 182, 624, 992, 728, 127, 1, 6, 48, 342, 1200, 3124, 4032, 2186, 255, 1, 4, 72, 448, 2400, 7502, 15624, 16256, 6560, 511, 1, 10, 72, 702, 3840
Offset: 1

Views

Author

N. J. A. Sloane, Jan 28 2001

Keywords

Examples

			Array begins:
  1,  1,  2,   2,   4,    2,    6,    4,   6,  4, 10, 4, ...
  1,  3,  8,  12,  24,   24,   48,   48,  72, 72, ...
  1,  7, 26,  56, 124,  182,  342,  448, 702, ...
  1, 15, 80, 240, 624, 1200, 2400, 3840, ...

References

  • Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
  • R. Sivaramakrishnan, "The many facets of Euler's totient. II. Generalizations and analogues", Nieuw Arch. Wisk. (4) 8 (1990), no. 2, 169-187.

Links

Crossrefs

See A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5). Columns give A000225, A024023, A020522, A024049, A059387, etc.
Main diagonal gives A067858.

Programs

  • Maple
    J := proc(n,k) local i,p,t1,t2; t1 := n^k; for p from 1 to n do if isprime(p) and n mod p = 0 then t1 := t1*(1-p^(-k)); fi; od; t1; end;
#alternative
A059379 := proc(n,k)
    add(d^k*numtheory[mobius](n/d),d=numtheory[divisors](n)) ;
end proc:
seq(seq(A059379(d-k,k),k=1..d-1),d=2..12) ; # R. J. Mathar, Nov 23 2018
  • Mathematica
    JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/#]&]/;(n>0)&&IntegerQ[n];
A004736[n_]:=Binomial[Floor[3/2+Sqrt[2*n]],2]-n+1;
A002260[n_]:=n-Binomial[Floor[1/2+Sqrt[2*n]],2];
A059379[n_]:=JordanTotient[A004736[n],A002260[n]]; (* Enrique Pérez Herrero, Dec 19 2010 *)
  • PARI
    jordantot(n,k)=sumdiv(n,d,d^k*moebius(n/d));
A002260(n)=n-binomial(floor(1/2+sqrt(2*n)),2);
A004736(n)=binomial(floor(3/2+sqrt(2*n)),2)-n+1;
A059379(n)=jordantot(A004736(n),A002260(n)); \\ Enrique Pérez Herrero, Jan 08 2011
  • Python
    from functools import cache
def MoebiusTrans(a, i):
    @cache
    def mb(n, d = 1):
          return d % n and -mb(d, n % d < 1) + mb(n, d + 1) or 1 // n
    def mob(m, n): return mb(m // n) if m % n == 0 else 0
    return sum(mob(i, d) * a(d) for d in range(1, i + 1))
def Jrow(n, size):
    return [MoebiusTrans(lambda m: m ** n, k) for k in range(1, size)]
for n in range(1, 8): print(Jrow(n, 13))
# Alternatively:
from sympy import primefactors as prime_divisors
from fractions import Fraction as QQ
from math import prod as product
def J(n: int, k: int) -> int:
    t = QQ(pow(k, n), 1)
    s = product(1 - QQ(1, pow(p, n)) for p in prime_divisors(k))
    return (t * s).numerator  # the denominator is always 1
for n in range(1, 8): print([J(n, k) for k in range(1, 13)])
# Peter Luschny, Dec 16 2023

Formula

J_k(n) = Sum_{d|n} d^k*mu(n/d). - Benoit Cloitre and Michael Orrison (orrison(AT)math.hmc.edu), Jun 07 2002
From Amiram Eldar, Jun 07 2025: (Start)
For a given k, J_k(n) is multiplicative with J_k(p^e) = p^(k*e) - p^(k*e-k).
For a given k, Dirichlet g.f. of J_k(n): zeta(s-k)/zeta(s).
Sum_{i=1..n} J_k(i) ~ n^(k+1) / ((k+1)*zeta(k+1)).
Sum_{n>=1} 1/J_k(n) = Product_{p prime} (1 + p^k/(p^k-1)^2) for k >= 2. (End)

A059380 Array of values of Jordan function J_k(n) read by antidiagonals (version 2).

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 7, 8, 2, 1, 15, 26, 12, 4, 1, 31, 80, 56, 24, 2, 1, 63, 242, 240, 124, 24, 6, 1, 127, 728, 992, 624, 182, 48, 4, 1, 255, 2186, 4032, 3124, 1200, 342, 48, 6, 1, 511, 6560, 16256, 15624, 7502, 2400, 448, 72, 4, 1, 1023, 19682
Offset: 1

Views

Author

N. J. A. Sloane, Jan 28 2001

Keywords

Examples

			Array begins:
1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, ...
1, 3, 8, 12, 24, 24, 48, 48, 72, 72, ...
1, 7, 26, 56, 124, 182, 342, 448, 702, ...
1, 15, 80, 240, 624, 1200, 2400, 3840, ...

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
  • R. Sivaramakrishnan, The many facets of Euler's totient. II. Generalizations and analogues, Nieuw Arch. Wisk. (4) 8 (1990), no. 2, 169-187

Links

Crossrefs

See A059379 and A059380 (triangle of values of J_k(n)), A000010 (J_1), A059376 (J_3), A059377 (J_4), A059378 (J_5). Columns give A000225, A024023, A020522, A024049, A059387, etc.
Main diagonal gives A067858.

Programs

  • Maple
    J := proc(n,k) local i,p,t1,t2; t1 := n^k; for p from 1 to n do if isprime(p) and n mod p = 0 then t1 := t1*(1-p^(-k)); fi; od; t1; end;
  • Mathematica
    JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/#]&]/;(n>0)&&IntegerQ[n];
A004736[n_]:=Binomial[Floor[3/2+Sqrt[2*n]],2]-n+1;
A002260[n_]:=n-Binomial[Floor[1/2+Sqrt[2*n]],2];
A059380[n_]:=JordanTotient[A002260[n],A004736[n]]; (* Enrique Pérez Herrero, Dec 19 2010 *)
  • PARI
    jordantot(n,k)=sumdiv(n,d,d^k*moebius(n/d));
A002260(n)=n-binomial(floor(1/2+sqrt(2*n)),2);
A004736(n)=binomial(floor(3/2+sqrt(2*n)),2)-n+1;
A059380(n)=jordantot(A002260(n),A004736(n)); \\ Enrique Pérez Herrero, Jan 08 2011

A321222 a(n) = Sum_{d|n} mu(d)*d^n.

Original entry on oeis.org

1, -3, -26, -15, -3124, 45864, -823542, -255, -19682, 9990233352, -285311670610, 2176246800, -302875106592252, 11111328602468784, 437893859848932344, -65535, -827240261886336764176, 101559568985784, -1978419655660313589123978, 99999904632567310800
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 06 2018

Keywords

Links

Crossrefs

Cf. A008683, A023900, A046970, A063453, A067858, A189922, A189923.

Programs

  • Mathematica
    Table[Sum[MoebiusMu[d] d^n, {d, Divisors[n]}], {n, 20}]
nmax = 20; Rest[CoefficientList[Series[Sum[MoebiusMu[k] (k x)^k/(1 - (k x)^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Product[1 - Boole[PrimeQ[d]] d^n, {d, Divisors[n]}], {n, 20}]
  • PARI
    a(n) = sumdiv(n, d, moebius(d)*d^n) \\ Andrew Howroyd, Nov 06 2018

Formula

G.f.: Sum_{k>=1} mu(k)*(k*x)^k/(1 - (k*x)^k).
a(n) = Product_{p|n, p prime} (1 - p^n).

A320974 a(n) = n^n * Product_{p|n, p prime} (1 + 1/p^n).

Original entry on oeis.org

1, 5, 28, 272, 3126, 47450, 823544, 16842752, 387440172, 10009766650, 285311670612, 8918294011904, 302875106592254, 11112685048647250, 437893920912786408, 18447025548686262272, 827240261886336764178, 39346558271492178663450, 1978419655660313589123980
Offset: 1

Views

Author

Ilya Gutkovskiy, Oct 25 2018

Keywords

Links

Crossrefs

Cf. A001615, A008683, A065958, A065959, A065960, A067858, A320973.

Programs

  • Mathematica
    Table[n^n Product[1 + Boole[PrimeQ[d]]/d^n, {d, Divisors[n]}], {n, 19}]
Table[SeriesCoefficient[Sum[MoebiusMu[k]^2 PolyLog[-n, x^k], {k, 1, n}], {x, 0, n}], {n, 19}]
Table[Sum[MoebiusMu[n/d]^2 d^n, {d, Divisors[n]}], {n, 19}]

Formula

a(n) = [x^n] Sum_{k>=1} mu(k)^2*PolyLog(-n,x^k), where PolyLog() is the polylogarithm function.
a(n) = Sum_{d|n} mu(n/d)^2*d^n.
a(n) = A320973(n,n).

A347251 a(n) = Sum_{d|n} mu(d)*mu(n/d)*d^n.

Original entry on oeis.org

1, -5, -28, 16, -3126, 47450, -823544, 0, 19683, 10009766650, -285311670612, -2176786432, -302875106592254, 11112685048647250, 437893920912786408, 0, -827240261886336764178, -101560344088905, -1978419655660313589123980, -100000000000001048576
Offset: 1

Views

Author

Seiichi Manyama, Aug 24 2021

Keywords

Links

Crossrefs

Diagonal of A347227.
Cf. A007427, A008683, A046099, A067858, A321222.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, MoebiusMu[#] * MoebiusMu[n/#] * #^n &]; Array[a, 20] (* Amiram Eldar, Aug 24 2021 *)
  • PARI
    a(n) = sumdiv(n, d, moebius(d)*moebius(n/d)*d^n);

Formula

If p is prime, a(p) = -1 - p^p.

A344210 a(n) = Sum_{d|n} mu(n/d) * d^n / phi(n).

Original entry on oeis.org

1, 3, 13, 120, 781, 22932, 137257, 4177920, 64566801, 2497558338, 28531167061, 2228476723200, 25239592216021, 1851888100411464, 54736732481116543, 2305807824841605120, 51702516367896047761, 6557709646516945221396, 109912203092239643840221
Offset: 1

Views

Author

Seiichi Manyama, May 12 2021

Keywords

Crossrefs

Main diagonal of A263950.
Cf. A000010, A067858, A008683, A000203.

Programs

  • Mathematica
    Table[DivisorSum[n,MoebiusMu[n/#]*#^n/EulerPhi[n]&],{n,20}] (* Giorgos Kalogeropoulos, May 13 2021 *)
  • PARI
    a(n) = sumdiv(n, d, moebius(n/d)*d^n)/eulerphi(n);

Formula

a(n) = J_n(n) / phi(n) = A067858(n) / A000010(n).
a(n) = (1/n) * Sum_{d|n} mu(n/d)*sigma(d^n). - Ridouane Oudra, Apr 03 2025

A332617 a(n) = Sum_{k=1..n} J_n(k), where J is the Jordan function, J_n(k) = k^n * Product_{p|k, p prime} (1 - 1/p^n).

Original entry on oeis.org

1, 4, 34, 336, 4390, 66312, 1197858, 24612000, 574002448, 14903406552, 427622607366, 13419501812640, 457579466056498, 16840326075104280, 665473192580864556, 28101209228393371200, 1262896789586657015796, 60182268296582518426368, 3031282541337682050032664
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 17 2020

Keywords

Links

Crossrefs

Cf. A000010, A002088, A007434, A059376, A059377, A059378, A059379, A059380, A067858, A319194, A321879.

Programs

  • Magma
    [&+[&+[MoebiusMu(k div d)*d^n:d in Divisors(k)]:k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 17 2020
  • Mathematica
    Table[Sum[Sum[MoebiusMu[k/d] d^n, {d, Divisors[k]}], {k, 1, n}], {n, 1, 19}]
Table[SeriesCoefficient[(1/(1 - x)) Sum[Sum[MoebiusMu[k] j^n x^(k j), {j, 1, n}], {k, 1, n}], {x, 0, n}], {n, 1, 19}]

Formula

a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} Sum_{j>=1} mu(k) * j^n * x^(k*j).
Showing 1-7 of 7 results.

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

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