A007619 -id:A007619 - cp's OEIS frontend

A239566 (Round(c^prime(n)) - 1)/prime(n), where c is the heptanacci constant (A118428).

7200, 25562, 332466, 16472758, 61145666, 3200477798, 45473543628, 172043098818, 2478186385762, 137291966046470, 7704742900338106, 29569459376703894, 1681851263230158754, 24987922624169214866, 96433670513455876108, 5566902760779797458210

Offset: 7

Peter J. C. Moses and Vladimir Shevelev, Mar 21 2014

nonn

For n>=7, round(c^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link. In particular, all terms are even.

S. Litsyn and V. Shevelev, Irrational Factors Satisfying the Little Fermat Theorem, International Journal of Number Theory, vol.1, no.4 (2005), 499-512.
V. Shevelev, A property of n-bonacci constant, Seqfan (Mar 23 2014)
Eric Weisstein's World of Mathematics, Heptanacci Constant

Cf. A007619, A007663, A238693, A238697, A238698, A238700, A239502, A239544, A239564, A239565.

All roots of the equation x^7-x^6-x^5-x^4-x^3-x^2-x-1 = 0
are the following: c=1.9919641966050350211,
-0.78418701799584451319 +/- 0.36004972226381653409*i,
-0.24065633852269642508 + /- 0.84919699909267892575*i,
  0.52886125821602342773 +/-  0.76534196109589443115*i.
Absolute values of all roots, except for septanacci constant c, are less than 1.
Conjecture. Absolute values of all roots of the equation x^n - x^(n-1) - ... -x - 1 = 0, except for n-bonacci constant c_n, are less than 1. If the conjecture is valid, then for sufficiently large k=k(n), for all m>=k, we have round(c_n^prime(m)) == 1 (mod 2*prime(m)) (cf. Shevelev link).

A250407 Near-Wilson primes (p = prime(n) satisfying (p-1)! == -1-A250406(n)*p (mod p^2)) with A250406(n) < 10.

Original entry on oeis.org

2, 3, 5, 7, 13, 61, 71, 79, 157, 281, 563, 1277, 1777, 2339, 6311, 8233, 8543, 11047, 22907, 27689

Offset: 1

Felix Fröhlich, Nov 22 2014

A250406(n) is essentially A007619(n) modulo A000040(n) (see Crandall et al. (1997), p. 442).

E. Costa, R. Gerbicz and D. Harvey, A search for Wilson primes, Math. Comp., 83 (2014), 3071-3091.
R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Math. Comp., 66 (1997), 433-449.

Cf. A007540, A246568, A250406.

forprime(p=1, 1e9, for(b=0, 9, if(Mod((p-1)!, p^2)==-1-b*p, print1(p, ", "); break({1}))))

A134295 a(n) = Sum_{k=1..n} ((n-k)!*(k-1)! - (-1)^k).

Original entry on oeis.org

2, 2, 6, 16, 65, 312, 1813, 12288, 95617, 840960, 8254081, 89441280, 1060369921, 13649610240, 189550368001, 2824077312000, 44927447040001, 760034451456000, 13622700994560001, 257872110354432000, 5140559166898176001

Offset: 1

Alexander Adamchuk, Oct 17 2007

nonn

According to the Generalized Wilson-Lagrange Theorem, a prime p divides (p-k)!*(k-1)! - (-1)^k for all integers k > 0. p divides a(p) for prime p. Quotients a(p)/p are listed in A134296(n) = {1, 2, 13, 259, 750371, 81566917, 2642791002353, 716984262871579, 102688143363690674087, ...}. p^2 divides a(p) for prime p = {7, 71}.

Cf. A007540, A007619 (Wilson quotients: ((p-1)!+1)/p).

Cf. A134296 (quotients a(p)/p).

Table[ Sum[ (n-k)!*(k-1)! - (-1)^k, {k,1,n} ], {n,1,30} ]

a(n) = Sum_{k=1..n} ((n-k)!*(k-1)! - (-1)^k).

A134296 Quotients A134295(p)/p = (1/p) * Sum_{k=1..p} ((p-k)!*(k-1)! - (-1)^k), where p = prime(n).

Original entry on oeis.org

1, 2, 13, 259, 750371, 81566917, 2642791002353, 716984262871579, 102688143363690674087, 21841112114495269555043222069, 17727866746681961093761724283871

Offset: 1

Alexander Adamchuk, Oct 17 2007

nonn

A134295(n) = Sum_{k=1..n} ((n-k)!*(k-1)! - (-1)^k) = {2, 2, 6, 16, 65, 312, 1813, 12288, 95617, 840960, 8254081, ...}. According to the Generalized Wilson-Lagrange Theorem, a prime p divides (p-k)!*(k-1)! - (-1)^k for all integers k > 0. a(n) = A134295(p)/p for p = prime(n). a(n) is prime for n = {2, 3, 7, 9, 37, ...}. Corresponding prime terms in a(n) are {2, 13, 2642791002353, 102688143363690674087, ...}.

Cf. A007540, A007619 (Wilson quotients: ((p-1)!+1)/p).

Cf. A134295 (Sum_{k=1..n} ((n-k)!*(k-1)! - (-1)^k)).

Table[ (Sum[ (Prime[n]-k)!*(k-1)! - (-1)^k, {k,1,Prime[n]} ]) / Prime[n], {n,1,20} ]

a(n) = (1/p) * Sum_{k=1..p} ((p-k)!*(k-1)! - (-1)^k) where p = prime(n).

A238692 a(n) is the quotient of the sum of (not necessarily distinct) integers i!+(prime(n)-1)!/i!, i=1,2,...,prime(n)-2, which are divisible by prime(n), and prime(n).

Original entry on oeis.org

0, 1, 7, 139, 365641, 39916801, 1317933016441, 355688356705921, 53128667010491295649, 10888872347627347035630931201, 8841761993746245283777145088001, 10333147966386144929666651337523200000001

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Mar 03 2014

nonn

a(n) is prime for n = {3,4,5,6,7,31,738}; a(738) ~ 7.1 * 10^18518. There are no others for n up to 1000. - Peter J. C. Moses, Mar 03 2014

			Let n=4, prime(n)=7. Consider integers i!+6!/i!, i=1,2,3,4,5: 721,362,126,54,126. Among them 721,126,126 are divisible by 7. So a(4)=(721 + 126 + 126)/7 = 139.

Cf. A238444, A007619.

Map[Total[Cases[Table[i!+(#-1)!/i!,{i,#-2}]/#,Integer]]&,Prime[Range[10]]] (* _Peter J. C. Moses, Mar 10 2014 *)

A261779 a(n) = ceiling((n-1)! / n).

Original entry on oeis.org

1, 1, 1, 2, 5, 20, 103, 630, 4480, 36288, 329891, 3326400, 36846277, 444787200, 5811886080, 81729648000, 1230752346353, 19760412672000, 336967037143579, 6082255020441600, 115852476579840000, 2322315553259520000, 48869596859895986087, 1077167364120207360000, 24817936069329577574400, 596585001666576384000000

Offset: 1

Franklin T. Adams-Watters, Aug 31 2015

nonn

Except for n = 4, this is ((n-1)!+1)/n for prime n, and (n-1)!/n otherwise.

Cf. A007619 (prime index elements).

Table[Ceiling[((n-1)!)/n],{n,30}] (* Harvey P. Dale, Oct 04 2018 *)

PARI
```
a(n) = ceil((n-1)!/n)
```

A280300 Primes such that the Wilson quotient and the Fermat quotient satisfy 2*((p-1)!+1)/p +(2^(p-1)-1)/p == 0 (mod p).

Original entry on oeis.org

3, 9511, 13691

Offset: 1

René Gy, Dec 31 2016

No new term less than 2000000. This sequence is included in A274994 because it can be shown that Sum_{k=1..(p-1)/2} (k^(p-2))*(k^(p-1)-1) == p*((2^(p-1)-1)/p)*(2*((p-1)!+1)/p +(2^(p-1)-1)/p) (mod p^2).

Cf. A274994, A001220, A007619.

A317507 Numbers k whose generalized Wilson quotient A157249(k) is prime.

Original entry on oeis.org

1, 5, 7, 8, 10, 11, 29, 62, 486, 614, 773, 1321, 1906, 2621

Offset: 1

Amiram Eldar, Sep 29 2018

The corresponding primes are 2, 5, 103, 13, 19, 329891, ...
Supersequence of A050299 (except for 1, the prime terms of this sequence).
No more terms below 10^4.

Cf. A007619, A050299, A157249.

p[n_] := Times @@ Select[Range[n], CoprimeQ[n, #] &]; e[1 | 2 | 4] = 1; e[n_] := (fi = FactorInteger[n]; If[MatchQ[fi, {{(p_)?OddQ, }} | {{2, 1}, {, }}], 1, -1]); a[n] := (p[n] + e[n])/n; n = 1; s={}; Do[If[PrimeQ[a[n]], AppendTo[s,n]], {n, 1, 1000}]; s (* after Jean-François Alcover at A157249 *)

phito(n) = prod(k=2, n-1, k^(gcd(k, n)==1)); \\ A001783
is(n) = if(n%2, isprimepower(n) || n==1, n==2 || n==4 || (isprimepower(n/2, &n) && n>2)); \\ A033948
e(n) = if (is(n), 1, -1);
gw(n) = (phito(n)+e(n))/n;
isok(n) = isprime(gw(n)); \\ Michel Marcus, Oct 28 2018

A239640 a(n) is the smallest number such that for n-bonacci constant c_n satisfies round(c_n^prime(m)) == 1 (mod 2*p_m) for every m>=a(n).

Original entry on oeis.org

3, 3, 4, 5, 7, 7, 10, 13, 14, 14, 19, 23, 23, 31, 34, 34, 46, 50, 60, 65, 73, 79, 88, 92, 107, 113, 126, 139, 149, 168, 182, 198, 210, 227, 244, 265, 276, 292, 317, 340, 369, 384, 408, 436, 444, 480, 516, 540, 565, 606, 628, 669, 704, 735, 759, 810, 829, 895, 925

Offset: 2

Vladimir Shevelev and Peter J. C. Moses, Mar 23 2014

nonn

The n-bonacci constant is a unique root x_1>1 of the equation x^n-x^(n-1)-...-x-1=0. So, for n=2 we have Fibonacci constant phi or golden ratio (A001622); for n=3 we have tribonacci constant (A058265); for n=4 we have tetranacci constant (A086088), for n=5 (A103814), for n=6 (A118427), etc.

			Let n=2, then c_2 = phi (Fibonacci constant). We have round(c_2^2)=3 is not == 1 (mod 4), round(c_2^3)=4 is not == 1 (mod 6), while round(c_2^5)=11 == 1 (mod 10) and one can prove that for p>=5, we have round(c_2^p) == 1 (mod 2*p). Since 5=prime(3), then a(2)=3.

S. Litsyn and V. Shevelev, Irrational Factors Satisfying the Little Fermat Theorem, International Journal of Number Theory, vol.1, no.4 (2005), 499-512.
V. Shevelev, A property of n-bonacci constant, Seqfan (Mar 23 2014)

Cf. A001622, A007619, A007663, A058265, A086088, A103814, A118427, A118428, A238693, A238697, A238698, A238700, A239502, A239544, A239564, A239565, A239566.

A277167 Prime numbers p such that (-1)^h + (h!)^2 == 0 (mod p^2) where h = (p-1)/2.

Original entry on oeis.org

3, 11, 31, 47, 53

Offset: 1

René Gy, Oct 01 2016

The above congruence is true modulo p for all odd primes. See A089043. But like for Wilson congruence, it is true modulo p^2, for a restricted number of primes. After 53, the next one (if any) seems very far away (>500000).

The fact that the congruence is true modulo p for all odd primes was proved by Lagrange in 1771. Using a theorem of Mathews (1892) and Eisenstein's logarithmetic rule for the Fermat quotient, the condition stated in the definition can be restated as W_p == -2q_p(2) (mod p), where W_p is the Wilson quotient of p (A007619) and q_p(2) is the Fermat quotient of p, base 2 (A007663). - John Blythe Dobson, Jul 31 2017

			(-1)^((11-1)/2)+(((11-1)/2)!)^2 = 14399 = 7*11^2*17.

Lagrange, "Démonstration d’un théoreme nouveau concernant les nombres premiers," Nouveaux Mémoires de l’Académie Royale des Sciences et Belles-Lettres [de Berlin], année 1771 (published 1783), 125-137.
G. B. Mathews, Theory of Numbers, part 1 [all published] (Cambridge, 1892), 318.

René Gy, Find the next "Wilson-like" prime.

Cf. A007540, A089043.

lista(nn) = forprime(p=3, nn, if ((((-1)^((p-1)/2)+(((p-1)/2)!)^2) % p^2) == 0, print1(p, ", "))); \\ Michel Marcus, Oct 02 2016

cp's OEIS Frontend

A239566 (Round(c^prime(n)) - 1)/prime(n), where c is the heptanacci constant (A118428).

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A250407 Near-Wilson primes (p = prime(n) satisfying (p-1)! == -1-A250406(n)*p (mod p^2)) with A250406(n) < 10.

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A134295 a(n) = Sum_{k=1..n} ((n-k)!*(k-1)! - (-1)^k).

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A134296 Quotients A134295(p)/p = (1/p) * Sum_{k=1..p} ((p-k)!*(k-1)! - (-1)^k), where p = prime(n).

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A238692 a(n) is the quotient of the sum of (not necessarily distinct) integers i!+(prime(n)-1)!/i!, i=1,2,...,prime(n)-2, which are divisible by prime(n), and prime(n).

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A261779 a(n) = ceiling((n-1)! / n).

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A280300 Primes such that the Wilson quotient and the Fermat quotient satisfy 2*((p-1)!+1)/p +(2^(p-1)-1)/p == 0 (mod p).

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A317507 Numbers k whose generalized Wilson quotient A157249(k) is prime.

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PARI

A239640 a(n) is the smallest number such that for n-bonacci constant c_n satisfies round(c_n^prime(m)) == 1 (mod 2*p_m) for every m>=a(n).

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A277167 Prime numbers p such that (-1)^h + (h!)^2 == 0 (mod p^2) where h = (p-1)/2.

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