Wadim Zudilin - cp's OEIS frontend

A179073 A179071 for p == 1 (mod 4).

-2, -18, -4, -70, -882, -32, -182, -29718, -1068, -500, -5604, -4030, -8890182, -776, -1744, -113582, -4832118, -1118, -1111225770, -1764132, -11018, -20000849130, -23156, -71011068, -16432, -2205718, -8920484118, -1063532

Offset: 1

Jonathan Sondow and Wadim Zudilin, Jun 29 2010

sign

It appears that a(n) is negative and even. (Added Aug 28 2011: This conjecture has been proved by Maxim Vsemirnov.)

M. Vsemirnov, On R. Chapman's "evil determinant": case p == 1 (mod 4), arXiv preprint arXiv:1108.4031 [math.NT], 2011-2012.

Cf. A179071, A179072, A179074.

a71[p_] := Module[{k}, k = (p-1)/2; Det @ Table[JacobiSymbol[j-i, p], {i, 1, k+1}, {j, 1, k+1}]];
a71 /@ Select[Range[1, 301, 4], PrimeQ] (* Jean-François Alcover, Dec 05 2018 *)

A179074 A179072 for p == 1 (mod 4).

Original entry on oeis.org

-2, -32, 256, -8192, -262144, 5242880, -33554432, -2684354560, 8589934592000, 932385860354048, 160159261748363264, -1125899906842624, -225179981368524800, 5260204364768739328, 43977037871723571052544

Offset: 1

Jonathan Sondow and Wadim Zudilin, Jun 29 2010

sign

It appears that a(n) is even.

Cf. A179071, A179072, A179073.

a72[p_] := Module[{k}, k = (p-1)/2; Det @ Table[JacobiSymbol[i+j, p], {i, 1, k}, {j, 1, k}]];
a72 /@ Select[Range[1, 201, 4], PrimeQ] (* Jean-François Alcover, Dec 05 2018 *)

A179072 Chapman's "evil" determinants II.

Original entry on oeis.org

-1, -2, 0, 0, -32, 256, 0, 0, -8192, 0, -262144, 5242880, 0, 0, -33554432, 0, -2684354560, 0, 0, 8589934592000, 0, 0, 932385860354048, 160159261748363264, -1125899906842624, 0, 0, -225179981368524800, 5260204364768739328, 0, 0

Offset: 2

Jonathan Sondow and Wadim Zudilin, Jun 29 2010

sign

Determinant of the k-by-k matrix with (i,j)-entry L((i+j)/p), where L(./p) denotes the Legendre symbol modulo p and p = p_n = 2k+1 is the n-th prime.
Guy says "Chapman has a number of conjectures which concern the distribution of quadratic residues." One is that if 3 < p_n == 3 (mod 4), then a(n) = 0.
It appears that a(n) is even, if p_n == 1 (mod 4).
For any odd prime p, (p+1)/2-i+(p+1)/2-j == -(i+j-1) (mod p) and hence we have L(-1/p)*|L((i+j)/p)|{i,j=1,...,(p-1)/2} = |L((i+j-1)/p)|{i,j=1,...,(p-1)/2}. Thus the value of a(n) was actually determined in the first reference of R. Chapman. - Zhi-Wei Sun, Aug 21 2013

			p_4 = 7 = 2*3 + 1 and the 3 X 3 matrix (L((i+j)/7)) is
   1, -1,  1
  -1,  1, -1
   1, -1, -1
which has determinant 0, so a(4) = 0.

Richard Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004, Section F5.

Robin Chapman, Determinants of Legendre symbol matrices, Acta Arith. 115 (2004), 231-244.
Robin Chapman, Steinitz classes of unimodular lattices, European J. Combin. 25 (2004), 487-493.
Robin Chapman, My evil determinant problem, 2009.
Maxim Vsemirnov, On R. Chapman's "evil determinant": case p=1 (mod 4), arXiv:1108.4031 [math.NT], 2011-2012.
Maxim Vsemirnov, On the evaluation of R. Chapman's "evil determinant", Linear Algebra Appl. 436(2012), 4101-4106.
Wikipedia, Legendre symbol

Cf. A179071 (Chapman's "evil" determinants I), A179073 (A179071 for p == 1 (mod 4)), A179074 (A179072 for p == 1 (mod 4)).

a[n_] := Module[{p, k}, p = Prime[n]; k = (p-1)/2; Det @ Table[JacobiSymbol[ i + j, p], {i, 1, k}, {j, 1, k}]];
Table[a[n], {n, 2, 32}] (* Jean-François Alcover, Nov 18 2018 *)

A179071 Chapman's "evil" determinants I.

Original entry on oeis.org

1, -2, 1, 1, -18, -4, 1, 1, -70, 1, -882, -32, 1, 1, -182, 1, -29718, 1, 1, -1068, 1, 1, -500, -5604, -4030, 1, 1, -8890182, -776, 1, 1, -1744, 1, -113582, 1, -4832118, 1, 1, -1118, 1, -1111225770, 1, -1764132, -11018, 1, 1, 1, 1, -20000849130, -23156, 1

Offset: 2

Jonathan Sondow and Wadim Zudilin, Jun 29 2010

sign

Determinant of the (k+1)X(k+1) matrix with (i,j)-entry L((j-i)/p), where L(./p) denotes the Legendre symbol modulo p and p = p_n = 2k+1 is the n-th prime.
Guy says "Chapman has a number of conjectures which concern the distribution of quadratic residues." One is that if p_n == 3 (mod 4), then a(n) = 1. Chapman also has a conjecture if p_n == 1 (mod 4), involving the fundamental unit and class number of the quadratic field Q(sqrt(p)). (Added Aug 23 2011: Both conjectures have been proved by Vsemirnov.)
It appears that a(n) is negative and even, if p_n == 1 (mod 4); see A179073. (Added Aug 28 2011: This conjecture has also been proved by Vsemirnov.)

			p_3 = 5 = 2*2+1 and the (2+1)-by-(2+1) matrix (L((j-i)/5)) is
   0,  1, -1
   1,  0,  1
  -1,  1,  0
which has determinant -2, so a(3) = -2.

Richard Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004, Section F5.

Robin Chapman, Determinants of Legendre symbol matrices, Acta Arith. 115 (2004), 231-244.
Robin Chapman, Steinitz classes of unimodular lattices, European J. Combin. 25 (2004), 487-493.
Robin Chapman, My evil determinant problem, 2009.
Maxim Vsemirnov, On R. Chapman's "evil determinant": case p=1 (mod 4), arXiv:1108.4031 [math.NT], 2011-2012.
Maxim Vsemirnov, On the evaluation of R. Chapman's "evil determinant", Linear Algebra Appl. 436(2012), 4101-4106.
Wikipedia, Legendre symbol

Cf. A179072 (Chapman's "evil" determinants II), A179073 (A179071 for p == 1 (mod 4)), A179074 (A179072 for p == 1 (mod 4)).

a[n_] := Module[{p, k}, p = Prime[n]; k = (p-1)/2; Det @ Table[JacobiSymbol[ j-i, p], {i, 1, k+1}, {j, 1, k+1}]];
Table[a[n], {n, 2, 52}] (* Jean-François Alcover, Nov 18 2018 *)

a(n) = my(p=prime(n), k=(p+1)/2); matdet(matrix(k, k, i, j, kronecker(j-i, p))); \\ Michel Marcus, Aug 25 2021

A157250 Wilson numbers: k such that the generalized Wilson quotient A157249(k) is divisible by k.

Original entry on oeis.org

1, 5, 13, 563, 5971, 558771, 1964215, 8121909, 12326713, 23025711, 26921605, 341569806, 399292158

Offset: 1

Jonathan Sondow and Wadim Zudilin, Feb 27 2009

A prime p is a Wilson prime if p divides its Wilson quotient A007619. A number n is a Wilson number if n divides its generalized Wilson quotient A157249.
The sequence contains all Wilson numbers <= 5 x 10^8. Heuristics suggest that #(Wilson numbers < N) is about (6/pi^2) log N, for large N.
A Wilson number is prime if and only if it is a Wilson prime A007540. Only three are known: 5, 13, 563.
The first composite Wilson number 5971 was discovered by Kloss, the others by Agoh, Dilcher, and Skula. Every known composite Wilson number n has at least two odd prime factors, so e(n) = -1.
For additional references and links, see A007540.

			A157249(13) = (A001783(13) + e(13))/13 = ((13-1)! + 1)/13 = 479001601/13 = 36846277 == 0 mod 13, so 13 is a member. A001783(5971) + e(5971) = A001783(5971) - 1 == 0 mod 5971^2, so 5971 is a member. But A157249(8) = (A001783(8) + e(8))/8 = (3*5*7 - 1)/8 = 13 ==/== 0 mod 8, so 8 is not a member.

L. E. Dickson, History of the Theory of Numbers, vol. 1, Divisibility and Primality, Chelsea, New York, 1966, p. 65.

T. Agoh, K. Dilcher, and L. Skula, Wilson quotients for composite moduli, Math. Comp. 67 (1998), 843-861.
K. E. Kloss, Some Number-Theoretic Calculations, J. Research of the National Bureau of Standards-B. Mathematics and Mathematical Physics, Vol. 69B, No. 4 (1965), 335-336.
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, in Proceedings of CANT 2011, arXiv:1110.3113 [math.NT], 2011-2012.
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., vol. 101 (2014), pp. 243-255.

Cf. Wilson quotient A007619, Wilson prime A007540, generalized Wilson quotient A157249, n-phi-torial A001783, numbers having a primitive root A033948.

f[n_] := Times @@ Select[Range[n], CoprimeQ[n, #]&];
e[1|2|4] = 1; e[n_] := If[MatchQ[FactorInteger[n], {{?OddQ, }} | {{2, 1}, {, }}], 1, -1];
WilsonQ[n_] := IntegerQ[(f[n] + e[n])/n^2];
Reap[For[k = 1, k < 10^7, k++, If[WilsonQ[k], Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Dec 11 2018 *)

A157249(n) == 0 mod n.

A001783(n) + e(n) == 0 mod n^2, where e(n) = +1 or -1 according as n does or does not have a primitive root.

A157249 Generalized Wilson quotients (or Wilson quotients for composite moduli).

Original entry on oeis.org

2, 1, 1, 1, 5, 1, 103, 13, 249, 19, 329891, 32, 36846277, 1379, 59793, 126689, 1230752346353, 4727, 336967037143579, 436486, 2252263619, 56815333, 48869596859895986087, 1549256, 1654529071288638505, 23390099351, 56463097772562963, 51860555558, 10513391193507374500051862069

Offset: 1

Jonathan Sondow and Wadim Zudilin, Feb 27 2009

nonn

By Wilson's Theorem, for prime p the Wilson quotient ((p-1)!+1)/p is an integer A007619. By Gauss's extension (see Dickson p. 65), the generalized Wilson quotient (P(n)+e(n))/n is an integer, where P(n) = n-phi-torial A001783 and e(n) = +1 or -1 according as n does or does not have a primitive root (see A033948).

For additional references and links, see A007540.

			P(8) = 3*5*7 = 105 and e(8) = -1, so a(8) = (105-1)/8 = 13.

L. E. Dickson, History of the Theory of Numbers, vol. 1, Divisibility and Primality, Chelsea, New York, 1966.

Seiichi Manyama, Table of n, a(n) for n = 1..200
T. Agoh, K. Dilcher, and L. Skula, Wilson quotients for composite moduli, Math. Comp. 67 (1998), 843-861.
K. E. Kloss, Some Number-Theoretic Calculations, J. Research of the National Bureau of Standards-B. Mathematics and Mathematical Physics, Vol. 69B, No. 4 (1965), 335-336.
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, in Proceedings of CANT 2011, arXiv:1110.3113 [math.NT], 2011-2012.

Cf. Wilson quotient A007619, Wilson prime A007540, Wilson number A157250, n-phi-torial A001783, numbers having a primitive root A033948.

Cf. A317507.

a := proc(n) local A001783,e,i;
A001783 := proc(n) local i; mul(i,i=select(k->igcd(k,n)=1,[$1..n]))end;
e := proc(n) local p,r,P; if n=1 or n=2 or n=4 then RETURN(1) fi;
P := select(isprime,[$3..n]); for p in P do r := p;
while r <= n do if n = r or n = 2*r then RETURN(1) fi;
r := r*p; od od; -1 end; (A001783(n)+e(n))/n end:
# Peter Luschny, Jul 19 2009

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; Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Sep 28 2011 *)

a(n) = (P(n)+e(n))/n, with P(n) = n-phi-torial = A001783(n) and e(n) = +1 if n = 1, 2, 4, p^k or 2p^k, where p is an odd prime and k > 0, and e(n) = -1 otherwise.

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User: Wadim Zudilin

A179073 A179071 for p == 1 (mod 4).

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A179074 A179072 for p == 1 (mod 4).

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A179072 Chapman's "evil" determinants II.

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A179071 Chapman's "evil" determinants I.

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A157250 Wilson numbers: k such that the generalized Wilson quotient A157249(k) is divisible by k.

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A157249 Generalized Wilson quotients (or Wilson quotients for composite moduli).

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