A067774 -id:A067774 - cp's OEIS frontend

A049579 Numbers k such that prime(k)+2 divides (prime(k)-1)!.

4, 6, 8, 9, 11, 12, 14, 15, 16, 18, 19, 21, 22, 23, 24, 25, 27, 29, 30, 31, 32, 34, 36, 37, 38, 39, 40, 42, 44, 46, 47, 48, 50, 51, 53, 54, 55, 56, 58, 59, 61, 62, 63, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95

Offset: 1

Labos Elemer

nonn

Numbers k such that prime(k+1) - prime(k) does not divide prime(k+1) + prime(k). These are the numbers k for which prime(k+1) - prime(k) > 2. - Thomas Ordowski, Mar 31 2022

If we prepend 1, the first differences are A251092 (see also A175632). The complement is A029707. - Gus Wiseman, Dec 03 2024

			prime(4) = 7, 6!+1 = 721 gives residue 1 when divided by prime(4)+2 = 9.

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

The first differences are A251092 except first term, run-lengths A373819.
The complement is A029707.
Runs of terms differing by one have lengths A027833, min A107770, max A155752.
A000040 lists the primes, differences A001223 (run-lengths A333254, A373821).
A038664 finds the first prime gap of difference 2n.
A046933 counts composite numbers between primes.
A071148 gives partial sums of odd primes.
For prime runs: A001359, A005381, A006512, A025584, A067774, A373403.
Cf. A006560, A006562, A037201, A122535, A175632, A176246, A373820.

pnmQ[n_]:=Module[{p=Prime[n]},Mod[(p-1)!+1,p+2]==1]; Select[Range[ 100],pnmQ] (* Harvey P. Dale, Jun 24 2017 *)

isok(n) = (((prime(n)-1)! + 1) % (prime(n)+2)) == 1; \\ Michel Marcus, Dec 31 2013

Definition edited by Thomas Ordowski, Mar 31 2022

A210361 Prime numbers p such that x^2 + x + p produces primes for x = 0..2 but not x = 3.

Original entry on oeis.org

107, 191, 311, 461, 821, 857, 881, 1301, 1871, 1997, 2081, 2237, 2267, 2657, 3251, 3461, 3671, 4517, 4967, 5231, 5477, 5501, 5651, 6197, 6827, 7877, 8087, 8291, 8537, 8861, 9431, 10427, 10457, 11171, 12917, 13001, 13691, 13757, 13877, 14081, 14321, 15641

Offset: 1

T. D. Noe, Apr 05 2012

nonn

The first term is A164926(3).

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A067774, A164926, A210360, A210362, A210363, A210364, A210365.

lookfor = 3; t = {}; n = 0; While[Length[t] < 50, n++; c = Prime[n]; i = 1; While[PrimeQ[i^2 + i + c], i++]; If[i == lookfor, AppendTo[t, c]]]; t

A210362 Prime numbers p such that x^2 + x + p produces primes for x = 0..3 but not x = 4.

Original entry on oeis.org

5, 101, 227, 1091, 1481, 1487, 3917, 4127, 4787, 8231, 9461, 10331, 11777, 12107, 14627, 16061, 20747, 25577, 27737, 29021, 32297, 33347, 35531, 35591, 36467, 38447, 39227, 41177, 42461, 44267, 44531, 49031, 59441, 69191, 77237, 79811, 80777, 93251, 93491

Offset: 1

T. D. Noe, Apr 05 2012

nonn

The first term is A164926(4).

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A067774, A164926, A210360, A210361, A210363, A210364, A210365.

lookfor = 4; t = {}; n = 0; While[Length[t] < 50, n++; c = Prime[n]; i = 1; While[PrimeQ[i^2 + i + c], i++]; If[i == lookfor, AppendTo[t, c]]]; t
Select[Prime[Range[10000]],AllTrue[#+{2,6,12},PrimeQ]&&!PrimeQ[#+20]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 26 2015 *)
Select[Prime[Range[10000]],Boole[PrimeQ[#+{2,6,12,20}]]=={1,1,1,0}&] (* Harvey P. Dale, Nov 17 2024 *)

A210363 Prime numbers p such that x^2 + x + p produces primes for x = 0..4 but not x = 5.

Original entry on oeis.org

347, 641, 1427, 2687, 4001, 4637, 4931, 19421, 21011, 22271, 23741, 26711, 27941, 32057, 43781, 45821, 55331, 55817, 68207, 71327, 91571, 128657, 165701, 167621, 172421, 179897, 191447, 193871, 205421, 234191, 239231, 258107, 258611, 259157, 278807, 290021

Offset: 1

T. D. Noe, Apr 05 2012

nonn

The first term is A164926(5).

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A067774, A164926, A210360, A210361, A210362, A210364, A210365.

lookfor = 5; t = {}; n = 0; While[Length[t] < 50, n++; c = Prime[n]; i = 1; While[PrimeQ[i^2 + i + c], i++]; If[i == lookfor, AppendTo[t, c]]]; t
Select[Prime[Range[26000]],AllTrue[#+{2,6,12,20},PrimeQ] && !PrimeQ[ #+30]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 13 2017 *)
Select[Prime[Range[26000]],Boole[PrimeQ[#+{2,6,12,20,30}]]=={1,1,1,1,0}&] (* Harvey P. Dale, May 29 2025 *)

A210364 Prime numbers p such that x^2 + x + p produces primes for x = 0..5 but not x = 6.

Original entry on oeis.org

1607, 3527, 13901, 31247, 33617, 55661, 68897, 97367, 166841, 195731, 221717, 347981, 348431, 354371, 416387, 506327, 548831, 566537, 929057, 954257, 1246367, 1265081, 1358801, 1505087, 1538081, 1595051, 1634441, 1749257, 2200811, 2322107, 2641547, 2697971

Offset: 1

T. D. Noe, Apr 05 2012

nonn

The first term is A164926(6).

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A067774, A164926, A210360, A210361, A210362, A210363, A210365.

lookfor = 6; t = {}; n = 0; While[Length[t] < 50, n++; c = Prime[n]; i = 1; While[PrimeQ[i^2 + i + c], i++]; If[i == lookfor, AppendTo[t, c]]]; t

A210365 Prime numbers p such that x^2 + x + p produces primes for x = 0..6 but not x = 7.

Original entry on oeis.org

1277, 28277, 113147, 421697, 665111, 1164587, 1615631, 2798921, 2846771, 3053747, 5071667, 5093507, 5344247, 5706641, 6383051, 8396777, 10732817, 10812407, 11920367, 13176587, 16197947, 16462541, 16655447, 16943471, 17807831, 18102101, 20488901, 23421311

Offset: 1

T. D. Noe, Apr 05 2012

nonn

The first term is A164926(7).

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A067774, A164926, A210360, A210361, A210362, A210363, A210364.

lookfor = 7; t = {}; n = 0; While[Length[t] < 30, n++; c = Prime[n]; i = 1; While[PrimeQ[i^2 + i + c], i++]; If[i == lookfor, AppendTo[t, c]]]; t

A373819 Run-lengths (differing by 0) of the run-lengths (differing by 2) of the odd primes.

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 4, 2, 3, 2, 4, 3, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 10, 2, 4, 1, 7, 1, 4, 1, 3, 1, 2, 1, 1, 1, 2, 1, 18, 3, 2, 1, 2, 1, 17, 2, 1, 2, 2, 1, 6, 1, 9, 1, 3, 1, 1, 1, 1, 1, 1, 1, 8, 1, 3, 1, 2, 2, 15, 1, 1, 1, 4, 1, 1, 1, 1, 1, 7, 1

Offset: 1

Gus Wiseman, Jun 20 2024

nonn

Run-lengths of A251092.

			The odd primes begin:
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, ...
with runs:
   3   5   7
  11  13
  17  19
  23
  29  31
  37
  41  43
  47
  53
  59  61
  67
  71  73
with lengths:
3, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, ...
which have runs beginning:
  3
  2 2
  1
  2
  1
  2
  1 1
  2
  1
  2
  1 1 1 1
  2 2
  1 1 1
with lengths a(n).

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

Run-lengths of A251092.
For antiruns we have A373820, run-lengths of A027833 (if we prepend 1).
Positions of first appearances are A373825, sorted A373824.
A000040 lists the primes.
A001223 gives differences of consecutive primes, run-lengths A333254, run-lengths of run-lengths A373821.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
A071148 gives partial sums of odd primes.
For prime runs: A001359, A006512, A025584, A067774, A373405, A373406.
For composite runs: A005381, A054265, A068780, A373403, A373404.
Cf. A005117, A006560, A006562, A029707, A037201, A038664, A076259, A176246.

Length/@Split[Length/@Split[Select[Range[3,1000], PrimeQ],#1+2==#2&]//Most]//Most

A373824 Sorted positions of first appearances in the run-lengths (differing by 0) of the run-lengths (differing by 2) of the odd primes.

Original entry on oeis.org

1, 2, 11, 13, 29, 33, 45, 51, 57, 59, 69, 75, 105, 129, 211, 227, 301, 313, 321, 341, 407, 413, 447, 459, 537, 679, 709, 767, 1113, 1301, 1405, 1411, 1429, 1439, 1709, 1829, 1923, 2491, 2543, 2791, 2865, 3301, 3471, 3641, 4199, 4611, 5181, 5231, 6345, 6555

Offset: 1

Gus Wiseman, Jun 21 2024

nonn

Sorted positions of first appearances in A373819.

			The runs of odd primes differing by 2 begin:
   3   5   7
  11  13
  17  19
  23
  29  31
  37
  41  43
  47
  53
  59  61
  67
  71  73
  79
with lengths:
3, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, ...
which have runs beginning:
  3
  2 2
  1
  2
  1
  2
  1 1
  2
  1
  2
  1 1 1 1
  2 2
  1 1 1
with lengths:
1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 4, 2, 3, 2, 4, 3,...
with sorted positions of first appearances a(n).

Sorted firsts of A373819 (run-lengths of A251092).
The unsorted version is A373825.
For antiruns we have A373826, unsorted A373827.
A000040 lists the primes.
A001223 gives differences of consecutive primes (firsts A073051), run-lengths A333254 (firsts A335406), run-lengths of run-lengths A373821.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
A071148 gives partial sums of odd primes.
A373820 gives run-lengths of antirun-lengths, run-lengths of A027833.
For prime runs: A001359, A006512, A025584, A067774, A373405, A373406.
For composite runs: A005381, A054265, A068780, A373403, A373404.
Cf. A006560, A006562, A029707, A037201, A038664, A069010, A122535, A176246, A373401, A373402.

t=Length/@Split[Length/@Split[Select[Range[3,10000],PrimeQ],#1+2==#2&]];
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]

A107986 Composite numbers of the form p+2 where p is prime.

Original entry on oeis.org

4, 9, 15, 21, 25, 33, 39, 45, 49, 55, 63, 69, 75, 81, 85, 91, 99, 105, 111, 115, 129, 133, 141, 153, 159, 165, 169, 175, 183, 195, 201, 213, 225, 231, 235, 243, 253, 259, 265, 273, 279, 285, 295, 309, 315, 319, 333, 339, 351, 355, 361, 369, 375, 381, 385, 391

Offset: 1

Cino Hilliard, Jun 13 2005

This sequence is analogous to the sequence formed by the Goldbach-Euler conjecture that every even number greater than 2 is the sum of 2 primes. If p + 2 is prime then p and p + 2 are twin primes. The number of terms in this sequence is infinite. This follows immediately from the proof that the number of primes p is infinite. Conjecture: The ratio of the number of terms in this sequence to Pi(n) tends to a limit < 1.

The first term in this sequence that is not also in A062721 is 45 = 3^2 * 5. - Alonso del Arte, May 03 2014

Cf. A062721, A067774, A107987.

Select[Range[4, 399], Not[PrimeQ[#]] && PrimeQ[# - 2] &] (* Alonso del Arte, May 03 2014 *)

a(n) = A067774(n) + 2. - Amiram Eldar, Jul 05 2024

A263091 Primes p for which A049820(x) = p has no solution.

Original entry on oeis.org

7, 13, 19, 37, 43, 67, 79, 103, 109, 113, 131, 163, 167, 193, 229, 241, 251, 257, 271, 293, 307, 313, 353, 359, 379, 383, 397, 401, 439, 463, 479, 487, 491, 499, 503, 509, 563, 571, 647, 653, 661, 673, 701, 739, 743, 757, 761, 773, 823, 859, 863, 883, 887, 911, 937, 941, 953, 967, 971, 977, 983, 1009, 1093, 1103, 1109, 1171, 1181, 1193, 1217, 1279, 1283, 1291, 1297, 1307, 1321, 1361

Offset: 1

Antti Karttunen, Oct 11 2015

nonn

Primes p that there is no such k for which k - d(k) = p, where d(k) is the number of divisors of k (A000005).

Antti Karttunen, Table of n, a(n) for n = 1..15383

Complement among primes: A263090.
Intersection of A000040 and A045765.
Subsequence of A067774 (A049591).
Cf. A000005, A049820, A060990.

lim = 10000; s = Select[Complement[Range@ lim, Sort@ DeleteDuplicates@ Table[n - DivisorSigma[0, n], {n, lim}]], PrimeQ]; Take[s, 76] (* Michael De Vlieger, Oct 13 2015 *)

allocatemem(123456789);
uplim1 = 2162160 + 320; \\ = A002182(41) + A002183(41).
v060990 = vector(uplim1);
for(n=3, uplim1, v060990[n-numdiv(n)]++);
A060990 = n -> if(!n,2,v060990[n]);
n=0; forprime(p=2, 524287, if((0 == A060990(p)), n++; write("b263091.txt", n, " ", p)));

;; With Antti Karttunen's IntSeq-library.
(define A263091 (MATCHING-POS 1 1 (lambda (n) (and (= 1 (A010051 n)) (zero? (A060990 n))))))

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A049579 Numbers k such that prime(k)+2 divides (prime(k)-1)!.

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A210361 Prime numbers p such that x^2 + x + p produces primes for x = 0..2 but not x = 3.

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A210362 Prime numbers p such that x^2 + x + p produces primes for x = 0..3 but not x = 4.

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A210363 Prime numbers p such that x^2 + x + p produces primes for x = 0..4 but not x = 5.

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A210364 Prime numbers p such that x^2 + x + p produces primes for x = 0..5 but not x = 6.

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A210365 Prime numbers p such that x^2 + x + p produces primes for x = 0..6 but not x = 7.

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A373819 Run-lengths (differing by 0) of the run-lengths (differing by 2) of the odd primes.

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A373824 Sorted positions of first appearances in the run-lengths (differing by 0) of the run-lengths (differing by 2) of the odd primes.

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A107986 Composite numbers of the form p+2 where p is prime.

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