A243103 Product of numbers m with 2 <= m <= n whose prime divisors all divide n.

1, 2, 3, 8, 5, 144, 7, 64, 27, 3200, 11, 124416, 13, 6272, 2025, 1024, 17, 35831808, 19, 1024000, 3969, 247808, 23, 859963392, 125, 346112, 729, 2809856, 29, 261213880320000000, 31, 32768, 264627, 18939904, 30625, 26748301344768, 37, 23658496, 369603, 32768000000, 41

Offset: 1

Author

Michael De Vlieger, Aug 19 2014

Keywords

• nonn

Comments

This sequence is the product of n-regular numbers.

A number m is said to be "regular" to n or "n-regular" if all the prime factors p of m also divide n.

The divisor is a special case of a regular m such that m also divides n in addition to all of its prime factors p | n.

Analogous to A007955 (Product of divisors of n).

If n is 1 or prime, a(n) = n.

If n is a prime power, a(n) = A007955(n).

Note: b-file ends at n = 4619, because a(4620) has more than 1000 decimal digits.

Product of the numbers 1 <= k <= n such that (floor(n^k/k) - floor((n^k - 1)/k)) = 1. - Michael De Vlieger, May 26 2016

Examples

			a(12) = 124416 since 1 * 2 * 3 * 4 * 6 * 8 * 9 * 12 = 124416. These numbers are products of prime factors that are the distinct prime divisors of 12 = {2, 3}.
From _David A. Corneth_, Feb 09 2015: (Start)
Let p# be the product of primes up to p, A002110. Then
a(13#) ~= 8.3069582 * 10 ^ 4133
a(17#) ~= 1.3953000 * 10 ^ 22689
a(19#) ~= 3.8258936 * 10 ^ 117373
a(23#) ~= 6.7960327 * 10 ^ 594048
a(29#) ~= 1.3276817 * 10 ^ 2983168
a(31#) ~= 2.8152792 * 10 ^ 14493041
a(37#) ~= 1.9753840 * 10 ^ 69927040
Up to n = 11# already in the table.
(End)
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..4619
• Encyclopedia Britannica, Regular Number (base-neutral definition)
• Eric W. Weisstein, Regular Number (decimal definition)
• Wikipedia, Regular Number (sexagesimal / Hamming number definition)

Crossrefs

Cf. A162306 (irregular triangle of regular numbers of n), A010846 (number of regular numbers of n), A244974 (sum of regular numbers of n), A007955, A244052 (record transform of regular numbers of n).

Cf. A123275, A283990.

Programs

Maple

A:= proc(n) local F, S, s, j, p;
  F:= numtheory:-factorset(n);
  S:= {1};
  for p in F do
    S:= {seq(seq(s*p^j, j=0..floor(log[p](n/s))), s=S)}
  od;
  convert(S,`*`)
end proc:
seq(A(n), n=1..100); # Robert Israel, Feb 09 2015

Mathematica

regularQ[m_Integer, n_Integer] := Module[{omega = First /@ FactorInteger @ m }, If[Length[Select[omega, Divisible[n, #] &]] == Length[omega], True, False]]; a20140819[n_Integer] := Times @@ Flatten[Position[Thread[regularQ[Range[1, n], n]], True]]; a20140819 /@ Range[41]
regulars[n_] := Block[{f, a}, f[x_] := First /@ FactorInteger@ x; a = f[n];{1}~Join~Select[Range@ n, SubsetQ[a, f@ #] &]]; Array[Times @@ regulars@ # &, 12] (* Michael De Vlieger, Feb 09 2015 *)
Table[Times @@ Select[Range@ n, (Floor[n^#/#] - Floor[(n^# - 1)/#]) == 1 &], {n, 41}] (* Michael De Vlieger, May 26 2016 *)

PARI

lista(nn) = {vf = vector(nn, n, Set(factor(n)[,1])); vector(nn, n, prod(i=1, n, if (setintersect(vf[i], vf[n]) == vf[i], i, 1)));} \\ Michel Marcus, Aug 23 2014

PARI

for(n=1, 100, print1(prod(k=1, n, k^(floor(n^k/k) - floor((n^k - 1)/k))),", ")) \\ Indranil Ghosh, Mar 22 2017

Python

from sympy import primefactors
def A243103(n):
    y, pf = 1, set(primefactors(n))
    for m in range(2,n+1):
        if set(primefactors(m)) <= pf:
            y *= m
    return y # Chai Wah Wu, Aug 28 2014

Scheme

;; A naive implementation, code for A123275bi given under A123275:
(define (A243103 n) (let loop ((k n) (m 1)) (cond ((= 1 k) m) ((= 1 (A123275bi n k)) (loop (- k 1) (* m k))) (else (loop (- k 1) m)))))
;; Antti Karttunen, Mar 22 2017

Formula

a(n) = product of terms of n-th row of irregular triangle A162306(n,k).

a(n) = Product_{k=1..n} k^( floor(n^k/k)-floor((n^k -1)/k) ). - Anthony Browne, Jul 06 2016

From Antti Karttunen, Mar 22 2017: (Start)

a(n) = Product_{k=2..n, A123275(n,k)=1} k.

For n >= 1, A046523(a(n)) = A283990(n).

(End)