A118071 -id:A118071 - cp's OEIS frontend

A038869 Primes p such that both p-2 and 2p-1 are prime.

7, 19, 31, 139, 199, 229, 271, 601, 619, 661, 811, 829, 1279, 1429, 1609, 2029, 2089, 2131, 2311, 2551, 2791, 3169, 3331, 3391, 3529, 3769, 4051, 4159, 4231, 4261, 4339, 4639, 4801, 5419, 5479, 5659, 5851, 6271, 6301, 6361, 6691, 6961, 7561, 7951, 8539

Offset: 1

Thomas Kellar

nonn

Primes p such that A(2*p) - 3*A(p) = 3 (7, 31, 661, 811, 2551, ...) and primes p such that 7*A(p) - A(2*p) = 21 (19, 139, 619, 1429, ...), where A=A288814, are both subsequences of A038869. - David James Sycamore, Aug 07 2017

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A005382, A118071.

[n: n in [0..10000]|IsPrime(n) and IsPrime(n-2) and IsPrime(2*n-1)]; // Vincenzo Librandi, Dec 18 2010

Transpose[Select[Partition[Prime[Range[1200]],2,1],#[[2]]-#[[1]]==2 && PrimeQ[2#[[2]]-1]&]][[2]] (* Harvey P. Dale, Jun 19 2014 *)

is(n)=n%6==1 && isprime(n-2) && isprime(n) && isprime(2*n-1) \\ Charles R Greathouse IV, Aug 09 2017

A157995 Primes which are the sum of 1 plus two consecutive not-twin primes, p1 and p2, (p2-p1)>2.

Original entry on oeis.org

19, 31, 43, 53, 79, 101, 113, 139, 163, 173, 199, 211, 223, 241, 269, 331, 353, 373, 463, 509, 521, 577, 601, 619, 631, 727, 773, 787, 811, 829, 853, 883, 907, 919, 947, 967, 991, 1013, 1031, 1181, 1193, 1231, 1291, 1301, 1361, 1429, 1447, 1483, 1531, 1543

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 11 2009

			19=7+11+1, 31=13+17+1, 43=19+23+1, 53=23+29+1, 79=37+41+1, 101=47+53+1, ...

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

Cf. A005385, A092738, A118071.

count:= 0: R:= NULL: p:= 2:
while count < 100 do
  q:= p; p:= nextprime(p);
  if p-q > 2 and isprime(p+q+1) then
     count:= count+1; R:= R, p+q+1
  fi
od:
R; # Robert Israel, May 13 2020

lst={};Do[p0=Prime[n];p1=Prime[n+1];a=p0+p1+1;If[PrimeQ[a]&&(p1-p0)>2,AppendTo[lst,a]],{n,6!}];lst
Select[Total[#]+1&/@Select[Partition[Prime[Range[200]],2,1],Last[#]-First[#]>2&],PrimeQ]  (* Harvey P. Dale, Mar 13 2011 *)

Definition corrected by Harvey P. Dale, Mar 13 2011

A157996 Primes which are sum of 1 and two nonconsecutive primes p1 and p2, p2 - p1 > 2.

Original entry on oeis.org

11, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 11 2009

nonn

Conjecture: for n > 1, a(n) = prime(n+5). - Charles R Greathouse IV, Mar 12 2012

A185154(n) is the smallest prime q, such that A049084(q) + 1 < A049084(a(n) - q - 1). - Reinhard Zumkeller, Mar 12 2012

			11=3+7+1, 17=5+11+1, 19=5+13+1, ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A076805, A005385, A092738, A118071, A157995, A065091, A006093.

a157996 n = a157996_list !! (n-1)
a157996_list = map (+ 1) $ filter f a006093_list where
   f x = g $ takeWhile (< x) a065091_list where
     g []  = False
     g [_] = False
     g (p:ps@(_:qs)) = (x - p) `elem` qs || g ps
-- Reinhard Zumkeller, Mar 12 2012

lst={};Do[p0=Prime[n];Do[px=Prime[n+k];If[PrimeQ[a=p0+px+1],AppendTo[lst,a]],{k,2,2*5!}],{n,6!}];Take[Union[lst],222]

is(n)=if(!isprime(n),return(0)); my(p=3,q=5); forprime(r=7,n-4, if(isprime(n-1-r) && n-1-r <= p, return(1)); p=q; q=r); 0 \\ Charles R Greathouse IV, Nov 05 2015

A269663 Semiprimes which are the product of a twin prime pair minus one.

Original entry on oeis.org

14, 34, 142, 898, 1762, 5182, 19042, 79522, 213442, 359998, 412162, 685582, 777922, 1192462, 1695202, 2585662, 4536898, 5143822, 5673922, 7225342, 12446782, 12659362, 12830722, 17040382, 17892898, 18818242, 20684302, 25100098, 32970562, 37601422, 46131262, 48441598

Offset: 1

K. D. Bajpai, Mar 02 2016

nonn

Subsequence of A103533 and A001358.

All the terms in this sequence, except a(1), are congruent to 1 (mod 3).

			a(1) = 14 = 2 * 7 that is semiprime. Also, 3 * 5 - 1 = 14 where {3,5} is a twin prime pair.
a(2) = 34 = 2 * 17 that is semiprime. Also, 5 * 7 - 1 = 34 where {5,7} is a twin prime pair.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A001097, A001358, A001359, A006512, A086870, A103533, A118071.

IsP2:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ s: n in [1..1000] | IsPrime(n) and IsPrime(n+2) and IsP2(s) where s is (n * (n+2) - 1)];

A269663:= proc() local a, b, d; a:= ithprime(n); b:=a+2; d:=(a*b)-1; if isprime(b)and bigomega(d)=2 then return (d): fi; end: seq(A269663 (n), n=1..1000);

A269663= {}; Do[a = Prime[n]; b = a + 2; c = a*b - 1; If[PrimeQ[b] && PrimeOmega[c] == 2, AppendTo[A269663, c]], {n, 1000}]; A269663
Select[Times @@ # - 1 & /@ Transpose@{#, 2 + #} &@ Select[Prime@ Range@ 900, NextPrime@ # == # + 2 &], PrimeOmega@ # == 2 &] (* Michael De Vlieger, Apr 01 2016 *)
Select[Times@@@Select[Partition[Prime[Range[1000]],2,1],#[[2]]-#[[1]]==2&]-1,PrimeOmega[ #]==2&] (* Harvey P. Dale, Mar 14 2023 *)

for(n = 1, 1000, p = prime(n); q = p + 2; c=(p*q) - 1; if(isprime(q) && bigomega(c)==2, print1(c, ", ")));

a(n) = 2*A086870(n). - Ray Chandler, Apr 04 2016

A269662 Semiprimes which are the sum of a twin prime pair plus one.

Original entry on oeis.org

9, 25, 85, 121, 145, 205, 217, 301, 361, 481, 565, 697, 841, 865, 1141, 1285, 1717, 1765, 2041, 2101, 2305, 2461, 2581, 2605, 2641, 2965, 2977, 3241, 3337, 3397, 3865, 3901, 3997, 4285, 4537, 4681, 4765, 5317, 5377, 5461, 5941, 6001, 6241, 6505, 6937, 7081, 7117

Offset: 1

K. D. Bajpai, Mar 02 2016

nonn

All the terms, except a(1), are congruent to 1 (mod 3).

			a(2) = 25 = 5 * 5 that is semiprime. Also, 25 = 11 + 13 + 1 where {11, 13} is a twin prime pair.
a(3) = 85 = 5 * 17 that is semiprime. Also, 55 = 41 + 43 + 1 where {41, 43} is a twin prime pair.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A001097, A001358, A001359, A006512, A054735, A118071.

IsP2:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ s: n in [1..1000] | IsPrime(n) and IsPrime(n+2) and IsP2(s) where s is (n + n+2 + 1)];

A269662 = {}; Do[a = Prime[n]; b = a + 2; c = a + b + 1; If[PrimeQ[b] && PrimeOmega[c] == 2, AppendTo[A269662, c]], {n, 1000}]; A269662
Select[Range[1, 7200, 2], And[PrimeOmega@ # == 2, And[PrimeQ@ #, NextPrime[#] - 2] == # &[(# - 1)/2 - 1]] &] (* Michael De Vlieger, Apr 01 2016 *)
Select[1+Total[#]&/@Select[Partition[Prime[Range[500]],2,1],#[[2]]-#[[1]] == 2&],PrimeOmega[#]==2&] (* Harvey P. Dale, Apr 10 2016 *)

for(n = 1, 1000, p=prime(n); q=p+2; s=p+q+1; if(isprime(q) && bigomega(s)==2, print1(s,", ")));

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A038869 Primes p such that both p-2 and 2p-1 are prime.

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A157995 Primes which are the sum of 1 plus two consecutive not-twin primes, p1 and p2, (p2-p1)>2.

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A157996 Primes which are sum of 1 and two nonconsecutive primes p1 and p2, p2 - p1 > 2.

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A269663 Semiprimes which are the product of a twin prime pair minus one.

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A269662 Semiprimes which are the sum of a twin prime pair plus one.

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