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# User:Enrique Pérez Herrero/Piltz

## Standard Piltz Functions

### Formulas:

• 1) ${\displaystyle \tau _{k}(n)=\sum _{d|n}^{}{\tau _{k-1}(d)}}$
• 2) ${\displaystyle \displaystyle \tau _{k}(n)=\prod _{i=1}^{\omega (n)}{\prod _{j=1}^{k-1}{\frac {\alpha _{i}+j}{j}}}=\prod _{i=1}^{\omega (n)}{\binom {\alpha _{i}+k-1}{k-1}};\;(k\geq 1)}$
• 3) ${\displaystyle \displaystyle \tau _{k}(s)=\prod _{i=1}^{\omega (s)}{{\binom {k}{k-1}}=k^{\omega (s)}}}$ , being ${\displaystyle s}$ an squarefree number: A005117.
• 4) ${\displaystyle \displaystyle \tau _{k+1}(n^{k})=\tau _{k}(n^{k})\cdot \tau _{2}(n);\;(k\geq 0)}$
• 5) ${\displaystyle \displaystyle \tau _{3}(n)^{2}=\left(\sum _{d|n}{\tau _{2}(d)}\right)^{2}}$, Sequence A097988
• 6) ${\displaystyle \displaystyle \tau _{k}(n)=\prod _{i=0}^{k-2}{\frac {\tau _{2}(n\cdot rad(n)^{i})}{\tau _{2}(rad(n)^{i})}};\;(k\geq 1)}$, where ${\displaystyle \displaystyle rad(n)}$ is A007947
• 7) ${\displaystyle \displaystyle \sum _{d|n}{\mu (d)\cdot \tau _{2}(d)}=(-1)^{\omega (n)}=\mu (rad(n))}$, Sequence A076479
• 8) ${\displaystyle \displaystyle \sum _{d|n}{{\mu (d)}^{2}\cdot {\tau _{2}(d)}^{k}}=\tau _{k}(rad(n))={\bigg (}{2^{k}+1}{\bigg )}^{\omega (n)}}$
• 9) ${\displaystyle \displaystyle |\sum _{d|n}{\mu (d)\cdot \tau _{3}(d^{2})}|=\sum _{d|n}{{\mu (d)}^{2}\cdot {\tau _{2}(d)}^{2}}=\tau _{5}(rad(n))=5^{\omega (n)}}$, Sequence A082476
• 10) ${\displaystyle \displaystyle \tau _{2}(\tau _{2}(rad(n)))-1=\omega (n)\;;\;(n>1)}$, Sequence A010553
• 11) ${\displaystyle \displaystyle \sum _{d|n}{\mu (d)^{2}\cdot \tau _{2}(d)}=|\sum _{d|n}{\mu (d)\cdot \tau _{2}(d^{3})}|=|\sum _{d|n}{\mu (d)\cdot \tau _{4}(d)}|=\tau _{3}(rad(n))=3^{\omega (n)}}$, Sequence A074816
• 12) ${\displaystyle \displaystyle |\sum _{d|n}{\mu (d)\cdot \tau _{k}(d^{m})}|={{\bigg (}{\binom {k+m-1}{m}}-1{\bigg )}}^{\omega (n)}}$
• 13) ${\displaystyle \displaystyle \sum _{d|n}{\mu (d)\cdot \tau _{k}(d^{m})}={{\bigg (}1-{\binom {k+m-1}{m}}{\bigg )}}^{\omega (n)}}$
• 14) ${\displaystyle \displaystyle \phi (n)=\sum _{i=1}^{n}{{\bigg \lfloor }{\frac {\tau _{k}(i\cdot n)}{\tau _{k}(i)\cdot \tau _{k}(n)}}{\bigg \rfloor }}}$ ,Sequence A000010
• 15) ${\displaystyle \displaystyle \pi (n)=\sum _{i=2}^{n}{{\bigg \lfloor }{\frac {k}{\tau _{k}(i)}}{\bigg \rfloor }}\;\;;(n>1)}$ ,Sequence A000720
• 16) ${\displaystyle \displaystyle \sigma _{0}^{*}(n)=|\sum _{d|n}{\mu (d)\cdot \tau _{3}(d)}|=\sum _{d|n}{{\mu (d)}^{2}}=\sum _{d|n}{\mu {\bigg (}{\frac {n}{d}}{\bigg )}\cdot \tau _{2}(d^{2})}=\sum _{d|n}{{\mu (d)}^{2}\cdot \tau _{2}(d^{2})}}$, Unitary divisor function: Sequence A034444
• 17) ${\displaystyle \displaystyle \tau _{k}(p^{\alpha })={\binom {\alpha +k-1}{k-1}}=P_{k-1}^{(\alpha ,\beta )}(1)\;(k>1)}$ , where ${\displaystyle \displaystyle P_{n}^{(\alpha ,\beta )}(z)}$ are the Jacobi Polynomials
• 18) ${\displaystyle \displaystyle \ \sum _{d|n}{k^{\omega (n)}}=\tau _{2}(n^{k})}$

### Standard Piltz Functions in OEIS

 Function Sequence Id Function Sequence Id ${\displaystyle \displaystyle \tau _{1}}$ A000012 ${\displaystyle \displaystyle \tau _{7}}$ A111217 ${\displaystyle \displaystyle \tau _{2}}$ A000005 ${\displaystyle \displaystyle \tau _{8}}$ A111218 ${\displaystyle \displaystyle \tau _{3}}$ A007425 ${\displaystyle \displaystyle \tau _{9}}$ A111219 ${\displaystyle \displaystyle \tau _{4}}$ A007426 ${\displaystyle \displaystyle \tau _{10}}$ A111220 ${\displaystyle \displaystyle \tau _{5}}$ A061200 ${\displaystyle \displaystyle \tau _{11}}$ A111221 ${\displaystyle \displaystyle \tau _{6}}$ A034695 ${\displaystyle \displaystyle \tau _{12}}$ A111306

### Other Sequences in OEIS Related to Divisor Function

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

## Frozen Piltz Functions

### Formulas:

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

 Function Sequence Id ${\displaystyle \displaystyle \tau _{0}}$ A063524 ${\displaystyle \displaystyle \tau _{-1}}$ A008683 ${\displaystyle \displaystyle \tau _{-2}}$ A007427 ${\displaystyle \displaystyle \tau _{-3}}$ 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;
(* 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;

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(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.");