A006512 -id:A006512 - cp's OEIS frontend

A029707 Numbers n such that the n-th and the (n+1)-st primes are twin primes.

2, 3, 5, 7, 10, 13, 17, 20, 26, 28, 33, 35, 41, 43, 45, 49, 52, 57, 60, 64, 69, 81, 83, 89, 98, 104, 109, 113, 116, 120, 140, 142, 144, 148, 152, 171, 173, 176, 178, 182, 190, 201, 206, 209, 212, 215, 225, 230, 234, 236, 253, 256, 262, 265, 268, 277

Offset: 1

N. J. A. Sloane, Dec 11 1999

nonn

Numbers m such that prime(m)^2 == 1 mod (prime(m) + prime(m + 1)). - Zak Seidov, Sep 18 2013

First differences are A027833. The complement is A049579. - Gus Wiseman, Dec 03 2024

Robert G. Wilson v, Table of n, a(n) for n = 1..86027
Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

Cf. A014574, A027833 (first differences), A007508. Equals PrimePi(A001359) (cf. A000720).
The complement is A049579, first differences A251092 except first term.
Lengths of runs of terms differing by 2 are A179067.
The first differences have run-lengths A373820 except first term.
A000040 lists the primes, differences A001223 (run-lengths A333254, A373821).
A038664 finds the first prime gap of 2n.
A046933 counts composite numbers between primes.
For prime runs: A005381, A006512, A025584, A067774.
Cf. A006560, A006562, A037201, A107770, A122535, A155752, A175632, A176246, A373819.

A029707 := proc(n)
    numtheory[pi](A001359(n)) ;
end proc:
seq(A029707(n),n=1..30); # R. J. Mathar, Feb 19 2017

Select[ Range@300, PrimeQ[ Prime@# + 2] &] (* Robert G. Wilson v, Mar 11 2007 *)
Flatten[Position[Flatten[Differences/@Partition[Prime[Range[100]],2,1]], 2]](* Harvey P. Dale, Jun 05 2014 *)

def A029707(n) :
   a = [ ]
   for i in (1..n) :
      if (nth_prime(i+1)-nth_prime(i) == 2) :
         a.append(i)
   return(a)
A029707(277) # Jani Melik, May 15 2014

a(n) = A107770(n) - 1. - Juri-Stepan Gerasimov, Dec 16 2009

A037074 Numbers that are the product of a pair of twin primes.

Original entry on oeis.org

15, 35, 143, 323, 899, 1763, 3599, 5183, 10403, 11663, 19043, 22499, 32399, 36863, 39203, 51983, 57599, 72899, 79523, 97343, 121103, 176399, 186623, 213443, 272483, 324899, 359999, 381923, 412163, 435599, 656099, 675683, 685583, 736163

Offset: 1

Felice Russo

Each entry is the product of p and p+2 where both p and p+2 are prime, i.e., the product of the lesser and greater of a twin prime pair.
Except for the first term, all entries have digital root 8. - Lekraj Beedassy, Jun 11 2004
The above statement follows from p > 3 => (p,p+2) = (6k-1,6k+1) => p*(p+2) = 36k^2 - 1 == 8 (mod 9), and A010888 === A010878 (mod 9). - M. F. Hasler, Jan 11 2013
Albert A. Mullin states that m is a product of twin primes iff phi(m)*sigma(m) = (m-3)*(m+1), where phi(m) = A000010(m) and sigma(m) = A000203(m). Of course, for a product of distinct primes p*q we know sigma(p*q) = (p+1)*(q+1) and if p, q, are twin primes, say q = p + 2, then sigma(p*q) = (p+1)*(q+1) = (p+1)*(p+3). - Jonathan Vos Post, Feb 21 2006
Also the area of twin prime rectangles. A twin prime rectangle is a rectangle whose sides are components of twin prime pairs. E.g., the twin prime pair (3,5) produces a 3 X 5 unit rectangle which has area 15 square units. - Cino Hilliard, Jul 28 2006
Except for 15, a product of twin primes is of the form 36k^2 - 1 (cf. A136017, A002822). - Artur Jasinski, Dec 12 2007
A072965(a(n)) = 1; A072965(m) mod A037074(n) > 0 for all m. - Reinhard Zumkeller, Jan 29 2008
The number of terms less than 10^(2n) is A007508(n). - Robert G. Wilson v, Feb 08 2012
If m is the product of twin primes, then sigma(m) = m + 1 + 2*sqrt(m + 1), phi(m) = m + 1 - 2*sqrt(m + 1). pmin(m) = sqrt(m + 1) - 1, pmax(m) = sqrt(m + 1) + 1. - Wesley Ivan Hurt, Jan 06 2013
Semiprimes of the form 4*k^2 - 1. - Vincenzo Librandi, Apr 13 2013

			a(2)=35 because 5*7=35, that is (5,7) is the 2nd pair of twin primes.

Albert A. Mullin, "Bicomposites, twin primes and arithmetic progression", Abstract 04T-11-48, Abstracts of AMS, Vol. 25, No. 4, 2004, p. 795.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A001359, A006512, A014574, A136017, A074480 (multiplicative closure), A209328.
Cf. A071700 (subsequence).
Cf. A075369.

a037074 = subtract 1 . a075369  -- Reinhard Zumkeller, Feb 10 2015
-- Reinhard Zumkeller, Feb 10 2015, Aug 14 2011

[p*(p+2): p in PrimesUpTo(1000) | IsPrime(p+2)];  // Bruno Berselli, Jul 08 2011

IsSemiprime:=func; [s: n in [1..500] | IsSemiprime(s) where s is 4*n^2-1]; // Vincenzo Librandi, Apr 13 2013

ZL:=[]: for p from 1 to 863 do if (isprime(p) and isprime(p+2) ) then ZL:=[op(ZL),(p*(p+2))]; fi; od; print(ZL); # Zerinvary Lajos, Mar 07 2007
for i from 1 to 150 do if ithprime(i+1) = ithprime(i) + 2 then print({ithprime(i)*ithprime(i+1)}); fi; od; # Zerinvary Lajos, Mar 19 2007

s = Select[ Prime@ Range@170, PrimeQ[ # + 2] &]; s(s + 2) (* Robert G. Wilson v, Feb 21 2006 *)
(* For checking large numbers, the following code is better. For instance, we could use the fQ function to determine that 229031718473564142083 is in this sequence. *) fQ[n_] := Block[{fi = FactorInteger[n]}, Last@# & /@ fi == {1, 1} && Differences[ First@# & /@ fi] == {2}]; Select[ Range[750000], fQ] (* Robert G. Wilson v, Feb 08 2012 *)
Times@@@Select[Partition[Prime[Range[500]],2,1],Last[#]-First[#]==2&] (* Harvey P. Dale, Oct 16 2012 *)

g(n) = for(x=1,n,if(prime(x+1)-prime(x)==2,print1(prime(x)*prime(x+1)","))) \\ Cino Hilliard, Jul 28 2006

a(n) = A001359(n)*A006512(n). A000010(a(n))*A000203(a(n)) = (a(n)-3)*(a(n)+1). - Jonathan Vos Post, Feb 21 2006
a(n) = (A014574(n))^2 - 1. a(n+1) = (6*A002822(n))^2 - 1. - Lekraj Beedassy, Sep 02 2006
a(n) = A075369(n) - 1. - Reinhard Zumkeller, Feb 10 2015
Sum_{n>=1} 1/a(n) = A209328. - Amiram Eldar, Nov 20 2020
A000010(a(n)) == 0 (mod 8). - Darío Clavijo, Oct 26 2022

More terms from Erich Friedman

A265759 Numerators of primes-only best approximates (POBAs) to 1; see Comments.

Original entry on oeis.org

3, 2, 5, 13, 11, 19, 17, 31, 29, 43, 41, 61, 59, 73, 71, 103, 101, 109, 107, 139, 137, 151, 149, 181, 179, 193, 191, 199, 197, 229, 227, 241, 239, 271, 269, 283, 281, 313, 311, 349, 347, 421, 419, 433, 431, 463, 461, 523, 521, 571, 569, 601, 599, 619, 617

Offset: 1

Clark Kimberling, Dec 15 2015

Suppose that x > 0. A fraction p/q of primes is a primes-only best approximate (POBA), and we write "p/q in B(x)", if 0 < |x - p/q| < |x - u/v| for all primes u and v such that v < q. Note that for some choices of x, there are values of q for which there are two POBAs. In these cases, the greater is placed first; e.g., B(3) = (7/2, 5/2, 17/5, 13/5, 23/7, 19/7, ...).
See A265772 and A265774 for definitions of lower POBA and upper POBA. In the following guide, for example, A001359/A006512 represents (conjecturally in some cases) the Lower POBAs p(n)/q(n) to 1, where p = A001359 and q = A006512 except for first terms in some cases. Every POBA is either a lower POBA or an upper POBA.
x        Lower POBA       Upper POBA       POBA
1        A001359/A006512  A006512/A001359  A265759/A265760
3/2      A104163/A158708  A162336/A158709  A265761/A222565
2        A005383/A005382  A005385/A005384  A079149/A120628
3        A091180/A088878  A094525/A023208  A265763/A265764
4        A162857/A062737  A090866/A023212  A265765/A120639
5        A265766/A158318  A265767/A023217  A265768/A265769
6        A227756/A158015  A051644/A007693  A265770/A265771
sqrt(2)  A265772/A265773  A265774/A265775  A265776/A265777
sqrt(3)  A265778/A265779  A265780/A265781  A265782/A265783
sqrt(5)  A265784/A265785  A265786/A265787  A265788/A265789
sqrt(8)  A265790/A265791  A265792/A265793  A265794/A265795
tau      A265796/A265797  A265798/A265799  A265800/A265801
1/tau    A265799/A265798  A265797/A265796  A265806/A265807
pi       A265808/A265809  A265810/A265811  A265812/A265813
e        A265814/A265815  A265816/A265817  A265818/A265819

			The POBAs for 1 start with 3/2, 2/3, 5/7, 13/11, 11/13, 19/17, 17/19, 31/29, 29/31, 43/41, 41/43, 61/59, 59/61. For example, if p and q are primes and q > 13, then 11/13 is closer to 1 than p/q is.

Cf. A000040, A001359, A006512, A265759, A265760.

x = 1; z = 200; p[k_] := p[k] = Prime[k];
t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265759/A265760 *)
Numerator[tL]   (* A001359 *)
Denominator[tL] (* A006512 *)
Numerator[tU]   (* A006512 *)
Denominator[tU] (* A001359 *)
Numerator[y]    (* A265759 *)
Denominator[y]  (* A265760 *)

A014092 Numbers that are not the sum of 2 primes.

Original entry on oeis.org

1, 2, 3, 11, 17, 23, 27, 29, 35, 37, 41, 47, 51, 53, 57, 59, 65, 67, 71, 77, 79, 83, 87, 89, 93, 95, 97, 101, 107, 113, 117, 119, 121, 123, 125, 127, 131, 135, 137, 143, 145, 147, 149, 155, 157, 161, 163, 167, 171, 173, 177, 179, 185, 187, 189, 191, 197, 203, 205, 207, 209

Offset: 1

N. J. A. Sloane

Suggested by the Goldbach conjecture that every even number larger than 2 is the sum of 2 primes.
Since (if we believe the Goldbach conjecture) all the entries > 2 in this sequence are odd, they are equal to 2 + an odd composite number (or 1).
Otherwise said, the sequence consists of 2 and odd numbers k such that k-2 is not prime. In particular there is no element from A006512, greater of a twin prime pair. - M. F. Hasler, Sep 18 2012
Values of k such that A061358(k) = 0. - Emeric Deutsch, Apr 03 2006
Values of k such that A073610(k) = 0. - Graeme McRae, Jul 18 2006

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Section 2.8 (for Goldbach conjecture).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Goldbach conjecture

Cf. A002372, A002373, A002374, A048974, A061358.
Cf. A010051, A000040, A051035 (composites).
Equivalent sequence for prime powers: A071331.
Numbers that can be expressed as the sum of two primes in k ways for k=0..10: this sequence (k=0), A067187 (k=1), A067188 (k=2), A067189 (k=3), A067190 (k=4), A067191 (k=5), A066722 (k=6), A352229 (k=7), A352230 (k=8), A352231 (k=9), A352233 (k=10).

a014092 n = a014092_list !! (n-1)
a014092_list = filter (\x ->
   all ((== 0) . a010051) $ map (x -) $ takeWhile (< x) a000040_list) [1..]
-- Reinhard Zumkeller, Sep 28 2011

g:=sum(sum(x^(ithprime(i)+ithprime(j)),i=1..j),j=1..50): gser:=series(g,x=0,230): a:=proc(n) if coeff(gser,x^n)=0 then n else fi end: seq(a(n),n=1..225); # Emeric Deutsch, Apr 03 2006

s1falsifiziertQ[s_]:= Module[{ip=IntegerPartitions[s, {2}], widerlegt=False},Do[If[PrimeQ[ip[[i,1]] ] ~And~ PrimeQ[ip[[i,2]] ], widerlegt = True; Break[]],{i,1,Length[ip]}];widerlegt]; Select[Range[250],s1falsifiziertQ[ # ]==False&] (* Michael Taktikos, Dec 30 2007 *)
Join[{1,2},Select[Range[3,300,2],!PrimeQ[#-2]&]] (* Zak Seidov, Nov 27 2010 *)
Select[Range[250],Count[IntegerPartitions[#,{2}],?(AllTrue[#,PrimeQ]&)]==0&] (* _Harvey P. Dale, Jun 08 2022 *)

isA014092(n)=local(p,i) ; i=1 ; p=prime(i); while(pA014092(a), print(n," ",a); n++)) \\ R. J. Mathar, Aug 20 2006

from sympy import prime, isprime
def ok(n):
    i=1
    x=prime(i)
    while xIndranil Ghosh, Apr 29 2017

Odd composite numbers + 2 (essentially A014076(n) + 2 ).

Equals {2} union A005408 \ A052147, i.e., essentially the complement of A052147 (or rather A048974) within the odd numbers A005408. - M. F. Hasler, Sep 18 2012

A027833 Distances between successive 2's in sequence A001223 of differences between consecutive primes.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 3, 6, 2, 5, 2, 6, 2, 2, 4, 3, 5, 3, 4, 5, 12, 2, 6, 9, 6, 5, 4, 3, 4, 20, 2, 2, 4, 4, 19, 2, 3, 2, 4, 8, 11, 5, 3, 3, 3, 10, 5, 4, 2, 17, 3, 6, 3, 3, 9, 9, 2, 6, 2, 6, 5, 6, 2, 3, 2, 3, 9, 4, 7, 3, 7, 20, 4, 7, 6, 5, 3, 7, 3, 20, 2, 14, 4, 10, 2, 3, 6, 4, 2, 2, 7, 2, 6, 3

Offset: 1

Jean-Marc MALASOMA (Malasoma(AT)entpe.fr)

nonn

a(n) = number of primes p such that A014574(n) < p < A014574(n+1). - Thomas Ordowski, Jul 20 2012
Conjecture: a(n) < log(A014574(n))^2. - Thomas Ordowski, Jul 21 2012
Conjecture: All positive integers are represented in this sequence. This is verified up to 184, by searching up to prime indexes of ~128000000. The rate of filling-in the smallest remaining gap among the integers, and the growth in the maximum value found, both slow down considerably relative to a fixed quantity of twin prime incidences examined in each pass. The maximum value found was 237. - Richard R. Forberg, Jul 28 2016
All positive integers below 312 are in this sequence. - Charles R Greathouse IV, Aug 01 2016
From Gus Wiseman, Jun 11 2024: (Start)
Also the length of the n-th maximal antirun of prime numbers > 3, where an antirun is an interval of positions at which consecutive terms differ by more than 2. These begin:
   5
   7  11
  13  17
  19  23  29
  31  37  41
  43  47  53  59
  61  67  71
  73  79  83  89  97 101
(End)

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

First differences of A029707 and A155752 = A029707 - 1. M. F. Hasler, Jul 24 2012
Positions of first appearances are A373401, sorted A373402.
Functional neighbors: A001359, A006512, A251092 or A175632, A373127 (firsts A373128, sorted A373200), A373403, A373405, A373409.
A000040 lists the primes, differences A001223.
A002808 lists the composite numbers, differences A073783.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
Cf. A025157, A038664, A071148,  A176246, A350842, A373406.

A027833 := proc(n)
    local plow,phigh ;
    phigh := A001359(n+1) ;
    plow := A001359(n) ;
    numtheory[pi](phigh)-numtheory[pi](plow) ;
end proc:
seq(A027833(n),n=1..100) ; # R. J. Mathar, Jan 20 2025

Differences[Flatten[Position[Differences[Prime[Range[500]]],2]]] (* Harvey P. Dale, Nov 17 2018 *)
Length/@Split[Select[Range[4,10000],PrimeQ[#]&],#1+2!=#2&]//Most (* Gus Wiseman, Jun 11 2024 *)

n=1; p=5; forprime(q=7,1e3, if(q-p==2, print1(n", "); n=1, n++); p=q) \\ Charles R Greathouse IV, Aug 01 2016

def A027833(n) :
   a = [ ]
   st = 2
   for i in (3..n) :
      if (nth_prime(i+1)-nth_prime(i) == 2) :
         a.append(i-st)
         st = i
   return(a)
A027833(496) # Jani Melik, May 15 2014

A040040 Average of twin prime pairs (A014574), divided by 2. Equivalently, 2a(n)-1 and 2a(n)+1 are primes.

Original entry on oeis.org

2, 3, 6, 9, 15, 21, 30, 36, 51, 54, 69, 75, 90, 96, 99, 114, 120, 135, 141, 156, 174, 210, 216, 231, 261, 285, 300, 309, 321, 330, 405, 411, 414, 429, 441, 510, 516, 525, 531, 546, 576, 615, 639, 645, 651, 660, 714, 726, 741, 744, 804, 810, 834, 849, 861, 894

Offset: 1

N. J. A. Sloane

Intersection of A005097 and A006254. - Zak Seidov, Mar 18 2005
The only possible pairs for 2a(n)+-1 are prime/prime (this sequence), not prime/not prime (A104278), prime/notprime (A104279) and not prime/prime (A104280), ... this sequence + A104280 + A104279 + A104278 = the odd numbers.
These numbers are never k mod (2k+1) or (k+1) mod (2k+1) with 2k+1 < a(n). - Jon Perry, Sep 04 2012
Excluding the first term, all remaining terms have digital root 3, 6 or 9. - J. W. Helkenberg, Jul 24 2013
Positive numbers x such that the difference between x^2 and adjacent squares are prime (both x^2-(x-1)^2 and (x+1)^2-x^2 are prime). - Doug Bell, Aug 21 2015

T. D. Noe, Table of n, a(n) for n = 1..10001

Cf. A001359, A006512, A014574, A054735, A111046, A045753 (even terms halved), A002822 (terms divided by 3).
Cf. A221310.
Cf. A260689, A264526.

a040040 = flip div 2 . a014574  -- Reinhard Zumkeller, Nov 17 2015

P := select(isprime,[$1..1789]): map(p->(p+1)/2, select(p->member(p+2,P),P)); # Peter Luschny, Mar 03 2011

Select[Range[900], And @@ PrimeQ[{-1, 1} + 2# ] &] (* Ray Chandler, Oct 12 2005 *)

p=2; forprime(b=3, 1e4, if(b-p==2, print1((p+1)/2", ")); p=b) \\ Altug Alkan, Nov 10 2015

a(n) = A014574(n)/2 = A054735(n+1)/4 = A111046(n+1)/8.
For n > 1, a(n) = 3*A002822(n-1). - Jason Kimberley, Nov 06 2015
A260689(a(n),1) = A264526(a(n)) = 1. - Reinhard Zumkeller, Nov 17 2015
From Michael G. Kaarhus, Aug 19 2022: (Start)
a(n) = (A001359(n) + 1)/2.
a(n) = (A006512(n) - 1)/2.
For n > 1, a(n) = A167379(n-1) * 3/2. (End)

More terms from Cino Hilliard, Oct 21 2002
Title corrected by Daniel Forgues, Jun 01 2009
Edited by Daniel Forgues, Jun 21 2009
Comment corrected by Daniel Forgues, Jul 12 2009

A054735 Sums of twin prime pairs.

Original entry on oeis.org

8, 12, 24, 36, 60, 84, 120, 144, 204, 216, 276, 300, 360, 384, 396, 456, 480, 540, 564, 624, 696, 840, 864, 924, 1044, 1140, 1200, 1236, 1284, 1320, 1620, 1644, 1656, 1716, 1764, 2040, 2064, 2100, 2124, 2184, 2304, 2460, 2556, 2580, 2604, 2640, 2856, 2904

Offset: 1

Enoch Haga, Apr 22 2000

(p^q)+(q^p) calculated modulo pq, where (p,q) is the n-th twin prime pair. Example: (599^601)+(601^599) == 1200 mod (599*601). - Sam Alexander, Nov 14 2003
El'hakk makes the following claim (without any proof): (q^p)+(p^q) = 2*cosh(q arctanh( sqrt( 1-((2/p)^2) ) )) + 2cosh(p arctanh( sqrt( 1-((2/q)^2) ) )) mod p*q. - Sam Alexander, Nov 14 2003
Also: Numbers N such that N/2-1 and N/2+1 both are prime. - M. F. Hasler, Jan 03 2013
Excluding the first term, all remaining terms have digital root 3, 6 or 9. - J. W. Helkenberg, Jul 24 2013
Except for the first term, this sequence is a subsequence of A005101 (Abundant numbers) and of A008594 (Multiples of 12). - Ivan N. Ianakiev, Jul 04 2021

			a(3) = 24 because the twin primes 11 and 13 add to 24.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 144.
El'hakk, Page of the time traveler [Archived copy on web.archive.org, as of Oct 28 2009.]

Cf. A001359, A006512, A014574, A040040, A111046.

a054735 = (+ 2) . (* 2) . a001359  -- Reinhard Zumkeller, Feb 10 2015

ZL:=[]:for p from 1 to 1451 do if (isprime(p) and isprime(p+2)) then ZL:=[op(ZL),p+(p+2)]; fi; od; print(ZL); # Zerinvary Lajos, Mar 07 2007
A054735 := proc(n)
2*A001359(n)+2;
end proc: # R. J. Mathar, Jan 06 2013

Select[Table[Prime[n] + 1, {n, 230}], PrimeQ[ # + 1] &] *2 (* Ray Chandler, Oct 12 2005 *)
Total/@Select[Partition[Prime[Range[300]],2,1],#[[2]]-#[[1]]==2&] (* Harvey P. Dale, Oct 23 2022 *)

is_A054735(n)={!bittest(n,0)&&isprime(n\2-1)&&isprime(n\2+1)} \\ M. F. Hasler, Jan 03 2013

pp=1;forprime(p=1,1482, if( p==pp+2, print1(p+pp,", ")); pp=p) \\ Following a suggestion by R. J. Cano, Jan 05 2013

a(n) = 2*A014574(n) = 4*A040040(n) = A111046(n)/2.

a(n) = 12*A002822(n-1) for all n > 1. - M. F. Hasler, Dec 12 2019

Additional comments from Ray Chandler, Nov 16 2003

Broken link fixed by M. F. Hasler, Jan 03 2013

A025584 Primes p such that p-2 is not a prime.

Original entry on oeis.org

2, 3, 11, 17, 23, 29, 37, 41, 47, 53, 59, 67, 71, 79, 83, 89, 97, 101, 107, 113, 127, 131, 137, 149, 157, 163, 167, 173, 179, 191, 197, 211, 223, 227, 233, 239, 251, 257, 263, 269, 277, 281, 293, 307, 311, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 389, 397, 401

Offset: 1

N. J. A. Sloane

Primes p (for p>=11) such that sigma(p-2) > p. - Benoit Cloitre, Feb 08 2002

Prime numbers that are not the sum of two smaller prime numbers. - Tomas Xordan, May 10 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Complement of A006512 in A000040.

Cf. A014092.

[p: p in PrimesUpTo(420) | not IsPrime(p-2)]; // Vincenzo Librandi, Jul 26 2013

Select[ Prime /@ Range[100], ! PrimeQ[# - 2] &] (* Jean-François Alcover, Apr 19 2013 *)

lista(nn) = forprime(p=2, nn, if (!isprime(p-2), print1(p, ", "));); \\ Michel Marcus, Dec 05 2015

a(n) ~ n log n. - Charles R Greathouse IV, Jan 10 2013

Better description from Vladeta Jovovic, Dec 14 2002

A052147 a(n) = prime(n) + 2.

Original entry on oeis.org

4, 5, 7, 9, 13, 15, 19, 21, 25, 31, 33, 39, 43, 45, 49, 55, 61, 63, 69, 73, 75, 81, 85, 91, 99, 103, 105, 109, 111, 115, 129, 133, 139, 141, 151, 153, 159, 165, 169, 175, 181, 183, 193, 195, 199, 201, 213, 225, 229, 231, 235, 241, 243, 253, 259

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk), Jan 24 2000

A048974, A052147, A067187 and A088685 are very similar after dropping terms less than 13. - Eric W. Weisstein, Oct 10 2003
A117530(n,2) = a(n) for n>1. - Reinhard Zumkeller, Mar 26 2006
a(n) = A000040(n) + 2 = A008864(n) + 1 = A113395(n) - 1 = A175221(n) - 2 = A175222(n) - 3 = A139049(n) - 4 = A175223(n) - 5 = A175224(n) - 6 = A140353(n) - 7 = A175225(n) - 8. - Jaroslav Krizek, Mar 06 2010
Left edge of the triangle in A065342. - Reinhard Zumkeller, Jan 30 2012
Union of A006512 and A107986. - David James Sycamore, Jul 08 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

A139690 is a subsequence.

Cf. A040976, A000040.

Filtered([1..300], k-> IsPrime(k) ) + 2; # G. C. Greubel, May 20 2019

a052147 = (+ 2) . a000040  -- Reinhard Zumkeller, Jul 03 2015

[p+2: p in PrimesUpTo(400)]; // Vincenzo Librandi, Nov 27 2013

seq(ithprime(n)+2,n=1..55); # Muniru A Asiru, Jul 08 2018

Prime[Range[70]]+2 (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

a(n)=prime(n)+2 \\ Charles R Greathouse IV, Jan 19 2017

[nth_prime(n) +2 for n in (1..70)] # G. C. Greubel, May 20 2019

A242719 Smallest even k such that lpf(k-3) > lpf(k-1) >= prime(n), where lpf=least prime factor (A020639).

Original entry on oeis.org

10, 26, 50, 170, 170, 362, 362, 842, 842, 1370, 1370, 1850, 1850, 2210, 3722, 3722, 3722, 4892, 5042, 7082, 7922, 7922, 7922, 10610, 10610, 10610, 11450, 13844, 16130, 16130, 17162, 19322, 19322, 24614, 24614, 25592, 29504, 29930, 29930, 36020, 36020

Offset: 2

Vladimir Shevelev, May 21 2014

nonn

The sequence is connected with a sufficient condition for the existence of an infinity of twin primes. In contrast to A242489, this sequence is nondecreasing.
All even numbers of the form A062326(n)^2 + 1 are in the sequence. All a(n)-1 are semiprimes. - Vladimir Shevelev, May 24 2014
a(n) <= A242489(n); a(n) >= prime(n)^2+1. Conjecture: a(n) <= prime(n)^4. - Vladimir Shevelev, Jun 01 2014
Conjecture: all numbers a(n)-3 are primes. Peter J. C. Moses verified this conjecture up to a(2001) (cf. with conjecture in A242720). - Vladimir Shevelev, Jun 09 2014

Jinyuan Wang, Table of n, a(n) for n = 2..5000 (terms 2..2001 from Peter J. C. Moses).

Cf. A001359, A006512, A062326, A137291, A242489, A242490, A242847, A243960, A245363, A246501, A246748, A246819, A247011.

lpf[k_] := FactorInteger[k][[1, 1]];
a[n_] := a[n] = For[k = If[n == 2, 10, a[n-1]], True, k = k+2, If[lpf[k-3] > lpf[k-1] >= Prime[n], Return[k]]];
Array[a, 50, 2] (* Jean-François Alcover, Nov 06 2018 *)

lpf(k) = factorint(k)[1,1];
vector(50, n, k=6; while(lpf(k-3)<=lpf(k-1) || lpf(k-1)Colin Barker, Jun 01 2014

Conjecturally, a(n) ~ (prime(n))^2, as n goes to infinity (cf. A246748, A246819). - Vladimir Shevelev, Sep 02 2014

a(n) = prime(n)^2 + 1 for and only for numbers n>=2 which are in A137291. - Vladimir Shevelev, Sep 04 2014

cp's OEIS Frontend

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A014092 Numbers that are not the sum of 2 primes.

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A027833 Distances between successive 2's in sequence A001223 of differences between consecutive primes.

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A040040 Average of twin prime pairs (A014574), divided by 2. Equivalently, 2*a(n)-1 and 2*a(n)+1 are primes.

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A040040 Average of twin prime pairs (A014574), divided by 2. Equivalently, 2a(n)-1 and 2a(n)+1 are primes.