Dimitri Papadopoulos - cp's OEIS frontend

A299016 Integer radii of circles over an integer lattice such that the number of unit squares whose centers are contained in the circle is less than the area of the circle.

2, 6, 12, 19, 24, 36, 40, 43, 48, 52, 53, 55, 60, 61, 65, 70, 74, 77, 89, 91, 108, 111, 116, 123, 125, 128, 129, 140, 141, 142, 146, 152, 154, 159, 166, 169, 171, 180, 181, 183, 184, 197, 198, 205, 209, 210, 212, 214, 222

Offset: 1

Dimitri Papadopoulos, Jun 19 2018

nonn

			For the circle with radius a(1) = 2, the point  (3/2, 3/2 ), i.e. the center of the unit square bounded by x = 1, x = 2, y = 1, y = 2, is outside the circle of radius 2 centered at the origin so there are 12 unit squares with centers inside the circle of radius 2, and 12 <  Pi *2 *2.

t = {}; For[n = 1, n < 223, n++, cnt = 0; For[x = -n, x < 0, x++, For[y = -n, y < 0, y++, If[N[Norm[{x + 1/2, y + 1/2}]] < n, cnt++]]] If[Pi*n*n > 4*cnt, t = Append[t, n]]] Print[t];

A267632 Triangle T(n, k) read by rows: the k-th column of the n-th row lists the number of ways to select k distinct numbers (k >= 1) from [1..n] so that their sum is divisible by n.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 2, 2, 1, 1, 1, 2, 4, 3, 1, 0, 1, 3, 5, 5, 3, 1, 1, 1, 3, 7, 9, 7, 3, 1, 0, 1, 4, 10, 14, 14, 10, 4, 1, 1, 1, 4, 12, 22, 26, 20, 12, 5, 1, 0, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1, 1, 5, 19, 42, 66, 76, 66, 43, 19, 5, 1, 0

Offset: 1

Dimitri Papadopoulos, Jan 18 2016

Row less the last element is palindrome for n=odd or n=power of 2 where n is the row number (observation-conjecture).
From Petros Hadjicostas, Jul 13 2019: (Start)
By reading carefully the proof of Lemma 5.1 (pp. 65-66) in Barnes (1959), we see that he actually proved a general result (even though he does not state it in the lemma).
According to the definition of this sequence, for 1 <= k <= n, T(n, k) is the number of unordered sets b_1, b_2, ..., b_k of k distinct integers from 1..n such that b_1 + b_2 + ... + b_k = 0 (mod n). The proof of Lemma 5.1 in Barnes (1959) implies that T(n, k) = (1/n) * Sum_{s | gcd(n, k)} (-1)^(k - (k/s)) * phi(s) * binomial(n/s, k/s) for 1 <= k <= n.
For fixed k >= 1, the g.f. of the column (T(n, k): n >= 1) (with T(n, k) = 0 for 1 <= n < k) is (x^k/k) * Sum_{s|k} phi(s) * (-1)^(k - (k/s)) / (1 - x^s)^(k/s), which generalizes Herbert Kociemba's formula from A032801.
Barnes' (1959) formula is a special case of Theorem 4 (p. 66) in Ramanathan (1944). If R(n, k, v) is the number of unordered sets b_1, b_2, ..., b_k of k distinct integers from 1..n such that b_1 + b_2 + ... + b_k = v (mod n), then he proved that R(n, k, v) = (1/n) * Sum_{s | gcd(n,k)} (-1)^(k - (k/s)) * binomial(n/s, k/s) * C_s(v), where C_s(v) = A054535(s, v) =  Sum_{d | gcd(s,v)} d * Moebius(s/d) is Ramanujan's sum (even though it was first discovered around 1900 by the Austrian mathematician R. D. von Sterneck).
Because C_s(v = 0) = phi(s), we get Barnes' (implicit) result; i.e., R(n, k, v=0) = T(n, k) for 1 <= k <= n.
For k=2, we have R(n, k=2, v=0) = T(n, k=2) = A004526(n-1) for n >= 1. For k=3, we have R(n, k=3, v=0) = T(n, k=3) = A058212(n) for n >= 1. For k=4, we have R(n, k=4, v=0) = A032801(n) for n >= 1. For k=5, we have R(n, k=5, v=0) = T(n, k=5) = A008646(n-5) for n >= 5.
The reason we have T(2*m+1, k) = A037306(2*m+1, k) = A047996(2*m+1, k) for m >= 0 and k >= 1 is the following. When n = 2*m + 1, all divisors s of gcd(n, k) are odd. In such a case, k - (k/s) is even for all k >= 1, and thus (-1)^(k - (k/s)) = 1, and thus T(n = 2*m+1, k) = (1/n) * Sum_{s | gcd(n, k)} phi(s) * binomial(n/s, k/s) = A037306(2*m+1, k) = A047996(2*m+1, k).
By summing the products of the g.f. of column k times y^k from k = 1 to k = infinity, we get the bivariate g.f. for the array: Sum_{n, k >= 1} T(n, k)*x^n*y^k = Sum_{s >= 1} (phi(s)/s) * log((1 - x^s + (-x*y)^s)/(1 - x^s)) = -x/(1 - x) - Sum_{s >= 1} (phi(s)/s) * log(1 - x^s + (-x*y)^s).
Letting y = 1 in the above bivariate g.f., we get the g.f. of the sequence (Sum_{1 <= k <= n} T(n, k): n >= 1) is -x/(1 - x) - Sum_{s >= 1} (phi(s)/s) * log(1 - x^s + (-x)^s) = -x/(1 - x) + Sum_{m >= 0} (phi(2*m + 1)/(2*m + 1)) * log(1 - 2*x^(2*m+1)), which is the g.f. of sequence A082550. Hence, sequence A082550 consists of the row sums.
There is another important interpretation of this array T(n, k) which is related to some of the references for sequence A047996, but because the discussion is too lengthy, we omit the details.
(End)

			For n = 5, there is one way to pick one number (5), two ways to pick two numbers (1+4, 2+3), two ways to pick 3 numbers (1+4+5, 2+3+5), one way to pick 4 numbers (1+2+3+4), and one way to pick 5 numbers (1+2+3+4+5) so that their sum is divisible by 5. Therefore, T(5,1) = 1, T(5,2) = 2, T(5,3) = 2, T(5,4) = 1, and T(5,5) = 1.
Table for T(n,k) begins as follows:
n\k 1 2   3    4    5    6    7    8    9   10
1   1
2   1 0
3   1 1   1
4   1 1   1    0
5   1 2   2    1    1
6   1 2   4    3    1    0
7   1 3   5    5    3    1    1
8   1 3   7    9    7    3    1    0
9   1 4  10   14   14   10    4    1    1
10  1 4  12   22   26   20   12    5    1    0
...

Alois P. Heinz, Rows n = 1..150, flattened
Eric Stephen Barnes, The construction of perfect and extreme forms I, Acta Arith., 5 (1959); see pp. 65-66.
Michiel Kosters, The subset sum problem for finite abelian groups, J. Combin. Theory Ser. A 120 (2013), 527-530.
Jiyou Li and Daqing Wan, Counting subset sums of finite abelian groups, J. Combin. Theory Ser. A 119 (2012), 170-182; see pp. 171-172.
K. G. Ramanathan, Some applications of Ramanujan's trigonometrical sum C_m(n), Proc. Indian Acad. Sci., Sect. A 20 (1944), 62-69; see p. 66.
R. D. von Sterneck, Ein Analogon zur additiven Zahlentheorie, Sitzungsber. Akad. Wiss. Sapientiae Math.-Naturwiss. Kl. 111 (1902), 1567-1601 (Abt. IIa).
R. D. von Sterneck, Über ein Analogon zur additiven Zahlentheorie, Jahresbericht der Deutschen Mathematiker-Vereinigung 12 (1903), 110-113.

Cf. A004526, A032801, A037306, A047996, A054532, A054533, A054534, A054535, A058212, A063776, A082550 (row sums), A309122.

A267632 := proc(n,k)
    local a,msel,p;
    a := 0 ;
    for msel in combinat[choose](n,k) do
        if modp(add(p,p=msel),n) = 0 then
            a := a+1 ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, May 15 2016
# second Maple program:
b:= proc(n, m, s) option remember; expand(`if`(n=0,
      `if`(s=0, 1, 0), b(n-1, m, s)+x*b(n-1, m, irem(s+n, m))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2, 0)):
seq(T(n), n=1..14);  # Alois P. Heinz, Aug 27 2018

f[k_, n_] :=  Length[Select[Select[Subsets[Range[n]], Length[#] == k &], IntegerQ[Total[#]/n] &]];MatrixForm[Table[{n, Table[f[k, n], {k, n}]}, {n, 10}]] (* Dimitri Papadopoulos, Jan 18 2016 *)

T(2n+1, k) = A037306(2n+1, k) = A047996(2n+1, k).
From Petros Hadjicostas, Jul 13 2019: (Start)
T(n, k) = (1/n) * Sum_{s | gcd(n, k)} (-1)^(k - (k/s)) * phi(s) * binomial(n/s, k/s) for 1 <= k <= n.
G.f. for column k >= 1: (x^k/k) * Sum_{s|k} phi(s) * (-1)^(k - (k/s)) / (1 - x^s)^(k/s).
Bivariate g.f.: Sum_{n, k >= 1} T(n, k)*x^n*y^k = -x/(1 - x) - Sum_{s >= 1} (phi(s)/s) * log(1 - x^s + (-x*y)^s).
(End)
Sum_{k=1..n} k * T(n,k) = A309122(n). - Alois P. Heinz, Jul 13 2019

A268397 a(n) is the smallest prime with at least n consecutive primitive roots.

Original entry on oeis.org

2, 5, 11, 37, 53, 83, 83, 269, 269, 467, 467, 1187, 1559, 1559, 1559, 6803, 6803, 6803, 10559, 10559, 10559, 35279, 38639, 38639, 38639, 38639, 38639

Offset: 1

Dimitri Papadopoulos, Feb 03 2016

			a(4)=37. 37 has the primitive roots 2, 5, 13, 15, 17, 18, 19, 20, 22, 24, 32, and 35 of which 17, 18, 19, and 20 are consecutive.

Cf. A060749, A261438 (has "exactly" instead of "at least").

PrimRoot[n_] :=Flatten[Position[Table[MultiplicativeOrder[i, n], {i, n - 1}],n - 1]];t = {};For[targ = 1, targ <= 22, targ++,flag = 0; For[n = 1, n < 1500, n++,prs = PrimRoot[Prime[n]];numprs = EulerPhi[Prime[n] - 1]; If[targ > numprs, ,For[m = 1, m <= numprs + 1 - targ, m++,temp = Take[prs, {m, m + targ - 1}];If[temp[[1]] + targ - 1 == temp[[targ]] && flag == 0,t = Append[t, Prime[n]]; flag = 1];If[flag == 1, Break[]];]; If[flag == 1, Break[]];];If[flag == 1, Break[]];]]; t
Join[{2},Module[{prl=Table[{p,Max[Length/@Select[Split[ Differences[ PrimitiveRootList[ p]]], #[[1]]==1&]]},{p,Prime[Range[1500]]}]},Table[ SelectFirst[ prl, #[[2]]>=k&],{k,20}]][[All, 1]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 23 2019 *)

More terms from Harvey P. Dale, Aug 23 2019

A267133 a(n) = (1/n)(2/n)(3/n)...((n-1)/n) where (k/n) is the Kronecker symbol, n >= 1.

Original entry on oeis.org

1, 1, -1, 0, 1, 0, -1, 0, 0, 0, -1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 1, 0, 0

Offset: 1

Dimitri Papadopoulos, Jan 10 2016

			a(3) = (1/3)(2/3) = (1)(-1) = -1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002313, A002145, A002808, A057077, A080339, A151763.

f:= proc(n) if not isprime(n) then 0 elif n mod 4 = 3 then -1 else 1 fi end proc:
f(1):= 1:
map(f, [$1..1000]); # Robert Israel, Jan 14 2016

Table[Product[JacobiSymbol[k, n], {k, n - 1}], {n, 75}] (* Michael De Vlieger, Jan 12 2016 *)

a(n) = prod(k=1, n-1, kronecker(k, n)); \\ Michel Marcus, Jan 11 2016

a(n)=if(isprime(n),(-1)^(n%4>2),n==1) \\ Charles R Greathouse IV, Jan 14 2016

A080339(n) = abs(a(n)) = a(n)^2.
a(c) = 0 if c is composite (A002808).
a(p) = 1 for primes p in A002313.
a(p) = -1 for primes p in A002145.
a(n) = A057077(n+3)*A080339(n) for n > 1. - Robert Israel, Jan 14 2016
a(n) = A151763(n), n > 2. - R. J. Mathar, Jan 17 2016

"Jacobi symbol" in Name changed to "Kronecker symbol" by Jianing Song, Dec 30 2018

A267010 Primes of the form p==3 (mod 4) such that the average of their primitive roots equals p/2.

Original entry on oeis.org

19, 307, 1451, 2179, 2251, 2683, 2843, 3259, 3907, 4447, 11863, 12907, 17623, 30763, 37963, 51059, 52543, 86131, 92467, 104851, 129763, 131203, 146683, 150151, 156151, 156703, 162523, 163819, 174007, 245899, 263827, 287731, 348643, 353611, 400123, 412831, 423091, 432587

Offset: 1

Dimitri Papadopoulos, Jan 08 2016

nonn

Most primes for which the average of the primitive roots=p/2 are of the form p==1(mod 4). Much rarer for primes of form p==3(mod 4) to have this property. (Observation)

			19 is a term because the primitive roots of 19 are 2, 3, 10, 13, 14, and 15. Their average is (2+3+10+13+14+15)/phi(18)=57/phi(18)=57/6=19/2.

Cf. A060749. Intersection of A002145 and A266987.

isA267010 := proc(n)
    if isprime(n) and modp(n,4) = 3 then
        isA266987(n) ;
    else
        false;
    end if;
end proc: # R. J. Mathar, Aug 14 2024

f[n_] := If[Total[Flatten[Position[Table[MultiplicativeOrder[i, Prime[n]], {i, Prime[n] - 1}],    Prime[n] - 1]]] == EulerPhi[Prime[n] - 1]*Prime[n]/2, 1, 0];
For[k = 1, k < 10000, k++, If[f[k] == 1 && Mod[Prime[k], 4] == 3, Print[k, "  ", Prime[k]]]]
Select[4*Range[1000] + 3, PrimeQ[#] && Mean[PrimitiveRootList[#]] == #/2 &] (* Amiram Eldar, Oct 11 2021 *)

vr(p) = j=0; r=vector(eulerphi(p-1)); pr=znprimroot(p); for(i=1, p-1, if(gcd(i, p-1)==1, r[j++]=lift(pr^i))); r; \\ after A060749
isok(p) = ((p % 4 == 3) && (vpr = vr(p)) && (vecsum(vpr) == #vpr*p/2)); \\ Michel Marcus, Jan 09 2016

a(16)-a(38) from Michel Marcus, Jan 09 2016

A267009 Primes p for which the average of the primitive roots is > p/2.

Original entry on oeis.org

3, 7, 11, 23, 47, 59, 71, 79, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 227, 239, 251, 263, 271, 283, 311, 331, 347, 359, 367, 383, 419, 431, 439, 443, 463, 467, 479, 487, 499, 503, 523, 547, 563, 571, 587, 599, 607, 647, 659, 691, 719, 727

Offset: 1

Dimitri Papadopoulos, Jan 08 2016

nonn

It appears that these primes are all congruent to 3 (mod 4).

			a(2) = 7 since the primitive roots of 7 are 3 and 5 and their average is (3+5)/2 = 8/2 > 7/2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008330, A060749, A088144, A266990.

A = Table[Total[Flatten[Position[Table[MultiplicativeOrder[i, Prime[k]], {i, Prime[k] - 1}], Prime[k] - 1]]]/(EulerPhi[Prime[k] - 1] Prime[k]/2), {k, 1, 1000}]; Prime[Flatten[Position[A, _?(# > 1 &)]]]
Select[Range[1000], PrimeQ[#] && Mean[PrimitiveRootList[#]] > #/2 &] (* Amiram Eldar, Oct 09 2021 *)

a(n) = prime(A266990(n)).

A266990 The indices of primes p for which the average of the primitive roots is > p/2.

Original entry on oeis.org

2, 4, 5, 9, 15, 17, 20, 22, 23, 27, 28, 31, 32, 34, 36, 38, 39, 41, 43, 46, 47, 49, 52, 54, 56, 58, 61, 64, 67, 69, 72, 73, 76, 81, 83, 85, 86, 90, 91, 92, 93, 95, 96, 99, 101, 103, 105, 107, 109, 111, 118, 120, 125, 128, 129, 131, 132, 133, 138, 141, 143, 144, 146, 150

Offset: 1

Dimitri Papadopoulos, Jan 08 2016

nonn

It appears that these primes are all congruent to 3 (mod 4).

			a(2) = 4 is a term since prime(a(2)) = prime(4) = 7, the primitive roots of 7 are 3 and 5 and their average is (3+5)/2 = 8/2 > 7/2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000720, A008330, A060749, A088144, A267009.

A = Table[Total[Flatten[Position[Table[MultiplicativeOrder[i, Prime[k]], {i, Prime[k] - 1}],Prime[k] - 1]]]/(EulerPhi[Prime[k] - 1] Prime[k]/2), {k, 1, 1000}]; Flatten[Position[A, _?(# > 1 &)]]
Select[Range[150], Mean[PrimitiveRootList[(p = Prime[#])]] > p/2 &] (* Amiram Eldar, Oct 09 2021 *)

a(n) = A000720(A267009(n)). - Amiram Eldar, Oct 09 2021

A266989 Primes for which the average of the primitive roots is < p/2.

Original entry on oeis.org

31, 43, 67, 223, 379, 491, 619, 631, 643, 683, 859, 883, 907, 1051, 1091, 1423, 1747, 1987, 2143, 2347, 2371, 2467, 2531, 2767, 3307, 3643, 3691, 3739, 3823, 3931, 4019, 4219, 4519, 4691, 4987, 5059, 5107, 5347, 5683, 5827, 6043

Offset: 1

Dimitri Papadopoulos, Jan 08 2016

nonn

These primes are congruent to 3 (mod 4).

			a(1)=31. The primitive roots of 31 are 3, 11, 12, 13, 17, 21, 22, and 24.
Their average is (3+11+12+13+17+21+22+24)/phi(30)=123/8<31/2.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A008330, A060749, A088144, A266987, A266988, A267010.

f:= proc(p) local g;
  if not isprime(p) then return false fi;
  g:= numtheory[primroot](p);
  evalb(add(g&^i mod p, i = select(t->igcd(t,p-1)=1, [$1..p-2]))
     < p/2 * numtheory:-phi(p-1))
end proc:
select(f, [seq(i,i=3..10000,4)]); # Robert Israel, Feb 09 2016

A = Table[Total[Flatten[Position[Table[MultiplicativeOrder[i, Prime[k]], {i, Prime[k] - 1}],Prime[k] - 1]]]/(EulerPhi[Prime[k] - 1] Prime[k]/2), {k, 1,100}]; Prime[Flatten[Position[A, _?(# < 1 &)]]]

ar(p) = my(r, pr, j); r=vector(eulerphi(p-1)); pr=znprimroot(p); for(i=1, p-1, if(gcd(i, p-1)==1, r[j++]=lift(pr^i))); vecsort(r) ;
isok(p) = my(vr = ar(p)); vecsum(vr)/#vr < p/2;
lista(nn) = forprime(p=2, nn, if (isok(p), print1(p, ", "))); \\ Michel Marcus, Feb 09 2016

a(n) = prime(A266988(n)).

A266988 The indices of primes for which the average of the primitive roots is < p/2.

Original entry on oeis.org

11, 14, 19, 48, 75, 94, 114, 115, 117, 124, 149, 153, 155, 177, 182, 224, 272, 300, 324, 348, 351, 365, 370, 403, 465, 510, 515, 522, 531, 546, 555, 578, 614, 634, 667, 677, 683, 707, 748, 765, 788, 795, 802, 808, 832, 850, 871, 876, 886, 888, 966, 980

Offset: 1

Dimitri Papadopoulos, Jan 08 2016

nonn

These primes are all of the form p==3 (mod 4). (conjecture)

			p(a[1])=p(11)=31. The primitive roots of 31 are 3, 11, 12, 13, 17, 21, 22, and 24.
Their average is (3+11+12+13+17+21+22+24)/phi(30)=123/8<31/2.

Cf. A008330, A060749, A088144, A266989.

A = Table[Total[Flatten[Position[Table[MultiplicativeOrder[i, Prime[k]], {i, Prime[k] - 1}],Prime[k] - 1]]]/(EulerPhi[Prime[k] - 1] Prime[k]/2), {k, 1, 1000}];Flatten[Position[A, _?(# < 1 &)]]

A266987 Primes p for which the average of the primitive roots equals p/2.

Original entry on oeis.org

2, 5, 13, 17, 19, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 307, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449

Offset: 1

Dimitri Papadopoulos, Jan 08 2016

nonn

From Robert Israel, Feb 01 2016: (Start)
Union of A002144 and A267010.
Contains A002144 because for each of these primes, x is a primitive root iff p-x is a primitive root. (End)

			a(13) = 13 since the primitive roots of 13 are 2, 6, 7, and 11 and the average of these primitive roots is (2+6+7+11)/phi(12) = 26/4 = 13/2.

Robert Israel, Table of n, a(n) for n = 1..8213

Cf. A002144, A008330, A060749, A088144, A266986, A267010.

proots := proc(n)
    local r,eulphi,m;
    if n = 1 then
        return {0} ;
    end if;
    eulphi := numtheory[phi](n) ;
    r := {} ;
    for m from 0 to n-1 do
        if numtheory[order](m,n) = eulphi then
            r := r union {m} ;
        end if;
    end do:
    return r;
end proc:
isA266987 := proc(n)
    local r;
    if isprime(n) then
        r := convert(proots(n),list) ;
        2*add(pr, pr=r)  = n*nops(r) ;
    else
        false;
    end if;
end proc:
for n from 1 to 500 do
    if isA266987(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Jan 12 2016
Filter:= proc(p) local x, s,js;
  if p mod 4 = 1 then return true fi;
  x:= numtheory:-primroot(p);
  js:= select(t -> igcd(t,p-1)=1, [$1..p-2]);
  s:= add(x&^ j mod p, j=js);
  evalb(s = p/2 * nops(js))
end proc:
select(Filter, [seq(ithprime(i),i=1..1000)]); # Robert Israel, Feb 01 2016

A = Table[Total[Flatten[Position[Table[MultiplicativeOrder[i, Prime[k]], {i, Prime[k] - 1}],Prime[k] - 1]]]/(EulerPhi[Prime[k] - 1] Prime[k]/2), {k, 1, 100}]; Prime[Flatten[Position[A, _?(# == 1 &)]]]
(* second program (version >= 10): *)
Select[Prime[Range[100]], Mean[PrimitiveRootList[#]] == #/2&] (* Jean-François Alcover, Jan 12 2016 *)

a(n) = prime(A266986(n)).

cp's OEIS Frontend

User: Dimitri Papadopoulos

A299016 Integer radii of circles over an integer lattice such that the number of unit squares whose centers are contained in the circle is less than the area of the circle.

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A267632 Triangle T(n, k) read by rows: the k-th column of the n-th row lists the number of ways to select k distinct numbers (k >= 1) from [1..n] so that their sum is divisible by n.

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A268397 a(n) is the smallest prime with at least n consecutive primitive roots.

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A267133 a(n) = (1/n)(2/n)(3/n)...((n-1)/n) where (k/n) is the Kronecker symbol, n >= 1.

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A267010 Primes of the form p==3 (mod 4) such that the average of their primitive roots equals p/2.

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Maple

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PARI

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A267009 Primes p for which the average of the primitive roots is > p/2.

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Mathematica

Formula

A266990 The indices of primes p for which the average of the primitive roots is > p/2.

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Mathematica

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A266989 Primes for which the average of the primitive roots is < p/2.

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