A075059 -id:A075059 - 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

A075061 Triangle in A075059 read by rows.

Original entry on oeis.org

2, 3, 4, 7, 8, 9, 13, 14, 15, 16, 61, 62, 63, 64, 65, 61, 62, 63, 64, 65, 66, 421, 422, 423, 424, 425, 426, 427, 841, 842, 843, 844, 845, 846, 847, 848, 2521, 2522, 2523, 2524, 2525, 2526, 2527, 2528, 2529, 2521, 2522, 2523, 2524, 2525, 2526, 2527, 2528, 2529

Offset: 1

Amarnath Murthy, Sep 08 2002

			2;
3 4;
7 8 9;
13 14 15 16;
61 62 63 64 65;
61 62 63 64 65 66;
...

Cf. A075059, A060401, A075062.

T(n,k+1)=T(n,k)+1. T(n,0)=A075059(n). - R. J. Mathar, Mar 01 2007

More terms from R. J. Mathar, Mar 01 2007

A075062 Row sums of triangle in A075059.

Original entry on oeis.org

2, 7, 24, 58, 315, 381, 2968, 6756, 22725, 25255, 304986, 332718, 4684771, 5045145, 5405520, 11531656, 208288233, 220540491, 4423058830, 4655851410, 4888643991, 5121436573, 123147264516, 128501493420, 669278610325, 696049754751

Offset: 1

Amarnath Murthy, Sep 08 2002

nonn

For odd n, a(n) is a multiple of n.

Cf. A075059, A060401, A075061.

a(n)=n*A075059(n)+A000217(n-1). - R. J. Mathar, Mar 20 2007

More terms from R. J. Mathar, Mar 20 2007

A060401 a(n) = minimal m such that m>n, n divides m, n-1 divides m-1, n-2 divides m-2 and so on down to 1 divides m-n+1.

Original entry on oeis.org

2, 4, 9, 16, 65, 66, 427, 848, 2529, 2530, 27731, 27732, 360373, 360374, 360375, 720736, 12252257, 12252258, 232792579, 232792580, 232792581, 232792582, 5354228903, 5354228904, 26771144425, 26771144426, 80313433227, 80313433228, 2329089562829, 2329089562830

Offset: 1

Christopher Burrows (cburrows(AT)math.upenn.edu), Apr 04 2001

nonn

A099427(a(n)) = n + 1. - Reinhard Zumkeller, Jul 02 2011

			a(5) = 65 because 5|65, 4|64, 3|63, 2|62, 1|61 and 65 is minimal.

Robert Israel, Table of n, a(n) for n = 1..2296

Rightmost diagonal of triangle in A075059. Cf. A075061, A075062.

Cf. A003418.

seq(n+ilcm($1..n),n=1..100); # Robert Israel, Jul 19 2016

a(n) = n + lcm(seq(i, i=1..n)).

a(n) = n + A003418(n). - Robert Israel, Jul 19 2016

A070858 Smallest prime == 1 mod L, where L = LCM of 1 to n.

Original entry on oeis.org

2, 3, 7, 13, 61, 61, 421, 2521, 2521, 2521, 55441, 55441, 4324321, 4324321, 4324321, 4324321, 85765681, 85765681, 232792561, 232792561, 232792561, 232792561, 10708457761, 10708457761, 26771144401, 26771144401, 401567166001, 401567166001, 18632716502401, 18632716502401

Offset: 1

Amarnath Murthy, May 16 2002

nonn

Beginning with 3, smallest prime p = a(n) such that p + k is divisible by k + 1 for each k = 1, 2, ..., n. For example: 61 --> 62, 63, 64, 65 and 66 are divisible respectively by 2, 3, 4, 5 and 6. - Robin Garcia, Jul 23 2012

Amiram Eldar, Table of n, a(n) for n = 1..1000 (terms 1..200 from R. J. Mathar)

Cf. A060357, A075059, A003418, A070844 to A070856, A035091.

A070858 := proc(n)
    local l,p;
    l := ilcm(seq(i,i=1..n)) ;
    for p from 1 by l do
        if isprime(p) then
            return p;
        end if;
    end do:
end proc; # R. J. Mathar, Jun 25 2013

a[n_] := Module[{m = 1, lcm = LCM @@ Range[n]}, While[!PrimeQ[m], m += lcm]; m]; Array[a, 30] (* Amiram Eldar, Mar 15 2025 *)

a(n)=my(L=lcm(vector(n,i,i)),k=1);while(!ispseudoprime(k+=L),); k \\ Charles R Greathouse IV, Jun 25 2013

More terms from Sascha Kurz, Feb 02 2003

A249051 The smallest integer > 1 of exactly n consecutive integers divisible respectively by the first n natural numbers (A000027), or 0 if no such number exists.

Original entry on oeis.org

2, 3, 7, 13, 0, 61, 421, 841, 0, 2521, 0, 27721, 0, 0, 360361, 720721, 0, 12252241, 0, 0, 0, 232792561, 0, 5354228881, 0, 26771144401, 0, 80313433201, 0, 2329089562801, 72201776446801, 0, 0, 0, 0, 144403552893601, 0, 0, 0, 5342931457063201, 0

Offset: 1

Robert G. Wilson v, Oct 30 2014

nonn

For all n > 1 and a(n) # 0, a(n) == 1 (mod p#), where p# are the primorial numbers (A034386).
When a(n) is not 0, a(n) = A075059(n).
a(n) = 0 when n is a member of A080765.

			a(3) = 7 because the smallest k such that 1|k, 2|k+1, 3|k+2, and 4 does not divide k+3 is 7.
a(4) = 13 because the smallest k such that 1|k, 2|k+1, 3|k+2, 4|k+3, and 5 does not divide k+4 is 13.

Cf. A075059, A034386, A080765.

f[n_] := Block[{lcm = LCM @@ Range@ n}, If[ lcm == LCM @@ Range[n + 1], 0, lcm + 1]]; Array[ f, 42] (* Robert G. Wilson v, Nov 13 2014 *)

a(5) corrected (0, not 181) by Jon Perry, Nov 05 2014

Sequence corrected by Robert G. Wilson v, Nov 13 2014

A254078 a(n) is the number of steps after which n variables with increasing value ranges all have equal values when the values of all variables are decreased by 1 at each step and the value is set to the maximum value again when the resulting value would be 0.

Original entry on oeis.org

4, 10, 58, 58, 418, 838, 2518, 2518, 27718, 27718, 360358, 360358, 360358, 720718, 12252238, 12252238, 232792558, 232792558, 232792558, 232792558, 5354228878, 5354228878, 26771144398, 26771144398, 80313433198, 80313433198, 2329089562798

Offset: 2

Felix Fröhlich, Jan 25 2015

nonn

The k-th variable can take k+1 different values.
From Charlie Neder, Oct 01 2018: (Start)
a(n) is the smallest k congruent to m-2 modulo m for 2 <= m <= n+1.
Proof: All variables will be equal for the first time precisely when they all are equal to 2, in which case each variable has changed from its maximum value m to 2. Additionally, this k is lcm(2,3,...,m) - 2, since advancing two more steps will return all variables to their maximum values.
Adding a variable that only takes one value {1} results in A070198 (LCM - 1). (End)

			In case of two variables, the first can take two values (1 and 2) and the second three values (1, 2 and 3). Performing the operation on the variables generates sequences of values 2, 1, 2, 1, 2, 1, ... for first variable and 3, 2, 1, 3, 2, 1, ... for second variable. After four steps, the value of both variables is 2, so a(2) = 4.

Felix Fröhlich, Visual representation of a(3)

Cf. A070198 (LCM - 1), A003418 (LCM), A075059 (LCM + 1).

a(n) = my(v=vector(n, x, x++), w=v, i=0); while(1, if(vecmax(v)==vecmin(v), return(i)); for(k=1, #v, if(v[k]==1, v[k]=w[k], v[k]--)); i++) \\ Felix Fröhlich, Feb 19 2017

from math import gcd
lcm = 2
for n in range(3,53):
  lcm *= n // gcd(lcm,n)
  print(n-1,lcm-2) # Charlie Neder, Oct 02 2018

a(n) = A003418(n+1) - 2. - Charlie Neder, Oct 02 2018

Value of a(6) corrected and more terms from Felix Fröhlich, Mar 25 2015
Illustration and program replaced with improved versions by Felix Fröhlich, Feb 19 2017
Corrected and extended by Charlie Neder, Oct 01 2018

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A075061 Triangle in A075059 read by rows.

Views

Author

Keywords

Examples

Crossrefs

Formula

Extensions

A075062 Row sums of triangle in A075059.

Views

Author

Keywords

Comments

Crossrefs

Formula

Extensions

A060401 a(n) = minimal m such that m>n, n divides m, n-1 divides m-1, n-2 divides m-2 and so on down to 1 divides m-n+1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A070858 Smallest prime == 1 mod L, where L = LCM of 1 to n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A249051 The smallest integer > 1 of exactly n consecutive integers divisible respectively by the first n natural numbers (A000027), or 0 if no such number exists.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A254078 a(n) is the number of steps after which n variables with increasing value ranges all have equal values when the values of all variables are decreased by 1 at each step and the value is set to the maximum value again when the resulting value would be 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

Formula

Extensions