A070251 -id:A070251 - cp's OEIS frontend

A074845 Numbers k such that S(k) = largest difference between consecutive divisors of k (ordered by size), where S(k) is the Kempner function (A002034).

6, 8, 9, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514

Offset: 1

Jason Earls, Sep 10 2002

It appears that terms > 6 are simply given by: composite k such that k^2 doesn't divide A000254(k). - Benoit Cloitre, Mar 09 2004
It appears that A011776(a(k)) = 2. - Gionata Neri, Jul 31 2017
It appears that this sequence consists of the numbers k such that A045763(k) > 0 and k does not divide A070251(k). - Isaac Saffold, Jun 01 2018

Michael De Vlieger, Table of n, a(n) for n = 1..670 (a(n) < 10^4, from b-file at A002034).

Cf. A002034, A060681.

Select[Range@ 514, Function[n, Module[{m = 1}, While[! Divisible[m!, n], m++]; m] == Max@ Differences@ Divisors@ n]] (* Michael De Vlieger, Jul 31 2017 *)

K(n) = my(s=1); while(s!%n>0, s++); s;
dd(n) = my(vd=divisors(n)); vecmax(vector(#vd-1, k, vd[k+1] - vd[k]));
isok(n) = K(n) == dd(n); \\ Michel Marcus, Aug 03 2017

A381674 a(n) = product of numbers k < n such that 1 < gcd(k,n) and rad(k) != rad(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 24, 1, 6, 6, 1920, 1, 17280, 1, 322560, 97200, 10080, 1, 58060800, 1, 1393459200, 51438240, 40874803200, 1, 536481792000, 3000, 25505877196800, 9797760, 535623421132800, 1, 40999294770610176000000, 1, 41845579776000, 51855036710400, 23310331287699456000

Offset: 1

Michael De Vlieger, Mar 15 2025

nonn

Terms are in A055932.

The only squarefree terms are 1 and 6.

			Table of n and a(n) for select n, showing exponents of prime factors of the latter and row n of A381094:
                                             1  1  1
   n                       a(n)  2  3  5  7  1  3  7   Row n of A381094
  ---------------------------------------------------------------------------------------
   6                        24   3, 1                  {2,3,4}
   8                         6   1, 1                  {6}
   9                         6   1, 1                  {6}
  10                      1920   7, 1, 1               {2,4,5,6,8}
  12                     17280   7, 3, 1               {2,3,4,8,9,10}
  14                    322560  10, 2, 1, 1            {2,4,6,7,8,10,12}
  15                     97200   4, 5, 2               {3,5,6,9,10,12}
  16                     10080   5, 2, 1, 1            {6,10,12,14}
  18                  58060800  12, 4, 2, 1            {2,3,4,8,9,10,14,15,16}
  20                1393459200  15, 5, 2, 1            {2,4,5,6,8,12,14,15,16,18}
  24              536481792000  15, 5, 3, 2, 1         {2,3,4,8,9,10,14,15,16,20,21,22}
  25                      3000   3, 1, 3               {10,15,20}
  30   40999294770610176000000  25,13, 6, 3, 1, 1      {2,3,4,5,6,8,9,10,12,14,..,28}
  32            41845579776000  16, 6, 3, 2, 1, 1      {6,10,12,14,18,20,22,24,26,28,30}
  36   11358323143857930240000  25,10, 4, 3, 2, 1, 1   {2,3,4,8,9,10,14,15,16,20,..,34}
a(n) = 6 for n = 8 or 9, since 6 is the only number less than n that shares a factor with n but rad(6) != rad(n).
a(6) = (2*3)*(4) = 24.
a(10) = (2*4*6*8)*(5) = 1920.
a(12) = (2*4*8*10)*(3*9) = 17280, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..599
Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..2^14, showing n that are prime powers in gold, n that are squarefree in green, and other n in blue and magenta, where magenta additionally represents powerful n that are not prime powers.
Michael De Vlieger, Plot prime(i)^m | a(n) at (x,y) = (n, i), n = 1..2048, 2X vertical exaggeration, with a color function representing m, where black represents m = 1, red m = 2, ..., magenta represents the largest m in the dataset, i.e., m = 2035.

Cf. A055932, A070251, A381094, A381497, A381675.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
Table[r = rad[n]; Times @@ Select[Range[n], Nor[CoprimeQ[#, n], rad[#] == r] &], {n, 120}]

a(n) is the product of row n of A381094.
a(n) = 1 for prime n and n = 4.
a(2*p) = p * 2^(p-1) * (p-1)! = A381675(n) for odd prime p = prime(n), n > 1.

cp's OEIS Frontend

A074845 Numbers k such that S(k) = largest difference between consecutive divisors of k (ordered by size), where S(k) is the Kempner function (A002034).

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