A048839 -id:A048839 - cp's OEIS frontend

A122144 Numbers k such that q(k) = M(k) where q(n) is the largest prime divisor of k and M(k) is the largest prime power divisor of k.

2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 99

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Aug 22 2006

nonn

Similar to A048839, the first difference occurs when n = 40.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006530, A034699, A048839, A122145, A122146.

Select[Range[2, 100], Max[Power @@@ (f = FactorInteger[#])] == f[[-1, 1]] &] (* Amiram Eldar, May 23 2024 *)

isok(k) = {my(f = factor(k), pm = 0); if(k > 1, for(i = 1, #f~, pm = max(pm, f[i, 1]^f[i, 2])); pm == f[#f~, 1], 0);} \\ Amiram Eldar, May 23 2024

Edited by Ray Chandler, Aug 23 2006

A371152 Array read by antidiagonals: row n lists numbers m >= 2 with A362333(m) = n.

Original entry on oeis.org

2, 3, 4, 5, 9, 8, 6, 12, 24, 16, 7, 18, 27, 48, 32, 10, 25, 54, 81, 96, 64, 11, 36, 72, 144, 243, 192, 128, 13, 45, 108, 162, 288, 576, 384, 256, 14, 49, 125, 324, 486, 729, 1152, 768, 512, 15, 50, 135, 405, 864, 1458, 2187, 2304, 1536, 1024

Offset: 1

Pontus von Brömssen, Mar 13 2024

			Table begins:
  n\k|    1    2    3     4     5     6      7      8      9     10
  ---+-------------------------------------------------------------
   1 |    2    3    5     6     7    10     11     13     14     15
   2 |    4    9   12    18    25    36     45     49     50     75
   3 |    8   24   27    54    72   108    125    135    216    250
   4 |   16   48   81   144   162   324    405    432    625    648
   5 |   32   96  243   288   486   864    972   1215   1944   2430
   6 |   64  192  576   729  1458  1728   2916   3645   5184   5832
   7 |  128  384 1152  2187  3456  4374   8748  10368  10935  17496
   8 |  256  768 2304  6561  6912 13122  20736  26244  32805  52488
   9 |  512 1536 4608 13824 19683 39366  41472  78732  98415 124416
  10 | 1024 3072 9216 27648 59049 82944 118098 236196 248832 295245

Cf. A048839 (first row), A057109 (terms in rows 2, 3, ...), A362333.

T(n,1) = 2^n.

A349093 a(n) is the smallest nonprime number m (m = A018252(t)) such that n divides the product P(t) of all nonprime numbers up to and including m (P(t) = A036691(t-1)).

Original entry on oeis.org

1, 4, 6, 4, 10, 6, 14, 6, 9, 10, 22, 6, 26, 14, 10, 8, 34, 9, 38, 10, 14, 22, 46, 6, 15, 26, 9, 14, 58, 10, 62, 8, 22, 34, 14, 9, 74, 38, 26, 10, 82, 14, 86, 22, 10, 46, 94, 8, 21, 15, 34, 26, 106, 9, 22, 14, 38, 58

Offset: 1

Lechoslaw Ratajczak, Mar 25 2022

nonn

a(n) >= 2*gpf(n) for n > 1, where gpf(n) denotes the greatest prime factor of n (A006530(n)).
Conjecture: the equation a(n) = a(n+1) has no solutions. This holds up to at least n = 10^7.
Consecutive solutions of the equation a(n) = 2*K(n) (where K(n) is the Kempner number A002034(n)) are consecutive terms of A048839.

			a(15) = 10 because:
15 does not divide 1=A036691(0)=1, 1*4=A036691(1)=4, 1*4*6=A036691(2)=24, 1*4*6*8=A036691(3)=192, 1*4*6*8*9=A036691(4)=1728 and does divide 1*4*6*8*9*10=A036691(5)=17280.

J. Sondow and E. W. Weisstein, MathWorld: Smarandache Function

Cf. A002034, A006530, A018252, A036691.

f(p,k):=(z:2, for m:2 while -1+sum(floor((p*m)/(p^t)),t,1,m)

a(p) = 2*p for prime p.
a(p_1*p_2*...*p_u) = 2*p_u, where p_i's are distinct primes and p_1 < p_2 < ... < p_u.
a(n) where n is factored as n = p_1^k_1*p_2^k_2*...*p_u^k_u is given by a(n) = max( a(p_1^k_1), a(p_2^k_2), ..., a(p_u^k_u) ), where a(p_i^k_i) = w*p_i and w is the smallest m >= 2 satisfying the inequality:
-1 + Sum_{t=1..m} floor((m*p_i)/(p_i)^t) >= k_i.

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A122144 Numbers k such that q(k) = M(k) where q(n) is the largest prime divisor of k and M(k) is the largest prime power divisor of k.

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A371152 Array read by antidiagonals: row n lists numbers m >= 2 with A362333(m) = n.

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A349093 a(n) is the smallest nonprime number m (m = A018252(t)) such that n divides the product P(t) of all nonprime numbers up to and including m (P(t) = A036691(t-1)).

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