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User:Enrique Pérez Herrero/Piltz

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PILTZ DIVISOR FUNCTIONS

Piltz Divisor Functions.gif

Standard Piltz Functions

Formulas:

  • 1)
  • 2)
  • 3) , being an squarefree number: A005117.
  • 4)
  • 5) , Sequence A097988
  • 6) , where is A007947
  • 7) , Sequence A076479
  • 8)
  • 9) , Sequence A082476
  • 10) , Sequence A010553
  • 11) , Sequence A074816
  • 12)
  • 13)
  • 14) ,Sequence A000010
  • 15) ,Sequence A000720
  • 16) , Unitary divisor function: Sequence A034444
  • 17) , where are the Jacobi Polynomials
  • 18)

Standard Piltz Functions in OEIS

Piltz Functions in OEIS
Function Sequence Id Function Sequence Id
A000012 A111217
A000005 A111218
A007425 A111219
A007426 A111220
A061200 A111221
A034695 A111306

Other Sequences in OEIS Related to Divisor Function

Sequence Related to Divisor Function
Function Sequence Id
A048691
A048785
A035116
A010553
A036450
A036452
A036453
A077592
A046798
A062319

Frozen Piltz Functions

Formulas:

1)


Frozen Piltz Functions in OEIS
Function Sequence Id
A063524
A008683
A007427
A007428

Mathematica: Programming Piltz Functions

(* Draft for Mathematica Package *)
tau[1,n_Integer]:=1; SetAttributes[tau, Listable];
tau[k_Integer,n_Integer]:=Plus@@(tau[k-1,Divisors[n]])/; k > 1;
tau[k_Integer,n_Integer]:=Plus@@(tau[k+1,Divisors[n]]*MoebiusMu[n/Divisors[n}}); k<1;

(* Standard Piltz Functions *)
A000012[n_]:=tau[1,n];
A000005[n_]:=tau[2,n];
A007425[n_]:=tau[3,n];
A007426[n_]:=tau[4,n];
A061200[n_]:=tau[5,n];
A034695[n_]:=tau[6,n];
A111217[n_]:=tau[7,n];
A111218[n_]:=tau[8,n];
A111219[n_]:=tau[9,n];
A111220[n_]:=tau[10,n];
A111221[n_]:=tau[11,n];
A111306[n_]:=tau[12,n]; 
 
(* Frozen Piltz Functions *)
A063524[n_]:=tau[0,n];
A008683[n_]:=tau[-1,n];
A007427[n_]:=tau[-2,n];
A007428[n_]:=tau[-3,n];
 
(*Other Sequences Related to Piltz Functions*)
A010553[n_]:=Nest[A000005,n,2];
A036450[n_]:=Nest[A000005,n,3];
A048691[n_]:=tau[2,n^2];
A048785[n_]:=tau[2,n^3];
A035116[n_]:=tau[2,n]^2;
A074816[n_]:=3^PrimeNu[n];
A082476[n_]:=5^PrimeNu[n];
(* Alternate code for A082476
A082476[n_]:=Abs[DivisorSum[n,MoebiusMu[#]*tau[3,#^2]&]];
*)
A097988[n_]:=tau[3,n]^2;
A046798[n_IntegerQ]:=DivisorSigma[0,1+2^n];
A062319[0]:=1;
A062319[n_]:=tau[2,n^n]; 

(*Adding Help Information on Piltz Functions *)
A000012::usage="A000012: The simplest sequence of positive numbers: the all 1's sequence.";
A000005::usage="A000005: d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.";
A007425::usage="A007425: d_3(n), or tau_3(n), the number of ordered factorizations of n as n = rst.";
A007426::usage="A007426: d_4(n), or tau_4(n), the number of ordered factorizations of n as n = rstu.";
A007427::usage="A007427: Moebius transform applied twice to sequence 1,0,0,0,....";
A007428::usage="A007428: Moebius transform applied thrice to sequence 1,0,0,0,....";
A008683::usage="A008683: Moebius (or Mobius) function mu(n).";
A010553::usage="A010553: tau(tau(n)).";
A034695::usage="A034695: Dirichlet convolution of number-of-divisors function (A000005) with A007426.";
A035116::usage="A035116: tau(n)^2, where tau(n) = A000005(n).";
A046798::usage="A046798: Number of divisors of 2^n+1.";
A048691::usage="A048691: tau(n^2), where tau = A000005.";
A048785::usage="A048785: tau(n^3), where tau = number of divisors (A000005).";
A061200::usage="A061200: tau_5(n) = number of ordered 5-factorizations of n.";
A062319::usage="A062319: Number of divisors of n^n, or of A000312.";
A063524::usage="A063524: Characteristic function of 1.";
A074816::usage="A074816: a(n) = sum(d|n, tau(d)*mu(d)^2 ).";
A082476::usage="A082476: a(n)=sum(d|n, mu(d)^2*tau(d)^2).";
A097988::usage="A097988: a(n)=Sum_(d dividing n){tau(d)}^3.";
A036450::usage="A036450: tau(tau(tau(n))).";
A111217::usage="A111217: d_7(n), tau_7(n), number of ordered factorizations of n as n = rstuvwx (7-factorizations).";
A111218::usage="A111218: d_8(n), tau_8(n), number of ordered factorizations of n as n = rstuvwxy (8-factorizations).";
A111219::usage="A111219: d_9(n), tau_9(n), number of ordered factorizations of n as n = rstuvwxyz (9-factorizations).";
A111220::usage="A111220: d_10(n), tau_10(n), number of ordered factorizations of n as n = rstuvwxyza (10-factorizations).";
A111221::usage="A111221: d_11(n), tau_11(n), number of ordered factorizations of n as n = rstuvwxyzab (11-factorizations).";
A111306::usage="A111306: d_12(n), tau_12(n), number of ordered factorizations of n as n = rstuvwxyzabc (12-factorizations).";

PARI/GP: Programming Piltz Functions

A000005(n)=numdiv(n);
A000012(n)=1;
A010553(n)=numdiv(numdiv(n));
A035116(n)=numdiv(n)^2; 

\\Adding help
addhelp(A000005,"A000005: d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.");
addhelp(A000012,"A000012: The simplest sequence of positive numbers: the all 1's sequence.");
addhelp(A007425,"A007425: d_3(n), or tau_3(n), the number of ordered factorizations of n as n = rst.");
addhelp(A007426,"A007426: d_4(n), or tau_4(n), the number of ordered factorizations of n as n = rstu.");
addhelp(A007427,"A007427: Moebius transform applied twice to sequence 1,0,0,0,....");
addhelp(A007428,"A007428: Moebius transform applied thrice to sequence 1,0,0,0,....");
addhelp(A008683,"A008683: Moebius (or Mobius) function mu(n).");
addhelp(A010553,"A010553: tau(tau(n)).");
addhelp(A034695,"A034695: Dirichlet convolution of number-of-divisors function (A000005) with A007426.");
addhelp(A035116,"A035116: tau(n)^2, where tau(n) = A000005(n).");
addhelp(A046798,"A046798: Number of divisors of 2^n+1.");
addhelp(A048691,"A048691: tau(n^2), where tau = A000005.");
addhelp(A048785,"A048785: tau(n^3), where tau = number of divisors (A000005).");
addhelp(A061200,"A061200: tau_5(n) = number of ordered 5-factorizations of n.");
addhelp(A062319,"A062319: Number of divisors of n^n, or of A000312.");
addhelp(A063524,"A063524: Characteristic function of 1.");
addhelp(A074816,"A074816: a(n) = sum(d|n, tau(d)*mu(d)^2 ).");
addhelp(A082476,"A082476: a(n)=sum(d|n, mu(d)^2*tau(d)^2).");
addhelp(A097988,"A097988: a(n)=Sum_(d dividing n){tau(d)}^3.");
addhelp(A109399,"A109399: tau(tau(tau(n))).");
addhelp(A111217,"A111217: d_7(n), tau_7(n), number of ordered factorizations of n as n = rstuvwx (7-factorizations).");
addhelp(A111218,"A111218: d_8(n), tau_8(n), number of ordered factorizations of n as n = rstuvwxy (8-factorizations).");
addhelp(A111219,"A111219: d_9(n), tau_9(n), number of ordered factorizations of n as n = rstuvwxyz (9-factorizations).");
addhelp(A111220,"A111220: d_10(n), tau_10(n), number of ordered factorizations of n as n = rstuvwxyza (10-factorizations).");
addhelp(A111221,"A111221: d_11(n), tau_11(n), number of ordered factorizations of n as n = rstuvwxyzab (11-factorizations).");
addhelp(A111306,"A111306: d_12(n), tau_12(n), number of ordered factorizations of n as n = rstuvwxyzabc (12-factorizations).");

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