A182238 -id:A182238 - cp's OEIS frontend

A356109 Numbers k such that k^2 + {1,3,7,13} are prime.

2, 4, 10, 5996, 8894, 11204, 14290, 23110, 30866, 37594, 43054, 64390, 74554, 83464, 93460, 109456, 111940, 132304, 151904, 184706, 238850, 262630, 265990, 277630, 300206, 315410, 352600, 355450, 376190, 404954, 415180, 462830, 483494, 512354, 512704, 566296

Offset: 1

Michel Lagneau, Jul 27 2022

nonn

Conjecture: the sequence is infinite.

			2^2 + {1,3,7,13} = {5,7,11,17} all prime.
4^2 + {1,3,7,13} = {17,19,23,29} all prime.

Intersection of A005574, A049422, A114270, A113536.

Subsequence of A182238.

q:= k-> andmap(j-> isprime(k^2+j), [1,3,7,13]):
select(q, [$0..1000000])[];  # Alois P. Heinz, Jul 27 2022

Select[Range[500000], AllTrue[#^2 + {1,3,7,13}, PrimeQ] &] (* Amiram Eldar, Jul 27 2022 *)

from sympy import isprime
def ok(n): return all(isprime(n*n+i) for i in {1,3,7,13})
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jul 27 2022

A182261 Numbers n such that n^2 + {1,3,7} are semiprimes.

Original entry on oeis.org

44, 46, 80, 88, 102, 104, 108, 226, 234, 238, 246, 272, 290, 308, 310, 328, 334, 358, 370, 426, 456, 480, 514, 526, 530, 586, 588, 614, 720, 766, 790, 842, 846, 848, 872, 880, 884, 896, 898, 900, 934, 940, 974, 980, 1040, 1076, 1078, 1088, 1110, 1160, 1208

Offset: 1

Jonathan Vos Post, Apr 21 2012

This is to A182238 as A001358 semiprimes are to A000040 primes.

			44 is in the sequence because (44^2) + 1 = 1937 = 13 * 149, (44^2) + 3 = 1939 = 7 * 277, and  (442) + 7 = 1943 = 29 * 67.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A001358, A182238.

IsSemiprime:=func; [n: n in [2..1225] | forall{n^2+i: i in [1,3,7] | IsSemiprime(n^2+i)}]; // Bruno Berselli, Apr 22 2012

a:= proc(n) option remember; local k;
      for k from 1+a(n-1) while map(x-> not isprime(k^2+x) and
          add(i[2], i=ifactors(k^2+x)[2])=2, [1, 3, 7])<>[true$3]
      do od; k
    end: a(0):=0:
seq(a(n), n=1..50);  # Alois P. Heinz, Apr 22 2012

okQ[n_] := AllTrue[n^2 + {1, 3, 7}, PrimeOmega[#] == 2&];
Select[Range[2000], okQ] (* Jean-François Alcover, Jun 01 2022 *)

{ n : {n^2+1, n^2+3, n^2+7} in A001358 }.

A356110 Numbers k such that k^2 + {1,3,7,13,31} are prime.

Original entry on oeis.org

4, 10, 14290, 43054, 109456, 315410, 352600, 483494, 566296, 685114, 927070, 1106116, 1248796, 1501174, 1997986, 2399204, 2501404, 2553100, 2726840, 2874680, 3291760, 4129394, 4473766, 4794520, 4901144, 6350306, 7444070, 7753456, 7892504, 8009536, 8069540

Offset: 1

Michel Lagneau, Jul 27 2022

nonn

Conjecture: the sequence is infinite.

			4^2 + {1,3,7,13,31} = {17,19,23,29,47} are all prime.

Cf. A005574, A049422, A114270, A113536, A182238, A356109.

q:= k-> andmap(j-> isprime(k^2+j), [1,3,7,13,31]):
select(q, [$0..1000000])[];  # Alois P. Heinz, Jul 27 2022

Select[Range[10^6], AllTrue[#^2 + {1,3,7,13,31}, PrimeQ] &] (* Amiram Eldar, Jul 27 2022 *)

from sympy import isprime
def ok(n): return all(isprime(n*n+i) for i in {1,3,7,13,31})
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jul 27 2022

A356175 Numbers k such that k^2 + {1,3,7,13,163} are prime.

Original entry on oeis.org

2, 4, 10, 14290, 64390, 74554, 83464, 93460, 132304, 238850, 262630, 277630, 300206, 352600, 376190, 404954, 415180, 610340, 806180, 984686, 1025650, 1047050, 1106116, 1382860, 2014624, 2440714, 2525870, 2538344, 2760026, 2826380, 3145600, 3508586, 3715156

Offset: 1

Jean-Marc Rebert, Jul 28 2022

nonn

For 14 <= m <= 999 and k <= A356110(31) = 8069560, the number of sets of primes of the form k^2 + {1,3,7,13,m} is the greatest for m = 163. There are 51 such terms. See b-file.

All terms are 2 or 4 modulo 6.

			2 is a term since 2^2 + {1,3,7,13,163} = {5,7,11,17,167} are all primes.

David A. Corneth, Table of n, a(n) for n = 1..6757 (first 237 terms from Jean-Marc Rebert, terms <= 10^10).

Cf. A356110.

Subsequence of A005574, A049422, A114270, A113536, A080149, A182238 and A356109.

q:= k-> andmap(j-> isprime(k^2+j), [1, 3, 7, 13, 163]):
select(q, [$0..1000000])[];  # Alois P. Heinz, Jul 28 2022

Select[Range[4*10^6], AllTrue[#^2 + {1, 3, 7, 13, 163}, PrimeQ] &] (* Amiram Eldar, Jul 28 2022 *)

is(k)=my(v=[1,3,7,13,163],ok=1);for(i=1,#v,if(!isprime(k^2+v[i]),ok=0;break));ok

from sympy import isprime
def ok(n): return all(isprime(n*n+i) for i in {1,3,7,13,163})
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jul 28 2022

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A356109 Numbers k such that k^2 + {1,3,7,13} are prime.

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A182261 Numbers n such that n^2 + {1,3,7} are semiprimes.

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A356110 Numbers k such that k^2 + {1,3,7,13,31} are prime.

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A356175 Numbers k such that k^2 + {1,3,7,13,163} are prime.

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