A220862 -id:A220862 - cp's OEIS frontend

A088192 Distance between prime(n) and the largest quadratic residue modulo prime(n).

1, 2, 1, 3, 2, 1, 1, 2, 5, 1, 3, 1, 1, 2, 5, 1, 2, 1, 2, 7, 1, 3, 2, 1, 1, 1, 3, 2, 1, 1, 3, 2, 1, 2, 1, 3, 1, 2, 5, 1, 2, 1, 7, 1, 1, 3, 2, 3, 2, 1, 1, 7, 1, 2, 1, 5, 1, 3, 1, 1, 2, 1, 2, 11, 1, 1, 2, 1, 2, 1, 1, 7, 3, 1, 2, 5, 1, 1, 1, 1, 2, 1, 7, 1, 3, 2, 1, 1, 1, 3, 2, 13, 3, 2, 2, 5, 1, 1, 2, 1

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu), Sep 22 2003

a(n) = smallest m>0 such that -m is a quadratic residue modulo prime(n).
a(n) = smallest m>0 such that prime(n) either splits or ramifies in the imaginary quadratic field Q(sqrt(-m)). Equals -A220862(n) except when n = 1. Cf. A220861, A220863. - N. J. A. Sloane, Dec 26 2012
The values are 1 or a prime number (easily provable!). The maximum occurring prime values increase very slowly: up to 10^5 terms the largest prime is 43. The primes do not appear in order.

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, Cor. 5.17, p. 105. - From N. J. A. Sloane, Dec 26 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Ferenc Adorjan, The sequence of largest quadratic residues modulo the primes.
J. A. Bergstra, I. Bethke, A negative result on algebraic specifications of the meadow of rational numbers, arXiv preprint arXiv:1507.00548 [math.RA], 2015-2016.

Records are (essentially) given by A147971.

Cf. A088190, A088191, A088193, A088194, A088195, A220861, A220862, A220863.

a[n_] := With[{p = Prime[n]}, If[JacobiSymbol[-1, p] > 0, 1, For[d = 2, True, d = NextPrime[d], If[JacobiSymbol[-d, p] >= 0, Return[d]]]]]; Array[a, 100] (* Jean-François Alcover, Feb 16 2018, after Charles R Greathouse IV *)

qrp_pm(fr,to)= {/* The distance of largest QR modulo the primes from the primes */ local(m,p,v=[]); for(i=fr,to,m=1; p=prime(i); j=2; while((j<=(p-1)/2)&&(m

do(p)=if(kronecker(-1,p)>0, 1, forprime(d=2, p, if(kronecker(-d, p) >= 0, return(d))))
apply(do, primes(100)) \\ Charles R Greathouse IV, Oct 31 2012

a(n) = A053760(n) unless -1 is a quadratic residue mod prime(n). - Charles R Greathouse IV, Oct 31 2012

Edited by Max Alekseyev, Oct 29 2012

A162299 Faulhaber's triangle: triangle T(k,y) read by rows, giving denominator of the coefficient [m^y] of the polynomial Sum_{x=1..m} x^(k-1).

Original entry on oeis.org

1, 2, 2, 6, 2, 3, 1, 4, 2, 4, 30, 1, 3, 2, 5, 1, 12, 1, 12, 2, 6, 42, 1, 6, 1, 2, 2, 7, 1, 12, 1, 24, 1, 12, 2, 8, 30, 1, 9, 1, 15, 1, 3, 2, 9, 1, 20, 1, 2, 1, 10, 1, 4, 2, 10, 66, 1, 2, 1, 1, 1, 1, 1, 6, 2, 11, 1, 12, 1, 8, 1, 6, 1, 8, 1, 12, 2, 12, 2730, 1, 3, 1, 10, 1, 7, 1, 6, 1, 1, 2, 13, 1, 420, 1, 12, 1, 20, 1, 28, 1, 60, 1, 12, 2, 14, 6, 1, 90, 1, 6, 1, 10, 1, 18, 1, 30, 1, 6, 2, 15

Offset: 0

Juri-Stepan Gerasimov, Jun 30 2009 and Jul 02 2009

There are many versions of Faulhaber's triangle: search the OEIS for his name. For example, A220862/A220963 is essentially the same as this triangle, except for an initial column of 0's. - N. J. A. Sloane, Jan 28 2017

			The first few polynomials:
    m;
   m/2  + m^2/2;
   m/6  + m^2/2 + m^3/3;
    0   + m^2/4 + m^3/2 + m^4/4;
  -m/30 +   0   + m^3/3 + m^4/2 + m^5/5;
  ...
Initial rows of Faulhaber's triangle of fractions H(n, k) (n >= 0, 1 <= k <= n+1):
    1;
   1/2,  1/2;
   1/6,  1/2,  1/3;
    0,   1/4,  1/2,  1/4;
  -1/30,  0,   1/3,  1/2,  1/5;
    0,  -1/12,  0,   5/12, 1/2,  1/6;
   1/42,  0,  -1/6,   0,   1/2,  1/2,  1/7;
    0,   1/12,  0,  -7/24,  0,   7/12, 1/2,  1/8;
  -1/30,  0,   2/9,   0,  -7/15,  0,   2/3,  1/2,  1/9;
  ...
The triangle starts in row k=1 with columns 1<=y<=k as
     1
     2   2
     6   2  3
     1   4  2  4
    30   1  3  2  5
     1  12  1 12  2  6
    42   1  6  1  2  2  7
     1  12  1 24  1 12  2  8
    30   1  9  1 15  1  3  2  9
     1  20  1  2  1 10  1  4  2 10
    66   1  2  1  1  1  1  1  6  2 11
     1  12  1  8  1  6  1  8  1 12  2 12
  2730   1  3  1 10  1  7  1  6  1  1  2 13
     1 420  1 12  1 20  1 28  1 60  1 12  2 14
     6   1 90  1  6  1 10  1 18  1 30  1  6  2 15
  ...
Initial rows of triangle of fractions:
    1;
   1/2, 1/2;
   1/6, 1/2,  1/3;
    0,  1/4,  1/2,  1/4;
  -1/30, 0,   1/3,  1/2,  1/5;
    0, -1/12,  0,   5/12, 1/2,  1/6;
   1/42, 0,  -1/6,   0,   1/2,  1/2,  1/7;
    0,  1/12,  0,  -7/24,  0,   7/12, 1/2,  1/8;
  -1/30, 0,   2/9,   0,  -7/15,  0,   2/3,  1/2,  1/9;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Mohammad Torabi-Dashti, Faulhaber’s Triangle, College Math. J., 42:2 (2011), 96-97.
Mohammad Torabi-Dashti, Faulhaber’s Triangle [Annotated scanned copy of preprint]
Eric Weisstein's MathWorld, Power Sum

Cf. A000367, A162298 (numerators).
See also A220962/A220963.
Cf. A053382, A053383.

A162299 := proc(k,y) local gf,x; gf := sum(x^(k-1),x=1..m) ; coeftayl(gf,m=0,y) ; denom(%) ; end proc: # R. J. Mathar, Jan 24 2011
# To produce Faulhaber's triangle of fractions H(n,k) (n >= 0, 1 <= k <= n+1):
H:=proc(n,k) option remember; local i;
if n<0 or k>n+1 then 0;
elif n=0 then 1;
elif k>1 then (n/k)*H(n-1,k-1);
else 1 - add(H(n,i),i=2..n+1); fi; end;
for n from 0 to 10 do lprint([seq(H(n,k),k=1..n+1)]); od:
for n from 0 to 12 do lprint([seq(numer(H(n,k)),k=1..n+1)]); od: # A162298
for n from 0 to 12 do lprint([seq(denom(H(n,k)),k=1..n+1)]); od: # A162299 # N. J. A. Sloane, Jan 28 2017

H[n_, k_] := H[n, k] = Which[n < 0 || k > n+1, 0, n == 0, 1, k > 1, (n/k)* H[n - 1, k - 1], True, 1 - Sum[H[n, i], {i, 2, n + 1}]];
Table[H[n, k] // Denominator, {n, 0, 14}, {k, 1, n + 1}] // Flatten (* Jean-François Alcover, Aug 04 2022 *)

Faulhaber's triangle of fractions H(n,k) (n >= 0, 1 <= k <= n+1) is defined by: H(0,1)=1; for 2<=k<=n+1, H(n,k) = (n/k)*H(n-1,k-1) with H(n,1) = 1 - Sum_{i=2..n+1}H(n,i). - N. J. A. Sloane, Jan 28 2017

Sum_{x=1..m} x^(k-1) = (Bernoulli(k,m+1)-Bernoulli(k))/k.

Offset set to 0 by Alois P. Heinz, Feb 19 2021

A220861 Choose smallest m>0 such that the n-th rational prime p ramifies in the imaginary quadratic extension field K = Q(sqrt(-m)); a(n) = discriminant(K).

Original entry on oeis.org

-4, -3, -20, -7, -11, -52, -68, -19, -23, -116, -31, -148, -164, -43, -47, -212, -59, -244, -67, -71, -292, -79, -83, -356, -388, -404, -103, -107, -436, -452, -127, -131, -548, -139, -596, -151, -628, -163, -167, -692, -179, -724, -191, -772, -788, -199

Offset: 1

N. J. A. Sloane, Dec 26 2012

sign

m=1 if p=2, otherwise m=p.

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, Cor. 5.17, p. 105.

Bruno Berselli, Table of n, a(n) for n = 1..1000

Cf. A088192, A220862, A220863.

Let p = prime(n). Then a(n) = -4 if p = 2, -p if p == 3 mod 4, -4p if p == 1 mod 4.

A220863 Choose smallest m>0 such that the n-th rational prime p splits in the imaginary quadratic extension field K = Q(sqrt(-m)); a(n) = discriminant(K).

Original entry on oeis.org

-7, -8, -4, -3, -8, -4, -4, -8, -20, -4, -3, -4, -4, -8, -20, -4, -8, -4, -8, -7, -4, -3, -8, -4, -4, -4, -3, -8, -4, -4, -3, -8, -4, -8, -4, -3, -4, -8, -20, -4, -8, -4, -7, -4, -4, -3, -8, -3, -8, -4, -4, -7, -4, -8, -4, -20, -4, -3, -4, -4, -8, -4, -8, -11, -4, -4, -8, -4, -8, -4, -4, -7, -3, -4, -8

Offset: 1

N. J. A. Sloane, Dec 26 2012

sign

a(n) = discriminant of extension field defined in A220862.

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989, Cor. 5.17, p. 105.

Cf. A088192, A220861, A220862.

Let i = A220862(n). Then a(n) = i if i == 1 (mod 4), otherwise 4i.

cp's OEIS Frontend

A088192 Distance between prime(n) and the largest quadratic residue modulo prime(n).

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A162299 Faulhaber's triangle: triangle T(k,y) read by rows, giving denominator of the coefficient [m^y] of the polynomial Sum_{x=1..m} x^(k-1).

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A220861 Choose smallest m>0 such that the n-th rational prime p ramifies in the imaginary quadratic extension field K = Q(sqrt(-m)); a(n) = discriminant(K).

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A220863 Choose smallest m>0 such that the n-th rational prime p splits in the imaginary quadratic extension field K = Q(sqrt(-m)); a(n) = discriminant(K).

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