Martin Becker - cp's OEIS frontend

A383574 Fourth column of A353077.

9, 14, 8, -1, 13, 7, 9, -1, 12, -1, 16, -1, -1, 7, 21, -1, 12, -1, -1, -1, 13, -1, 33, -1, 9, -1, 12, -1, 13, 7, -1, -1, -1, -1, 19, -1, -1, -1, 8, -1, 10, -1, -1, -1, 10, -1, 25, -1, -1, -1, 15, -1, -1, -1, -1, -1, 8, -1, 16, -1, -1, 7, -1, -1, 12, -1, -1

Offset: 4

Martin Becker, May 03 2025

sign

Integers 0 to 6 are not in the sequence: For n > 5, the first three columns of A353077 are necessarily -1, -1, -1 or 0, 1, 3, and the fourth column is -1 or > 3, respectively. It is actually > 6 in the second case, as 4 - 3 = 1 - 0, 5 - 3 = 3 - 1, 6 - 3 = 3 - 0, respectively, would violate the distinctness of differences in perfect difference sets.
For n = 2^m + 1, m > 2, a(n) = 7, because 2 is a multiplier of such sets, therefore perfect difference sets containing 1, 2, 4, and 8 with translated sets containing 0, 1, 3, and 7 exist.
If n-1 is a prime power, a(n) != -1, as then there exist Singer type perfect difference sets.
If 4 <= n < 2*10^10 and n-1 is not a prime power, a(n) = -1. Cf. Gordon (2020).
Empirical observations further suggest that:
For n = 3^m + 1, m >= 1, a(n) = 9.
The most frequent positive value is 10.
11 is not in the sequence.

			For n = 4, there are 4 perfect difference sets containing 0 and 1: {0, 1, 3, 9}, {0, 1, 4, 6}, {0, 1, 5, 11}, and {0, 1, 8, 10}. The lexically earliest is {0, 1, 3, 9}. Its fourth element is 9, thus a(4) = 9.
There are no perfect difference sets with 7 elements. Thus a(7) = -1.

Martin Becker, Table of n, a(n) for n = 4..5000
Daniel Gordon, The Prime Power Conjecture is true for n < 2,000,000, 1994.
Daniel Gordon, On difference sets with small lambda, arXiv:2007.07292 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Perfect Difference Set
Eric Weisstein's World of Mathematics, Prime Power Conjecture

Fourth column of A353077.

Cf. A000961, A333852, A373514.

A373514 Number of simple difference sets of the Singer type (m^2 + m + 1, m + 1, 1) that are a superset of {0, 1, 3} with m = m(n) = A000961(n), for n >= 1.

Original entry on oeis.org

0, 1, 1, 0, 2, 1, 1, 1, 4, 3, 1, 6, 3, 8, 2, 3, 9, 6, 2, 8, 14, 10, 14, 4, 14, 20, 10, 2, 14, 24, 15, 18, 6, 27, 30, 19, 34, 21, 26, 22, 33, 10, 13, 30, 5, 44, 38, 30, 41, 26, 36, 25, 56, 17, 58, 52, 38, 51, 40, 63, 45, 41, 46, 76, 47, 70, 72, 55, 15, 80, 6

Offset: 1

Martin Becker, Jun 07 2024

nonn

			For n=5, m=5, there are 2 Singer type planar difference sets of order 5 containing 0, 1, and 3: {0,1,3,8,12,18} and {0,1,3,10,14,26}. Thus a(5) = 2.
For n=11, m=16, there is only 1 such set: {0,1,3,7,15,31,63,90,116,127,136,181,194,204,233,238,255}. Thus a(11) = 1.

Martin Becker, Table of n, a(n) for n = 1..400

Cf. A335866, A000961, A373946. Counts sets in A333852 with the property that 3 is also in the set.

A373946 Number of primitive polynomials of third degree over GF(m) with vanishing quadratic term with m = m(n) = A000961(n), for n >= 2.

Original entry on oeis.org

1, 1, 0, 4, 3, 18, 8, 16, 18, 48, 48, 27, 80, 48, 108, 108, 72, 300, 144, 224, 180, 308, 192, 336, 560, 240, 648, 420, 576, 540, 648, 768, 1080, 1200, 912, 1360, 1008, 1352, 1188, 1584, 960, 2340, 1620, 4410, 2112, 2432, 1980, 2952, 1560, 2592, 2025, 4592, 2448, 4872, 4576

Offset: 2

Martin Becker, Jun 23 2024

nonn

Apparently, a(n) = A373514(n) * A000010( 3 * A000961(n) - 3 ) * A025474(n) / 2, for n >= 2.

			For n=5, m=5, there are 20 primitive polynomials over GF(5) of the form x^3+a*x^2+b*x+c. Among these, 4 polynomials have a=0: x^3+3*x+2, x^3+3*x+3, x^3+4*x+2, and x^3+4*x+3. Thus, a(5) = 4.

Martin Becker, Table of n, a(n) for n = 2..400

Cf. A000961, A319213, A373514.

is_max_o = (x1, x0, m, e)-> {
  for(i = 1, #e, if(x1^e[i] == x0, return(0))); x1^m == x0;
}
count_them = (q)-> {
  z = ffprimroot(ffgen(q, 'c));
  m = q^3 - 1; f = factor(m); d = #f~;
  e = vector(d, i, m/f[d + 1 - i, 1]);
  co = vector(q - 1, i, z^(i - 1));
  r = 0;
  for(a = 1, q - 1,
    for(b = 1, q - 1,
      p = co[1]*x^3 + co[a]*x + co[b];
      x1 = Mod(x, p); x0 = x1^0;
      if(is_max_o(x1, x0, m, e) && polisirreducible(p), r += 1)
    )
  );
  r;
}
print1(count_them(2));
for(q = 3, 64, if(isprimepower(q), print1(", ", count_them(q))))

A344448 Square array read by antidiagonals upwards: T(n,k) for integer k >= 0 is the n-th prime p such that p^(2*3^k) + p^(3^k) + 1 is prime.

Original entry on oeis.org

2, 3, 2, 5, 3, 2, 17, 11, 11, 191, 41, 191, 263, 311, 4457, 59, 269, 557, 557, 5867, 3803, 71, 383, 761, 659, 7001, 13859, 1889, 89, 509, 797, 887, 7019, 22961, 16829, 17, 101, 809, 863, 1607, 7541, 31223, 62549, 69677, 113921, 131, 827, 977, 2309, 8609, 44351, 67103, 102647, 176459, 24071

Offset: 1

Martin Becker, May 19 2021

T(n,k)^(3^k), for all n >= 1, k >= 0, arranged by increasing values, is A342690. It is conjectured that all columns are infinite. If 3^k was replaced by k in the definition, all additional columns would be empty, as x^(2*k) + x^k + 1 is reducible if k has prime factors other than 3. For checking the property, Pocklington-Lehmer type primality tests seem particularly effective, as n-1 always has a large smooth factor p^(3^k), cf. the paper of Brillhart, Lehmer and Selfridge (1975), Theorem 5.

This array describes the essence of A342690 and A342691 in much more terse form. T(1, 8) = 113921 matches the 33177-digit value q = 113921^3^8 in A342690 and the 66353-digit prime q^2+q+1 in A342691.

			Array begins:
===============================================================
n\k |   0    1    2    3     4     5      6      7      8     9
----+----------------------------------------------------------
  1 |   2    2    2  191  4457  3803   1889     17 113921 24071
  2 |   3    3   11  311  5867 13859  16829  69677 176459 ...
  3 |   5   11  263  557  7001 22961  62549 102647 ...
  4 |  17  191  557  659  7019 31223  67103 164963 ...
  5 |  41  269  761  887  7541 44351 181931 170669 ...
  6 |  59  383  797 1607  8609 45737 188333 207923 ...
  7 |  71  509  863 2309  8627 61751 205433 235679 ...
  8 |  89  809  977 2621 21773 63377 210407 342833 ...
  9 | 101  827 1091 2687 22871 79481 219761 459209 ...

J. Brillhart, D. H. Lehmer and J. L. Selfridge, New primality criteria and factorizations of 2^m+-1, Math. Compl. 29 (1975) 620-647.

The first column T(n,0) is A053182(n). The second column T(n,1) is A066100(n).

Cf. A342690, A342691.

N=5; K=2; m=matrix(N, K+1); for(k=0, K, i=0; forprime(p=2, , q=p^3^k;if(isprime(q^2+q+1, 1), i+=1; m[i,k+1]=p; if(i==N, break)))); m

A342691 Primes of the form (p^k)^2 + p^k + 1 with prime p and positive integer k.

Original entry on oeis.org

7, 13, 31, 73, 307, 757, 1723, 3541, 5113, 8011, 10303, 17293, 28057, 30103, 86143, 147073, 262657, 459007, 492103, 552793, 579883, 598303, 684757, 704761, 735307, 830833, 1191373, 1204507, 1353733, 1395943, 1424443, 1482307, 1772893, 1886503, 2037757, 2212657

Offset: 1

Martin Becker, May 18 2021

nonn

Also, primes of the form (p^3^m)^2 + p^3^m + 1 with prime p and nonnegative integer m, since k must be a power of 3, from the theory of cyclotomic polynomials.

			31 = (5^1)^2 + 5^1 + 1 is in the sequence as 31 is prime and 5 is prime and 1 is a positive integer.
73 = (2^3)^2 + 2^3 + 1 is in the sequence as it is prime and 2 is prime and 3 is a positive integer.

Martin Becker, Table of n, a(n) for n = 1..20000

Contains A053183 and A063784.
Intersection of A335865 and A000040 minus {3}.
Cf. A342690, A344448.

Select[Table[q^2 + q + 1, {q, Select[Range[1500], PrimePowerQ[#] &]}], PrimeQ] (* Amiram Eldar, Aug 16 2024 *)

for(q=2,2048,if(isprimepower(q),m=q^2+q+1;if(isprime(m),print1(m, ", "))))

A342690 Prime powers q in A246655 such that q^2 + q + 1 is prime.

Original entry on oeis.org

2, 3, 5, 8, 17, 27, 41, 59, 71, 89, 101, 131, 167, 173, 293, 383, 512, 677, 701, 743, 761, 773, 827, 839, 857, 911, 1091, 1097, 1163, 1181, 1193, 1217, 1331, 1373, 1427, 1487, 1559, 1583, 1709, 1811, 1847, 1931, 1973, 2129, 2273, 2309, 2339, 2411, 2663

Offset: 1

Martin Becker, May 18 2021

nonn

Also, prime powers q = p^(3^k) with prime p and nonnegative integer k and the property that q^2 + q + 1 is prime, since the exponent must be a power of 3, from the theory of cyclotomic polynomials. 17^(3^7) is in the sequence, generating a 5382-digit prime.

			5 = 5^1 is a term: 5^2 + 5 + 1 = 31 is prime.
8 = 2^3 is a term: 8^2 + 8 + 1 = 73 is prime.

Martin Becker, Table of n, a(n) for n = 1..20000

Intersection of A246655 and A002384.

Cf. A342691, A344448.

Select[Range@2000,PrimePowerQ@#&&PrimeQ[#^2+#+1]&] (* Giorgos Kalogeropoulos, May 18 2021 *)

N=50; i=0; a=vector(N); for(q=2, oo, if(isprimepower(q) && isprime(q^2+q+1), i+=1; a[i]=q; if(i==N, break))); a

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User: Martin Becker

A383574 Fourth column of A353077.

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A373514 Number of simple difference sets of the Singer type (m^2 + m + 1, m + 1, 1) that are a superset of {0, 1, 3} with m = m(n) = A000961(n), for n >= 1.

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A373946 Number of primitive polynomials of third degree over GF(m) with vanishing quadratic term with m = m(n) = A000961(n), for n >= 2.

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A344448 Square array read by antidiagonals upwards: T(n,k) for integer k >= 0 is the n-th prime p such that p^(2*3^k) + p^(3^k) + 1 is prime.

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A342691 Primes of the form (p^k)^2 + p^k + 1 with prime p and positive integer k.

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A342690 Prime powers q in A246655 such that q^2 + q + 1 is prime.

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