A057825 -id:A057825 - cp's OEIS frontend

A049537 Values of k for which A075059(k) = A003418(k) + 1 is prime.

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 19, 20, 21, 22, 25, 26, 31, 47, 48, 89, 90, 91, 92, 93, 94, 95, 96, 127, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 1369, 1370, 1371, 1372, 2251, 2252, 2253, 2254, 2255, 2256, 2257, 2258, 2259, 2260, 2261, 2262, 2263, 2264, 2265, 2266, 3271, 3272, 3273, 3274, 3275, 3276, 3277, 3278, 3279, 3280, 3281, 3282, 3283, 3284, 3285, 3286, 3287

Offset: 1

Labos Elemer

nonn

			8 is not in the sequence because A075059(8) = 1 + A003418(8) = 1 + lcm(1, 2, ..., 8) = 841 = 29^2 is not prime.
127 is in the sequence because A075059(127) = 1 + A003418(127) = 1 + lcm(1, 2, ..., 127) = 6676878045498705789701874602220118271269436344024536001 is prime.

Jinyuan Wang, Table of n, a(n) for n = 1..150

Cf. A000040, A003418, A049536, A057825, A075059.

Join[{0}, Select[Range[250], PrimeQ[LCM@@Range[#]+1]&]] (* Harvey P. Dale, Nov 15 2011 *)

isok(n) = isprime(lcm(vector(n, i, i))+1); \\ Michel Marcus, Feb 25 2014

a(43)-a(57) from Ray Chandler, Jan 16 2009

a(1)=0 prepended and a(58)-a(86) added by Max Alekseyev, Sep 04 2015

A070198 Smallest nonnegative number m such that m == i (mod i+1) for all 1 <= i <= n.

Original entry on oeis.org

0, 1, 5, 11, 59, 59, 419, 839, 2519, 2519, 27719, 27719, 360359, 360359, 360359, 720719, 12252239, 12252239, 232792559, 232792559, 232792559, 232792559, 5354228879, 5354228879, 26771144399, 26771144399, 80313433199, 80313433199

Offset: 0

Benoit Cloitre, May 06 2002

Also, smallest k such that, for 0 <= i < n, i+1 divides k-i.
Suggested by Chinese Remainder Theorem. This sequence can generate others: smallest b(n) such that b(n) == i (mod (i+2)), 1 <= i <= n, gives b(1)=1 and b(n) = a(n+1)-1 for n > 1; smallest c(n) such that c(n) == i (mod (i+3)), 1 <= i <= n, gives c(1)=1, c(2)=17 and c(n) = a(n+2) - 2 for n > 2; smallest d(n) such that c(n) == i (mod (i+4)), 1 <= i <= n, gives d(1)=1, d(2)=26, d(3)=206 and d(n) = a(n+3) - 3 for n > 3, etc.
A208768(n) occurs A057820(n) times. - Reinhard Zumkeller, Mar 01 2012
From Kival Ngaokrajang, Oct 10 2013: (Start)
A070198(n-1) is m such that max(Sum_{i=1..n} m (mod i)) = A000217(n-1).
Example for n = 3:
  m\i = 1  2  3  sum
  1     0  1  1   2
  2     0  0  2   2
  3     0  1  0   1
  4     0  0  1   1
  5     0  1  2   3 <--max remainder sum = 3 = A000217(2)
  6     0  0  0   0  first occurs at m = 5 = A070198(2)
(End)

			a(3) = 11 because 11 == 1 (mod 2), 11 == 2 (mod 3) and 11 == 3 (mod 4).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Chinese Remainder Theorem
Wikipedia, Chinese remainder theorem
Index entries for sequences related to lcm's

Cf. A053664, A072562.
Cf. A057825 (indices of primes). - R. J. Mathar, Jan 14 2009
Cf. A116151. - Zak Seidov, Mar 11 2014

a070198 n = a070198_list !! n
a070198_list = map (subtract 1) $ scanl lcm 1 [2..]
-- Reinhard Zumkeller, Mar 01 2012

[Exponent(SymmetricGroup(n))-1 : n in [1..30]]; /* Vincenzo Librandi, Oct 31 2014 - after Arkadiusz Wesolowski in A003418 */

seq(ilcm($1..n) - 1, n=1..100); # Robert Israel, Nov 03 2014

f[n_] := ChineseRemainder[ Range[0, n - 1], Range[n]]; Array[f, 28] (* or *)
f[n_] := LCM @@ Range@ n - 1; Array[f, 28] (* Robert G. Wilson v, Oct 30 2014 *)

from math import lcm
def A070198(n): return lcm(*range(1,n+2))-1 # Chai Wah Wu, May 02 2023

a(n) = lcm(1, 2, 3, ..., n+1) - 1 = A003418(n+1) - 1.

Edited by N. J. A. Sloane, Nov 18 2007, at the suggestion of Max Alekseyev

A154524 Primes p such that lcm(1,2,3,...,p-2,p-1,p) - 1 is prime.

Original entry on oeis.org

3, 5, 7, 19, 23, 29, 47, 61, 97, 181, 233, 307, 401, 887, 1021, 1087, 1361, 1481, 2053, 2293, 5407, 5857, 11059, 14281, 27277, 27803, 36497, 44987, 53017

Offset: 1

Lekraj Beedassy, Jan 11 2009

nonn

a(28) > 42000. - Daniel Suteu, Oct 06 2018

a(30) > 100000. - Michael S. Branicky, Jul 04 2025

			7 is in the sequence because it is prime and also lcm(1,2,3,4,5,6,7)-1 = 420-1 = 419 is prime. - _Emeric Deutsch_, Jan 16 2009

Cf. A056604, A154525, A154526.

P := proc(n) options operator, arrow: ilcm(seq(j, j = 1 .. n)) end proc: a := proc(n) if isprime(n) and isprime(P(n)-1) then n else end if end proc: seq(a(n), n = 1 .. 3000); # Emeric Deutsch, Jan 16 2009

A057825 INTERSECT A000040. - R. J. Mathar, Jan 14 2009

a(8)-a(17) from Ray Chandler, Jan 16 2009
a(18)-a(22) from Emeric Deutsch, Jan 16 2009
a(23)-a(27) from Daniel Suteu, Oct 06 2018
a(28)-a(29) from Michael S. Branicky, Jul 03 2025

A385564 Prime powers k such that lcm(1, 2, 3, ..., k)-1 is prime.

Original entry on oeis.org

3, 4, 5, 7, 8, 19, 23, 29, 32, 47, 61, 97, 181, 233, 307, 401, 887, 1021, 1087, 1361, 1481, 2053, 2293, 5407, 5857, 11059, 14281, 27277, 27803, 36497, 44987, 53017

Offset: 1

Jeppe Stig Nielsen, Jul 03 2025

nonn

The prime associated with each a(n) is A057824(n).

a(33) > 10^5. Up to 10^5, contains 4, 8, 32 not in subsequence A154524. - Michael S. Branicky, Jul 04 2025

			k=32 is a prime power, so point at which A003418 attains a new value, namely lcm(1, 2, 3, ..., 32) = 144403552893600, and by subtracting one we get 144403552893599 which is a prime number, so 32 is a member of the sequence.

Intersection of A057825 and A000961.
Supersequence of A154524.
Cf. A057824, A051453, A003418.

Select[Range[6000],PrimePowerQ[#]&&PrimeQ[Fold[LCM,Range[#]]-1]&] (* James C. McMahon, Jul 09 2025 *)

L=1;for(k=2,6000,!isprimepower(k,&p)&&next();L*=p;ispseudoprime(L-1)&&print1(k,", "))

a(31)-a(32) from Michael S. Branicky, Jul 03 2025

cp's OEIS Frontend

A049537 Values of k for which A075059(k) = A003418(k) + 1 is prime.

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A070198 Smallest nonnegative number m such that m == i (mod i+1) for all 1 <= i <= n.

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A154524 Primes p such that lcm(1,2,3,...,p-2,p-1,p) - 1 is prime.

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A385564 Prime powers k such that lcm(1, 2, 3, ..., k)-1 is prime.

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