A014556 -id:A014556 - cp's OEIS frontend

A332207 a(n) is the number of 3-factorizations of n.

1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 1, 3, 2, 3, 2, 4, 1, 4, 3, 3, 2, 4, 2, 5, 3, 3, 2, 5, 2, 6, 2, 3, 3, 6, 2, 6, 3, 4, 3, 6, 1, 6, 4, 5, 3, 5, 2, 7, 4, 5, 3, 6, 2, 8, 3, 4, 3, 8, 2, 8, 4, 4, 4, 8, 3, 7, 2, 6, 4, 9, 2, 8, 3, 5, 5, 6, 2, 10, 6, 6, 3, 7, 2, 10, 4, 5, 3, 10

Offset: 1

Michel Marcus, Feb 07 2020

nonn

See Bingham link for definitions.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Aram Bingham, Ternary arithmetic, factorization, and the class number one problem, arXiv:2002.02059 [math.NT], 2020. See Table 1 p. 14.

Cf. A014556.

a[n_] := Block[{c=0, z=1}, While[3 z^2 - 3 z + 1 <= n, c += Length@ Solve[ x y z - (x-1) (y-1) (z-1) == n && x >= y >= z, {x, y}, Integers]; z++]; c]; Array[a, 88] (* Giovanni Resta, Feb 07 2020 *)

More terms from Giovanni Resta, Feb 07 2020

A356751 Positive integers m such that x^2 - x + m contains more than m/2 prime numbers for x = 1, 2, ..., m.

Original entry on oeis.org

3, 5, 7, 11, 17, 41, 47, 59, 67, 101, 107, 161, 221, 227, 347, 377

Offset: 1

Marco Ripà, Aug 25 2022

This sequence is related to A188424, since we are considering only the addends m := 2n - 1 of k^2 + k + 2n - 1 such that A188424(n)/(2n - 1) > 1/2.
Furthermore, it is conjectured that the present sequence consists of only 16 terms (it has been checked by brute force that there are only 16 terms which are smaller than 20000) and that they are all prime or semiprime (e.g., a(12) = 161, a(13) = 221, and a(16) = 377 are semiprime). Lastly, we trivially point out that all terms must be odd, since if m is even, then x^2 - x + m is also even (and x^2 - x + 2 has only one prime for x <= 2).
For an explanation of the abundance of primes of the form x^2 - x + m, for some given m, see Goudsmit's paper in Links.
Stronger conjecture: for every real number e > 0 and every integer m > 0, there are finitely many integer polynomials P(x) = Ax^2 + Bx + C with at least e*m primes in P(1), ..., P(m) and max(|A|, |B|, |C|) <= m. - Charles R Greathouse IV, Sep 11 2022
Altering the bounds for x in the definition to 0 <= x <= m-1 (and counting the same prime twice for x=0 and x=1 if m is prime) would result in an additional term 2. Conjecturally, there would be no more additional terms. - Pontus von Brömssen, Jun 20 2024

			7 is a term since x^2 - x + 7 is prime for x = 1, 3, 4, and 6, which is 4 values of x, and 4 > 7/2.

S. A. Goudsmit, Unusual Prime Number Sequences, Nature Vol. 214 (1967), 1164.
Brady Haran and Matt Parker, Caboose Numbers, Youtube video, June 2024.

Cf. A005846, A007641, A007635, A057604, A188424, A331940, A356756.

Cf. A014556 (Euler's "Lucky" numbers).

q[k_] := Count[Range[k], ?(PrimeQ[#^2 - # + k] &)] > k/2; Select[Range[400], q] (* _Amiram Eldar, Aug 26 2022 *)

isok(m) = sum(k=1, m, isprime(k^2 - k + m)) > m/2; \\ Michel Marcus, Aug 26 2022

from sympy import isprime
def ok(m): return 2*sum(1 for x in range(1, m+1) if isprime(x**2-x+m)) > m
print([m for m in range(1, 400) if ok(m)]) # Michael S. Branicky, Aug 26 2022

A118124 Triangle T(n,m) = (n+m)^2+n+m+41, read by rows.

Original entry on oeis.org

41, 43, 47, 47, 53, 61, 53, 61, 71, 83, 61, 71, 83, 97, 113, 71, 83, 97, 113, 131, 151, 83, 97, 113, 131, 151, 173, 197, 97, 113, 131, 151, 173, 197, 223, 251, 113, 131, 151, 173, 197, 223, 251, 281, 313, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 151, 173

Offset: 0

Roger L. Bagula, May 12 2006

Defined to display A005846(d) in the d-th antidiagonal, d sufficiently small.

			41;
43, 47;
47, 53, 61;
53, 61, 71, 83;
61, 71, 83, 97, 113;
71, 83, 97, 113, 131, 151;
83, 97, 113, 131, 151, 173, 197;
97, 113, 131, 151, 173, 197, 223, 251;
113, 131, 151, 173, 197, 223, 251, 281, 313;
131, 151, 173, 197, 223, 251, 281, 313, 347, 383;
151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461;

Cf. A014556.

f[n_] = n^2 + n + 41 t[n_, m_] = f[n + m] a = Table[Table[t[n, m], {n, 0, m}], {m, 0, 10}] c = Flatten[a]

Keyword:less and reference to A005846, A014556 added - The Assoc. Eds. of the OEIS, Oct 20 2010

A217396 Prime indices of Euler's lucky numbers.

Original entry on oeis.org

1, 2, 3, 5, 7, 13

Offset: 1

Raphie Frank, Oct 02 2012

a(n) = A000720(A014556(n)). - Alexander R. Povolotsky, Sep 22 2022

A276476 a(n) is the number of distinct integers of the form x^2-x-prime(n) for 0<=x<=prime(n)+1 whose absolute value is prime.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 9, 13, 11, 17, 20, 17, 10, 32, 16, 23, 26, 30, 25, 21, 55, 38, 30, 27, 25, 34, 57, 19, 83, 49, 44, 40, 39, 60, 37, 77, 54, 57, 27, 43, 79, 67, 45, 110, 42, 93, 79, 79, 43, 85, 46, 90, 96, 41, 54, 96, 127, 107, 63, 78, 181, 67, 78, 72, 189, 51, 77, 103

Offset: 1

Charles Kusniec, Sep 12 2016

nonn

			a(2)=2 because prime(2)=3 and x^2 - x - 3 generates {-3, -3, -1, 3, 9}. This contains two integers, -3 and 3, whose absolute value is prime.
a(14)=32 because prime(14)=43 and x^2 - x - 43 generates 32 prime numbers for x = 0..44.

Cf. A014556, A253827.

isaprime(x) = isprime(x) || isprime(-x);
nbp(n) = {v = vector(prime(n)+2, x, x--; x^2-x-prime(n)); vp = select(x->isaprime(x), v); vp = Set(vp); #vp;} \\ Michel Marcus, Sep 13 2016

More terms from Michel Marcus, Sep 13 2016

A279241 Let f(n) = 4n^2 + 2n + 41. The values |f(n)| are primes for all n in the range -20 to 19 (but not for n=-21 or 20). The sequence lists this maximal run of primes in the order in which they appear.

Original entry on oeis.org

1601, 1447, 1301, 1163, 1033, 911, 797, 691, 593, 503, 421, 347, 281, 223, 173, 131, 97, 71, 53, 43, 41, 47, 61, 83, 113, 151, 197, 251, 313, 383, 461, 547, 641, 743, 853, 971, 1097, 1231, 1373, 1523

Offset: 1

Charles Kusniec, Dec 08 2016

This same list will also appear for 0<=x<=39 using the form 4x^2-158x+1601.

The substitution 2n = m changes this quadratic form into Euler's famous quadratic form m^2+m+41 (see A005846). Concerning the conjectured extremal properties of these forms, one should note the comment from T. D. Noe in A005846. For another quadratic form similar to this one, see A145096. - N. J. A. Sloane, Dec 17 2016

Cf. A005846, A014556, A145096.

s1:=[]; f:=n->4*n^2+2*n+41;
for n from -20 to 19 do if isprime(abs(f(n))) then s1:=[op(s1), abs(f(n))]; fi; od:
s1; # From N. J. A. Sloane, Dec 17 2016. This does nothing more than produce the primes mentioned in the definition

Edited by N. J. A. Sloane, Dec 17 2016

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A332207 a(n) is the number of 3-factorizations of n.

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A356751 Positive integers m such that x^2 - x + m contains more than m/2 prime numbers for x = 1, 2, ..., m.

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A118124 Triangle T(n,m) = (n+m)^2+n+m+41, read by rows.

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A217396 Prime indices of Euler's lucky numbers.

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A276476 a(n) is the number of distinct integers of the form x^2-x-prime(n) for 0<=x<=prime(n)+1 whose absolute value is prime.

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PARI

Extensions

A279241 Let f(n) = 4*n^2 + 2*n + 41. The values |f(n)| are primes for all n in the range -20 to 19 (but not for n=-21 or 20). The sequence lists this maximal run of primes in the order in which they appear.

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A279241 Let f(n) = 4n^2 + 2n + 41. The values |f(n)| are primes for all n in the range -20 to 19 (but not for n=-21 or 20). The sequence lists this maximal run of primes in the order in which they appear.