User:Enrique Pérez Herrero/Piltz

PILTZ DIVISOR FUNCTIONS

Standard Piltz Functions

Formulas:

• 1) $\tau _{k}(n)=\sum _{d|n}^{}{\tau _{k-1}(d)}$

• 2) $\displaystyle \tau _{k}(n)=\prod _{i=1}^{\omega (n)}{\prod _{j=1}^{k-1}{\frac {\alpha _{i}+j}{j}}}=\prod _{i=1}^{\omega (n)}{\binom {\alpha _{i}+k-1}{k-1}};\;(k\geq 1)$

• 3) $\displaystyle \tau _{k}(s)=\prod _{i=1}^{\omega (s)}{{\binom {k}{k-1}}=k^{\omega (s)}}$ , being $s$ an squarefree number: A005117.

• 4) $\displaystyle \tau _{k+1}(n^{k})=\tau _{k}(n^{k})\cdot \tau _{2}(n);\;(k\geq 0)$

• 5) $\displaystyle \tau _{3}(n)^{2}=\left(\sum _{d|n}{\tau _{2}(d)}\right)^{2}$, Sequence A097988

• 6) $\displaystyle \tau _{k}(n)=\prod _{i=0}^{k-2}{\frac {\tau _{2}(n\cdot rad(n)^{i})}{\tau _{2}(rad(n)^{i})}};\;(k\geq 1)$, where $\displaystyle rad(n)$ is A007947

• 7) $\displaystyle \sum _{d|n}{\mu (d)\cdot \tau _{2}(d)}=(-1)^{\omega (n)}=\mu (rad(n))$, Sequence A076479

• 8) $\displaystyle \sum _{d|n}{{\mu (d)}^{2}\cdot {\tau _{2}(d)}^{k}}=\tau _{k}(rad(n))={\bigg (}{2^{k}+1}{\bigg )}^{\omega (n)}$

• 9) $\displaystyle |\sum _{d|n}{\mu (d)\cdot \tau _{3}(d^{2})}|=\sum _{d|n}{{\mu (d)}^{2}\cdot {\tau _{2}(d)}^{2}}=\tau _{5}(rad(n))=5^{\omega (n)}$, Sequence A082476

• 10) $\displaystyle \tau _{2}(\tau _{2}(rad(n)))-1=\omega (n)\;;\;(n>1)$, Sequence A010553

• 11) $\displaystyle \sum _{d|n}{\mu (d)^{2}\cdot \tau _{2}(d)}=|\sum _{d|n}{\mu (d)\cdot \tau _{2}(d^{3})}|=|\sum _{d|n}{\mu (d)\cdot \tau _{4}(d)}|=\tau _{3}(rad(n))=3^{\omega (n)}$, Sequence A074816

• 12) $\displaystyle |\sum _{d|n}{\mu (d)\cdot \tau _{k}(d^{m})}|={{\bigg (}{\binom {k+m-1}{m}}-1{\bigg )}}^{\omega (n)}$

• 13) $\displaystyle \sum _{d|n}{\mu (d)\cdot \tau _{k}(d^{m})}={{\bigg (}1-{\binom {k+m-1}{m}}{\bigg )}}^{\omega (n)}$

• 14) $\displaystyle \phi (n)=\sum _{i=1}^{n}{{\bigg \lfloor }{\frac {\tau _{k}(i\cdot n)}{\tau _{k}(i)\cdot \tau _{k}(n)}}{\bigg \rfloor }}$ ,Sequence A000010

• 15) $\displaystyle \pi (n)=\sum _{i=2}^{n}{{\bigg \lfloor }{\frac {k}{\tau _{k}(i)}}{\bigg \rfloor }}\;\;;(n>1)$ ,Sequence A000720

• 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

• 17) $\displaystyle \tau _{k}(p^{\alpha })={\binom {\alpha +k-1}{k-1}}=P_{k-1}^{(\alpha ,\beta )}(1)\;(k>1)$ , where $\displaystyle P_{n}^{(\alpha ,\beta )}(z)$ are the Jacobi Polynomials

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

2) $\displaystyle \tau _{-k}(p)=-k;\;\;(k>0)$

3) $\displaystyle \tau _{-k}(s)=(-k)^{\omega (s)}$ , being $s$ a squarefree number: A005117.

Frozen Piltz Functions in OEIS

Function	Sequence Id
$\displaystyle \tau _{0}$	A063524
$\displaystyle \tau _{-1}$	A008683
$\displaystyle \tau _{-2}$	A007427
$\displaystyle \tau _{-3}$	A007428
$\displaystyle \tau _{-4}$	A247343
$\displaystyle \tau _{-5}$	A341831
$\displaystyle \tau _{-6}$	A341832
$\displaystyle \tau _{-7}$	A341833
$\displaystyle \tau _{-8}$	A341834
$\displaystyle \tau _{-9}$	A341835
$\displaystyle \tau _{-10}$	A341836

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).";

• Download Mathematica Piltz Functions Package

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).");

• Wrong comments in sequence: http://oeis.org/A061391

Bibliography

• Joe Roberts, The Lure of Integers, MAA, 1992, Integer 3, pages 8-9

• William J. LeVeque, Fundamentals of Number Theory, Dover Publications Inc, 1977, p.125

• Psychedelic Geometry Blogspot: SPLITTING FTA FUNCTIONS (II)-PILTZ DIVISOR FUNCTIONS (1)

• Psychedelic Geometry Blogspot: SPLITTING FTA FUNCTIONS (III)-PILTZ DIVISOR FUNCTIONS (2)

• Psychedelic Geometry Blogspot: TOTIENT CARNIVAL