A049591 -id:A049591 - cp's OEIS frontend

A067774 Primes p such that p+2 is not a prime.

2, 7, 13, 19, 23, 31, 37, 43, 47, 53, 61, 67, 73, 79, 83, 89, 97, 103, 109, 113, 127, 131, 139, 151, 157, 163, 167, 173, 181, 193, 199, 211, 223, 229, 233, 241, 251, 257, 263, 271, 277, 283, 293, 307, 313, 317, 331, 337, 349, 353, 359, 367, 373, 379, 383, 389

Offset: 1

Benoit Cloitre, Feb 06 2002

Primes n such that n!*B(n+1) is an integer where B(k) are the Bernoulli numbers.

All primes except for the lower members of twin primes - i.e. remove 3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, ... - Gerard Schildberger, Feb 13 2005

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

Cf. A049591.

Complement of A001359 in A000040, A025584, A007510.

[p: p in PrimesUpTo(400) | NextPrime(p)-p ne 2]; // Bruno Berselli, Apr 09 2013

f[n_]:=PrimeQ[n+2]; lst={}; Do[p=Prime[n]; If[ !f[p], AppendTo[lst,p]], {n,5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 17 2009 *)

is(n)=isprime(n)&&!isprime(n+2) \\ Charles R Greathouse IV, Jul 01 2013

Except for a(1)=2, a(n+1)=A049591(n).

a(n) ~ n log n. - Charles R Greathouse IV, Jul 01 2013

Better description from Vladeta Jovovic, Dec 14 2002

A136798 First term in a sequence of at least 3 consecutive composite integers.

Original entry on oeis.org

8, 14, 20, 24, 32, 38, 44, 48, 54, 62, 68, 74, 80, 84, 90, 98, 104, 110, 114, 128, 132, 140, 152, 158, 164, 168, 174, 182, 194, 200, 212, 224, 230, 234, 242, 252, 258, 264, 272, 278, 284, 294, 308, 314, 318, 332, 338, 350, 354, 360, 368, 374, 380, 384, 390, 398

Offset: 1

Enoch Haga, Jan 21 2008

The meaning of "first" is that the run of composites is started with this term, that is, it is the one after a prime.
The number of terms in any run of composites is odd, because the difference between the relevant consecutive primes is even.
Composite numbers m such that m+1 is also composite, but m-1 is not. - Reinhard Zumkeller, Aug 04 2015

			a(1)=8 because 8 is the first term in a sequential run of 3 composites, 8,9,10

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Carlos Rivera, Puzzle 430, Grimm's Conjecture, Prime puzzles and problems connection.

Cf. A136799, A136800, A136801.
Cf. A049591, A010051.
a(n) = 2 * A104280(n).

import Data.List (elemIndices)
a136798 n = a136798_list !! (n-1)
a136798_list = tail $ map (+ 1) $ elemIndices 1 $
   zipWith (*) (0 : a010051_list) $ map (1 -) $ tail a010051_list
-- Reinhard Zumkeller, Aug 04 2015

Prime/@Flatten[Position[Differences[Prime[Range[80]]],?(#>2&)]]+1 (* _Harvey P. Dale, Jun 19 2013 *)

a(n) = A049591(n)+1. - R. J. Mathar, Jan 23 2008

A010051(a(n)-1) * (1 - A010051(a(n)) - A010051(a(n)+1)) = 1. - Reinhard Zumkeller, Aug 04 2015

Edited by R. J. Mathar, May 27 2009

A067775 Primes p such that p + 4 is composite.

Original entry on oeis.org

2, 5, 11, 17, 23, 29, 31, 41, 47, 53, 59, 61, 71, 73, 83, 89, 101, 107, 113, 131, 137, 139, 149, 151, 157, 167, 173, 179, 181, 191, 197, 199, 211, 227, 233, 239, 241, 251, 257, 263, 269, 271, 281, 283, 293, 311, 317, 331, 337, 347, 353, 359, 367, 373, 383, 389

Offset: 1

Benoit Cloitre, Feb 06 2002

nonn

Primes n such that n!*B(n+3) is an integer where B(k) are the Bernoulli numbers B(1) = -1/2, B(2) = 1/6, B(4) = -1/30, ..., B(2m+1) = 0 for m > 1.
If n is prime n!*B(n-1) is always an integer. Note that if Goldbach's conjecture (2n = p1 + p2 for all n >= 2) is false and K is the smallest value of n for which it fails, then for 2(K-2) = p3 + p4, the primes p3 and p4 must be taken from this list. See similar comment for A140555. - Keith Backman, Apr 06 2012
Complement of A023200 (primes p such that p + 4 is also prime) with respect to A000040 (primes). For p > 2: primes p such that there is no prime of the form r^2 + p where r is prime, subsequence of A232010. Example: the prime 7 is not in the sequence because 2^2 + 7 = 11 (prime). A232009(a(n)) = 0 for n > 1 . - Jaroslav Krizek, Nov 22 2013

Klaus Brockhaus, Table of n, a(n) for n = 1..1026
Eric Weisstein's World of Mathematics, Bernoulli Number

Cf. A049591, A140555, A232009, A232010, A232012, A023200.

A067775 = {}; Do[p = Prime@ n; If[ IntegerQ[ p! BernoulliB[p + 3]], AppendTo[A067775, p]], {n, 77}]; A067775 (* Robert G. Wilson v, Aug 19 2008 *)
Select[Prime[Range[80]], Not[PrimeQ[# + 4]] &] (* Alonso del Arte, Apr 02 2014 *)

lista(nn) = {forprime(p=1, nn, if (! isprime(p+4), print1(p, ", ")););} \\ Michel Marcus, Nov 22 2013

a(n) ~ n log n. - Charles R Greathouse IV, Nov 22 2013

New name from Klaus Brockhaus at the suggestion of Michel Marcus, Nov 22 2013

A105399 Largest prime <= numbers of the form 6k+3 (duplicates removed).

Original entry on oeis.org

3, 7, 13, 19, 23, 31, 37, 43, 47, 53, 61, 67, 73, 79, 83, 89, 97, 103, 109, 113, 127, 131, 139, 151, 157, 163, 167, 173, 181, 193, 199, 211, 223, 229, 233, 241, 251, 257, 263, 271, 277, 283, 293, 307, 313, 317, 331, 337, 349, 353, 359, 367, 373, 379, 383, 389

Offset: 1

Giovanni Teofilatto, May 01 2005

Apart from the initial 3, the same as A049591. [Proof from T. Khovanova, Jan 23 2008: True for primes up to 5 by inspection. Higher primes must be of the form 6k+1 or 6k+5 since 6k+2 and 6k+4 are divisible by 2 and 6k+3 is divisible by 3. So searching the prime p backwards from the composite, odd 6k+3 in steps of 2 implies that p+2, skipped during that scan, is composite. So p is not in A001359 but in A049591.] - R. J. Mathar, Jan 28 2008

			7 is in the sequence because 7 is the largest prime < 9=6*1+3.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A106002.

Cf. A049591.

pp[n_] := Block[{k = n},While[ ! PrimeQ[k], k-- ];k];Union[Table[pp[6n + 3], {n, 0, 65}]] (* Ray Chandler, Oct 17 2006 *)
Union[If[PrimeQ[#],#,NextPrime[#,-1]]&/@(6*Range[0,70]+3)] (* Harvey P. Dale, Aug 20 2021 *)

Edited, corrected and extended by Ray Chandler, Oct 17 2006

A297925 Even numbers k such that k - 5 is prime but k - 3 is not prime.

Original entry on oeis.org

12, 18, 24, 28, 36, 42, 48, 52, 58, 66, 72, 78, 84, 88, 94, 102, 108, 114, 118, 132, 136, 144, 156, 162, 168, 172, 178, 186, 198, 204, 216, 228, 234, 238, 246, 256, 262, 268, 276, 282, 288, 298, 312, 318, 322, 336, 342, 354, 358, 364, 372, 378, 384, 388, 394, 402, 406, 414, 426, 438, 444, 448, 454

Offset: 1

David James Sycamore, Jan 08 2018

nonn

Even numbers that are the sum of 5 and another prime, but not the sum of 3 and another prime. For n >= 1, a(n) - 5 = A049591(n), a(n) - 3 = A107986(n+1).
Let r(n) = a(n) - 5, Then r(n) is the greatest prime < a(n), and therefore A056240(a(n)) = 5*r(n). Furthermore, since r(n) + 2 must be composite, A056240(a(n)) = 5*A049591(n).
The terms in this sequence, combined with those in A298366 and A298252 form a partition of A005843(n);n>=3 (nonnegative even numbers>=6). This is because any even integer n>=6 satisfies either (i) n-3 is prime, (ii) n-5 is prime but n-3 is composite, or (iii) both n-5 and n-3 are composite.

			12 is a term because 12 - 5 = 7 is prime, and 12 - 3 = 9 is composite. Also A049591(1)+5=7+5=12 and A107986(2)+3=9+3=12.
18 is a term because 18 - 5 = 13 is prime, and 18 - 3 = 15 is composite.
16 is not a term because 16 - 5 = 11 and 16 - 3 = 13 are both prime.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Similar to A130038. Subsequence of A175222.

Cf. A049591, A107986.

Filtered([8..500], k-> IsPrime(k-5) and not IsPrime(k-3) and (k mod 2)=0); # G. C. Greubel, May 21 2019

[n: n in [3..500] | IsPrime(n-5) and not IsPrime(n-3) and (n mod 2) eq 0]; // G. C. Greubel, May 21 2019

N:=100
for n from 8 to N by 2 do
if isprime(n-5) and not isprime(n-3) then print (n);
end if
end do

Select[Range[6, 500, 2], And[PrimeQ[# - 5], ! PrimeQ[# - 3]] &] (* Michael De Vlieger, Jan 10 2018 *)
Select[Range[6, 500, 2], Boole[PrimeQ[# -{5, 3}]] == {1, 0} &] (* Harvey P. Dale, Jan 30 2024 *)

isok(n) = !(n % 2) && isprime(n-5) && !isprime(n-3); \\ Michel Marcus, Jan 09 2018

[n for n in (3..500) if is_prime(n-5) and not is_prime(n-3) and (mod(n, 2)==0)] # G. C. Greubel, May 21 2019

a(n) = A049591(n) + 5 = A107986(n+1) + 3 for all n >= 1.

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))))))

A069231 Numbers n such that there are exactly 3 primes p satisfying the inequality n < p < n + tau(n)^2 where tau(n) = A000005(n).

Original entry on oeis.org

4, 9, 21, 51, 55, 62, 74, 77, 82, 86, 87, 91, 106, 122, 123, 129, 134, 142, 143, 145, 146, 155, 158, 159, 161, 177, 183, 214, 215, 217, 237, 249, 254, 259, 265, 274, 278, 298, 299, 301, 309, 334, 335, 339, 341, 343, 358, 365, 371, 377, 382, 386, 394, 395, 407

Offset: 1

Benoit Cloitre, Apr 13 2002

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

Cf. A000005, A049591, A069230, A069232, A069233.

filter:= n -> nops(select(isprime, [$(n+1) .. (n+numtheory:-tau(n)^2-1)]))=3:
select(filter, [$1..1000]); # Robert Israel, Jan 05 2018

fQ[n_] := Block[{r = Range[n, n + DivisorSigma[0, n]^2]}, If[ PrimeQ@ n, r = Rest@ r]; If[ PrimeQ[ r[[-1]]], r = Most@ r]; Length@ Select[r, PrimeQ] == 3]; Select[Range@410, fQ] (* Robert G. Wilson v, Jan 05 2018 *)

isok(n) = #select(x->isprime(x), vector(numdiv(n)^2-1, k, k+n)) == 3; \\ Michel Marcus, Jun 18 2017

A069232 Numbers n such that there are exactly 2 primes p satisfying the inequality n < p < n + tau(n)^2 where tau(n) = A000005(n).

Original entry on oeis.org

2, 25, 85, 118, 119, 133, 141, 194, 209, 213, 235, 247, 253, 323, 326, 327, 329, 355, 362, 381, 391, 393, 398, 411, 413, 415, 422, 445, 466, 473, 481, 482, 493, 501, 502, 511, 514, 515, 517, 519, 533, 535, 537, 538, 542, 543, 545, 551, 553, 573, 579, 581, 583

Offset: 1

Benoit Cloitre, Apr 13 2002

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000005, A049591, A069230, A069231, A069233.

Select[Range[900],Length[Select[Range[#+1,#+DivisorSigma[0,#]^2-1],PrimeQ]] == 2&] (* Harvey P. Dale, Sep 20 2020 *)

isok(n) = #select(x->isprime(x), vector(numdiv(n)^2-1, k, k+n)) == 2; \\ Michel Marcus, Jun 18 2017

Missing term 2 added by Michel Marcus, Jun 18 2017

A069233 Numbers k such that there is exactly 1 prime p satisfying the inequality k < p < k + tau(k)^2 where tau(k) = A000005(k).

Original entry on oeis.org

3, 5, 11, 17, 29, 41, 49, 59, 71, 101, 107, 111, 115, 121, 137, 149, 169, 179, 191, 197, 201, 202, 203, 205, 206, 227, 239, 269, 281, 287, 289, 291, 295, 311, 314, 319, 321, 347, 361, 403, 419, 431, 461, 469, 471, 505, 521, 526, 527, 569, 599, 617, 622, 623

Offset: 1

Benoit Cloitre, Apr 13 2002

Numbers k such that A069230(k) = 1. - Amiram Eldar, Jan 29 2025

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

Cf. A000005, A049591, A069230, A069231, A069232.

q[k_] := Count[k + Range[1, DivisorSigma[0, k]^2 - 1], ?PrimeQ] == 1; Select[Range[640], q] (* _Amiram Eldar, Jan 29 2025 *)

isok(n) = #select(x->isprime(x), vector(numdiv(n)^2-1, k, k+n)) == 1; \\ Michel Marcus, Jun 18 2017

A297150 Let b(k) denote A292081(k); the sequence lists numbers b(2n) where for all m > n, b(2m) > b(2n).

Original entry on oeis.org

35, 65, 95, 115, 155, 185, 215, 235, 265, 305, 335, 365, 395, 415, 445, 485, 515, 545, 565, 635, 655, 695, 755, 785, 815, 835, 865, 905, 965, 995, 1055, 1115, 1145, 1165, 1205, 1255, 1285, 1315, 1355, 1385, 1415, 1465, 1535, 1565, 1585, 1655, 1685, 1745, 1765, 1795, 1835, 1865, 1895, 1915, 1945, 1985

Offset: 1

David James Sycamore, Dec 26 2017

nonn

This is also an ascending subsequence of the even-indexed terms of A056240(2n) (of which A292081 is a subsequence). For n >= 1, a(n) is a semiprime of the form a(n)=5*A049591(n), and the index m in A056240 of any term in this sequence belongs to the sequence of even numbers m such that m-5 is prime and m-3 is not prime (A297925). See A297925 for explanation.

			a(1)=5*A049591(1)=5*7=35. Also A056240(A297925(1))=A056240(12)=35.
a(17)=5*A049591(17)=5*103=515. Also A056240(A297925(17))=A056240(108)=515.

Vincenzo Librandi, Table of n, a(n) for n = 1..1690

Similar to A288313.

Cf. A292081, A288313, A056240, A049591, A297925.

[5*p: p in PrimesInInterval(3, 500) | not IsPrime(p + 2)]; // Vincenzo Librandi, Nov 12 2018

5 Select[Prime[Range[3, 100]], ! PrimeQ[(# + 2)] &] (* Vincenzo Librandi, Nov 12 2018 *)

a(n) = 5*A049591(n) = A056240(A297925(n)).

cp's OEIS Frontend

A067774 Primes p such that p+2 is not a prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A136798 First term in a sequence of at least 3 consecutive composite integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

Extensions

A067775 Primes p such that p + 4 is composite.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A105399 Largest prime <= numbers of the form 6k+3 (duplicates removed).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A297925 Even numbers k such that k - 5 is prime but k - 3 is not prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Scheme

A069231 Numbers n such that there are exactly 3 primes p satisfying the inequality n < p < n + tau(n)^2 where tau(n) = A000005(n).

Views

Author