A054116 -id:A054116 - cp's OEIS frontend

A100083 Numbers n such that n divides Sum_{m=1..n} (m+1)!.

1, 2, 4, 8, 31, 62, 124, 248, 373, 746, 1492, 2984, 11563, 23126, 46252, 92504

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 08 2004

n | Sum_{m=1..n} (m+1)! => n | Sum_{m=2..n+1} m! => n | Sum_{m=2..n-1} m! for n>2 => Sum_{m=2..n-1+k} m! == 0 (mod n) for all k>=0. If n is present and even, then n/2 is present. - Robert G. Wilson v, Nov 11 2004
The terms occur in groups of 4, in a series of n, 2n, 4n, and 8n. Is there any way of calculating the next term in the reduced series: 1, 31, 373, 11563? - Harvey P. Dale, Jun 11 2013
If n is in the sequence and k|n, then k is also in the sequence. In the other direction, if s and t is in the sequence and gcd(s,t)=1, then n=s*t is also in the sequence. Therefore, we need to check only the prime powers, after which we can easily build the rest of the sequence. The prime powers in the sequence begin with 2, 4, 8, 31, 373, ... - Robert Gerbicz, Jun 11 2013
For any terms in this sequence, their LCM also belongs to this sequence. If a(17) exists, it is prime. - Max Alekseyev, Jun 11 2013
a(17) > 74*10^7. - Lars Blomberg, Jun 15 2013
Integers n such that n | (A003422(n) - 2). - David W. Wilson, Jul 20 2013

			The first few partial sums of (m+1)!, starting with m=1 are 2, 8, 32, 152, 872, 5912, 46232, 409112. Of these, 2 is divisible by 1; 8 is divisible by 2; 152 is divisible by 4; but 32 is not divisible by 3. Therefore the first few terms of this sequence are 1, 2, 4.

R. Gerbicz (and others), Re: A100083, SeqFan list, Jun 11 2013

Cf. A057245, A064384

s = -1; Do[s = s + n!; If[ Mod[s, n] == 0, Print[n]], {n, 50000}] (* Robert G. Wilson v, Nov 15 2004 *)
Take[Flatten[Select[MapIndexed[List,Accumulate[Range[2,24000]!]], Divisible[#[[1]],#[[2,1]]]&]],{2,-1,2}] (* Harvey P. Dale, Jun 11 2013 *)

s=0:for(n=1,5000,s=s+(n+1)!: if(s%n==0,print(n)))

is(n)=my(t=Mod(1,n));sum(m=2,n+1,t*=m)==0 \\ Charles R Greathouse IV, Jun 11 2013

from itertools import count, islice
def A100083_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        a, c = 0, 1
        for m in range(2,n):
            c = c*m%n
            if c==0:
                break
            a = (a+c)%n
        if not a:
            yield n
A100083_list = list(islice(A100083_gen(),20)) # Chai Wah Wu, Apr 16 2024

Numbers n such that n | (A007489(n+1)-1), also n | (A003422(n+2)-2), n | A054116(n+1).

a(13)-a(14) from Robert G. Wilson v, Nov 15 2004
a(15) from Harvey P. Dale, Jun 11 2013
a(16) from Giovanni Resta, confirmed by Charles R Greathouse IV and Robert G. Wilson v, Jun 11 2013
Edited by Max Alekseyev, Mar 27 2015

A267123 Integers n such that n+2!, n+2!+3!, n+2!+3!+4!, n+2!+3!+4!+5!, n+2!+3!+4!+5!+6!, and n+2!+3!+4!+5!+6!+7! are all prime.

Original entry on oeis.org

11, 15, 99, 231, 351, 455, 725, 3525, 4935, 5405, 5709, 7575, 7641, 12545, 12891, 13749, 16065, 19859, 20475, 23969, 27791, 28049, 28571, 42459, 54615, 58755, 61979, 64481, 71835, 81011, 86261, 88649

Offset: 1

Emre APARI, Jan 10 2016

nonn

			99+2!=101 (is prime)
99+2!+3!=107 (is prime)
99+2!+3!+4!=131 (is prime)
99+2!+3!+4!+5!=251 (is prime)
99+2!+3!+4!+5!+6!=971 (is prime)
99+2!+3!+4!+5!+6!+7!=6011 (is prime)

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A054116, A267125.

r = {2, 8, 32, 152, 872, 5912}; fQ[n_] := Union[ PrimeQ[n + r]] == {True}; Select[ Range@ 100000, fQ] (* Robert G. Wilson v, Jan 10 2016 *)

is(n)=for(k=2,7,if(!isprime(n+=k!), return(0))); 1 \\ Charles R Greathouse IV, Feb 23 2016

list(lim)=my(v=List(),p=2,q=3,g,n); forprime(r=5,lim+8, g=q-p; if(g>6 || (g<6 && r-p>6), p=q;q=r; next); n=p+6; for(k=4,7, if(!isprime(n+=k!), p=q;q=r;next(2))); listput(v,p-2);p=q;q=r); Vec(v) \\ Charles R Greathouse IV, Feb 23 2016

A267125 Numbers n such that n+2!, n+2!+3!, n+2!+3!+4!, n+2!+3!+4!+5!, n+2!+3!+4!+5!+6!, n+2!+3!+4!+5!+6!+7!, n+2!+3!+4!+5!+6!+7!+8!, n+2!+3!+4!+5!+6!+7!+8!+9!, and n+2!+3!+4!+5!+6!+7!+8!+9!+10! are all prime.

Original entry on oeis.org

3525, 58755, 2171625, 3711201, 4612811, 4657289, 6714495, 7075271, 7687071, 9330381, 10523045, 11904249, 14060501, 16634171, 17191839, 22909971, 32351711, 35723709, 43992879, 45377325, 49031165, 56682171, 60219615, 64348635, 83743601, 86669615, 94265805

Offset: 1

Emre APARI, Jan 10 2016

nonn

			3525+2!=3527 (is prime)
3525+2!+3!=3533 (is prime)
3525+2!+3!+4!=3557 (is prime)
3525+2!+3!+4!+5!=3677 (is prime)
3525+2!+3!+4!+5!+6!=4397 (is prime)
3525+2!+3!+4!+5!+6!+7!=9437 (is prime)
3525+2!+3!+4!+5!+6!+7!+8!=49757 (is prime)
3525+2!+3!+4!+5!+6!+7!+8!+9!=412637 (is prime)
3525+2!+3!+4!+5!+6!+7!+8!+9!+10!=4041437 (is prime)

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A054116, A267123.

r = Accumulate@ Array[#! &, 9, 2]; fQ[n_] := Union[ PrimeQ[n + r]] == {True}; k = 1; lst = {}; While[k < 10^8, If[ fQ@ k, AppendTo[lst, k]]; k += 2]; lst (* Robert G. Wilson v, Jan 10 2016 *)

is(n)=for(k=2,10,if(!isprime(n+=k!), return(0))); 1 \\ Charles R Greathouse IV, Feb 23 2016

list(lim)=my(v=List(),p=2,q=3,g,n); forprime(r=5,lim+8, g=q-p; if(g>6 || (g<6 && r-p>6), p=q;q=r; next); n=p+6; for(k=4,10, if(!isprime(n+=k!), p=q;q=r;next(2))); listput(v,p-2);p=q;q=r); Vec(v) \\ Charles R Greathouse IV, Feb 23 2016

a(11) onward from Robert G. Wilson v, Jan 10 2016

A345889 Number of tilings of an n-cell circular array with rectangular tiles of any size, and where the number of possible colors of a tile is given by the smallest cell covered.

Original entry on oeis.org

1, 4, 16, 76, 436, 2956, 23116, 204556, 2018956, 21977356, 261478156, 3374988556, 46964134156, 700801318156, 11162196262156, 189005910310156, 3390192763174156, 64212742967590156, 1280663747055910156, 26826134832910630156, 588826498721714470156

Offset: 1

Lara Pudwell, Jun 28 2021

nonn

Jonathan Beagley and Lara Pudwell, Colorful Tilings and Permutations, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.4.

Partial differences give A001710.

Cf. A003422, A014288, A054116.

Accumulate@ Array[#!/2 &, 21, 2] (* Michael De Vlieger, Jun 28 2021 *)

a(n) = sum(k=2, n+1, k!/2); \\ Michel Marcus, Jun 29 2021

a(n) = Sum_{k=2..n+1} k!/2.
a(n) = A054116(n+1)/2.
a(n) = a(n-1) + A001710(n+1).
a(n) = A014288(n+1) - 1 = A003422(n+2)/2 - 1. - Alois P. Heinz, Jun 28 2021
a(n) ~ n*n!/2. - Stefano Spezia, Jun 29 2021

cp's OEIS Frontend

A100083 Numbers n such that n divides Sum_{m=1..n} (m+1)!.

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A267123 Integers n such that n+2!, n+2!+3!, n+2!+3!+4!, n+2!+3!+4!+5!, n+2!+3!+4!+5!+6!, and n+2!+3!+4!+5!+6!+7! are all prime.

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A267125 Numbers n such that n+2!, n+2!+3!, n+2!+3!+4!, n+2!+3!+4!+5!, n+2!+3!+4!+5!+6!, n+2!+3!+4!+5!+6!+7!, n+2!+3!+4!+5!+6!+7!+8!, n+2!+3!+4!+5!+6!+7!+8!+9!, and n+2!+3!+4!+5!+6!+7!+8!+9!+10! are all prime.

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Keywords

Examples

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Crossrefs

Programs

Mathematica

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Extensions

A345889 Number of tilings of an n-cell circular array with rectangular tiles of any size, and where the number of possible colors of a tile is given by the smallest cell covered.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula