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User:Enrique Pérez Herrero/Piltz

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PILTZ DIVISOR FUNCTIONS

Image:Piltz Divisor Functions.gif

Standard Piltz Functions

Formulas:

  • 1)  \tau_{k}(n)=\sum_{d|n}^{}{\tau_{k-1}(d)}
  • 2) \displaystyle \tau_{k}(n)=\prod_{i=1}^{\omega(n)}{\prod_{j=1}^{k-1}\frac{\alpha_{i}+j}{j}}=\prod_{i=1}^{\omega(n)}{\binom{\alpha_{i}+k-1}{k-1}}; \; (k \ge 1)
  • 3) \displaystyle \tau_{k}(s)= \prod_{i=1}^{\omega(s)}{\binom{k}{k-1}=k^{\omega(s)}} , being s an squarefree number: A005117.
  • 4) \displaystyle \tau_{k+1}(n^k)=\tau_{k}(n^k)\cdot \tau_2(n); \; (k \ge 0)
  • 8) \displaystyle \sum_{d|n}{{\mu(d)}^{2}\cdot{\tau_{2}(d)}^{k}}=\tau_k(rad(n))=\bigg({2^{k}+1}\bigg)^{\omega(n)}
  • 12) \displaystyle |\sum_{d|n}{\mu(d)\cdot\tau_{k}(d^{m})}|={\bigg(\binom{k+m-1}{m}-1 \bigg)}^{\omega(n)}
  • 13) \displaystyle \sum_{d|n}{\mu(d)\cdot\tau_{k}(d^{m})}={\bigg(1-\binom{k+m-1}{m}\bigg) }^{\omega(n)}
  • 16) \displaystyle\sigma_{0}^{*}(n)=|\sum_{d|n}{\mu(d)\cdot\tau_{3}(d)}|=\sum_{d|n}{{\mu(d)}^2}=\sum_{d|n}{\mu\bigg(\frac{n}{d}\bigg)\cdot\tau_{2}(d^{2})}=\sum_{d|n}{{\mu(d)}^2\cdot\tau_{2}(d^{2})}, Unitary divisor function: Sequence A034444
  • 18) \displaystyle\ \sum_{d|n}{k^{\omega(n)}}=  \tau_{2}(n^{k})

Standard Piltz Functions in OEIS

Piltz Functions in OEIS
Function Sequence Id Function Sequence Id
\displaystyle \tau_1 A000012 \displaystyle\tau_7 A111217
\displaystyle\tau_2 A000005 \displaystyle\tau_8 A111218
\displaystyle\tau_3 A007425 \displaystyle\tau_9 A111219
\displaystyle\tau_4 A007426 \displaystyle\tau_{10} A111220
\displaystyle\tau_5 A061200 \displaystyle\tau_{11} A111221
\displaystyle\tau_6 A034695 \displaystyle\tau_{12} A111306

Other Sequences in OEIS Related to Divisor Function

Sequence Related to Divisor Function
Function Sequence Id
\displaystyle \tau_{2}(n^{2}) A048691
\displaystyle \tau_{2}(n^{3}) A048785
\displaystyle \tau_{2}(n)^{2}  A035116
\displaystyle \tau_{2}\left(\tau_{2}(n)\right) A010553
\displaystyle \tau_{2}\left(\tau_{2}\left(\tau_{2}(n)\right)\right) A036450
\displaystyle \tau_{2}\left(\tau_{2}\left(\tau_{2}\left(\tau_{2}(n)\right)\right)\right) A036452
\displaystyle \tau_{2}\left(\tau_{2}\left(\tau_{2}\left(\tau_{2}\left(\tau_{2}(n)\right)\right)\right)\right) A036453
\displaystyle \tau_{k}(n) A077592
\displaystyle \tau_{2}(2^{n}+1) A046798
\displaystyle \tau_{2}(n^{n}) A062319

Frozen Piltz Functions

Formulas:

1) \displaystyle \tau_{-k}(n^{k})={\mu(n)}^{k};\;\;(k>0)


Frozen Piltz Functions in OEIS
Function Sequence Id
\displaystyle\tau_0 A063524
\displaystyle\tau_{-1} A008683
\displaystyle\tau_{-2} A007427
\displaystyle\tau_{-3} 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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