A063637 -id:A063637 - cp's OEIS frontend

A241732 Primes p such that p^3 + 2 and p^3 - 2 are semiprime.

2, 11, 13, 17, 41, 89, 101, 239, 271, 331, 571, 641, 719, 1051, 1231, 1321, 1549, 1559, 1721, 1741, 1831, 1993, 1999, 2029, 2311, 2459, 2749, 2837, 2861, 2939, 3389, 3467, 3671, 4049, 4111, 4273, 4787, 4919, 4969, 5657, 5689, 5861, 6221, 6679, 6691, 6829, 7109

Offset: 1

K. D. Bajpai, Apr 27 2014

nonn

			11 is prime and appears in the sequence because 11^3 + 2 = 1333 = 31 * 43 and 11^3 - 2 = 1329 = 3 * 443, both are semiprime.
41 is prime and appears in the sequence because 41^3 + 2 = 68923 = 157 * 439 and 41^3 - 2 = 68919 = 3 * 22973, both are semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..2310

Cf. A001358, A063637, A063638, A072381, A082919, A145292, A228183, A237627, A241483, A241493, A241659.

with(numtheory): KD:= proc() local k; k:=ithprime(n); if bigomega(k^3+2)=2 and bigomega(k^3-2)=2 then k; fi; end: seq(KD(), n=1..2000);

A241732 = {}; Do[t = Prime[n]; If[PrimeOmega[t^3 + 2] == 2 && PrimeOmega[t^3 - 2] == 2, AppendTo[A241732, t]], {n, 500}]; A241732
Select[Prime[Range[1000]],PrimeOmega[#^3+2]==PrimeOmega[#^3-2]==2&] (* Harvey P. Dale, Jun 20 2019 *)

A242343 Triangular numbers T such that (T+2) is semiprime.

Original entry on oeis.org

36, 55, 91, 120, 153, 276, 300, 325, 435, 595, 903, 1035, 1225, 1653, 1711, 1891, 2016, 2145, 2485, 2556, 3003, 3240, 3741, 4095, 4465, 4560, 4851, 5253, 5460, 5565, 5995, 6105, 6216, 6441, 6555, 6903, 7021, 7140, 7260, 8001, 8256, 8911, 9045, 9180, 9591, 10585

Offset: 1

K. D. Bajpai, May 11 2014

nonn

The n-th triangular number T(n) = n*(n+1)/2 = A000217(n).
Triangular numbers of the form p*q - 2, where p and q are primes.
The indices of these triangular numbers are 8, 10, 13, 15, 17, 23, 24, 25, 29, 34, 42, 45, 49, 57, 58, 61, 63, 65, 70, 71, 77, 80, 86, 90, 94, 95, 98, 102, 104, 105, 109, 110, 111, 113, 114, 117, 118, 119, 120, 126, 128, 133, 134, 135, 138, 145, ... - Wolfdieter Lang, May 13 2014

			a(1) = 36 = 8*(8+1)/2 = 36 + 2 = 38 = 2 * 19 is semiprime.
a(2) = 55 = 10*(10+1)/2 = 55 + 2 = 57 = 3 * 19 is semiprime.

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

Cf. A001358, A000217, A063637, A063638.

with(numtheory): A242343:= proc()local t; t:=x/2*(x+1); if bigomega(t+2)=2 then  RETURN (t); fi;end: seq(A242343 (),x=1..200);

Select[Table[n/2*(n + 1), {n, 200}], PrimeOmega[# + 2] == 2 &]
Select[Accumulate[Range[200]],PrimeOmega[#+2]==2&] (* Harvey P. Dale, Dec 25 2024 *)

A289250 Primes p such that p + 4 is a semiprime.

Original entry on oeis.org

2, 5, 11, 17, 29, 31, 47, 53, 61, 73, 83, 89, 107, 137, 139, 151, 157, 173, 179, 181, 197, 199, 211, 233, 263, 283, 317, 331, 337, 367, 373, 389, 409, 433, 443, 449, 467, 523, 541, 547, 569, 577, 587, 593, 607, 619, 631, 677, 683, 691, 709, 719, 727, 733, 751, 787, 809, 811, 827

Offset: 1

Zak Seidov, Jun 29 2017

nonn

Except for case p=5, p+4 is never a perfect square.

For p = {2, 11, 31, 73, 139, 433, 1759, 2017} p+4 is a product of two consecutive primes.

			2+4=6=2*3, 5+4=9=3*3, 11+4=15=3*5 (all semiprimes).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001358, A063637, A082919, A241484,

Select[Prime@ Range@ 150, PrimeOmega[# + 4] == 2 &] (* Michael De Vlieger, Jun 29 2017 *)

issemi(n)=bigomega(n)==2
is(n)=isprime(n) && issemi(n+4) \\ Charles R Greathouse IV, Jul 02 2017

A104485 Primes p = p(k) such that prime(k) + 2 and prime(k+1) + 2 are both semiprimes.

Original entry on oeis.org

19, 31, 47, 83, 109, 113, 127, 199, 251, 257, 293, 353, 401, 443, 467, 479, 487, 491, 499, 503, 557, 571, 577, 647, 677, 743, 761, 787, 829, 863, 911, 937, 941, 947, 971, 977, 983, 1109, 1187, 1193, 1291, 1327, 1361, 1381, 1399, 1459, 1499, 1553, 1559

Offset: 1

Giovanni Teofilatto, Apr 19 2005

			19 is a term because prime(8) + 2 = 19 + 2 = 21 = 3*7 and prime(9) + 2 = 25 = 5*5.

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

Cf. A063637.

fQ[n_] := Plus @@ Flatten[ Table[ #[[2]], {1}] & /@ FactorInteger[n]] == 2; Prime /@ Select[ Range[ 270], fQ[ Prime[ # ] + 2] && fQ[ Prime[ # + 1] + 2] &] (* Robert G. Wilson v, Apr 19 2005 *)
Select[Prime[Range[250]],PrimeOmega[#+2]==PrimeOmega[NextPrime[#]+2]==2&] (* Harvey P. Dale, Apr 01 2024 *)

Corrected and extended by Robert G. Wilson v, Apr 19 2005

A106667 a(n) = 1 if prime(n) + 2 is a prime, a(n) = -1 if prime(n) + 2 is a semiprime, otherwise 0.

Original entry on oeis.org

-1, 1, 1, -1, 1, -1, 1, -1, -1, 1, -1, -1, 1, 0, -1, -1, 1, 0, -1, 1, 0, 0, -1, -1, 0, 1, 0, 1, -1, -1, -1, -1, 1, -1, 1, 0, -1, 0, -1, 0, 1, -1, 1, 0, 1, -1, -1, 0, 1, 0, -1, 1, 0, -1, -1, -1, 1, 0, 0, 1, 0, -1, -1, 1, 0, -1, 0, -1, 1, 0, -1, -1, 0, 0, -1, 0, -1, 0, -1, -1, 1, 0, 1, 0, 0, -1, -1, 0, 1, 0, -1, -1, -1, -1

Offset: 1

Giovanni Teofilatto, May 13 2005

			a(1) = -1 because prime(1) = 2 and 2 + 2 = 4 is a semiprime;
a(2) = 1 because prime(2) = 3 and 3 + 2 = 5 is a prime;
a(14) = 0 because prime(14) = 43 and 43 + 2 = 45 is neither prime nor semiprime.

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

Cf. A001359, A063637.

p:= 1:
for n from 1 to 100 do
  p:= nextprime(p);
  if isprime(p+2) then A[n]:=1
  elif numtheory:-bigomega(p+2)=2 then A[n]:=-1
  else A[n]:= 0
  fi
od:
seq(A[n],n=1..100); # Robert Israel, Aug 29 2018

Corrected, and definition clarified, by Robert Israel, Aug 29 2018

A242344 Triangular numbers T such that T-2 is semiprime.

Original entry on oeis.org

6, 28, 36, 120, 136, 171, 276, 300, 325, 528, 561, 780, 820, 903, 1081, 1128, 1176, 1275, 1540, 1596, 1653, 2080, 2211, 2415, 2485, 2556, 2775, 3160, 3240, 3403, 3655, 3828, 4371, 4851, 5151, 5253, 5356, 5460, 5995, 6105, 6328, 6441, 6903, 7381, 7503, 8001, 8256

Offset: 1

K. D. Bajpai, May 11 2014

nonn

The n-th triangular number T(n) = n*(n+1)/2.

Triangular numbers of the form p*q + 2, where p and q are primes (not necessarily distinct).

			a(2) = 28 = 7*(7+1)/2 = 28 - 2 = 26 = 2 * 13 is semiprime.
a(3) = 36 = 8*(8+1)/2 = 36 - 2 = 34 = 2 * 17 is semiprime.

K. D. Bajpai and N. J. A. Sloane, Table of n, a(n) for n = 1..27939 [First 10000 terms from K. D. Bajpai]

Cf. A001358, A000217, A063637, A063638.

with(numtheory): A242344:= proc()local t; t:=x*(x+1)/2;if bigomega(t-2)=2 then  RETURN (t); fi;end: seq(A242344(),x=1..200);

Select[Table[n*(n + 1)/2, {n, 200}], PrimeOmega[# - 2] == 2 &]
Select[Accumulate[Range[200]],PrimeOmega[#-2]==2&] (* Harvey P. Dale, Feb 21 2023 *)

A242356 Triangular numbers T such that both (T+2) and (T-2) are semiprimes.

Original entry on oeis.org

36, 120, 276, 300, 325, 903, 1653, 2485, 2556, 3240, 4851, 5253, 5460, 5995, 6105, 6441, 6903, 8001, 8256, 8911, 9591, 10585, 12561, 12720, 14365, 20301, 21115, 22791, 23436, 24753, 26335, 26565, 26796, 27495, 29161, 30381, 31375, 34191, 34980, 37401, 40755

Offset: 1

K. D. Bajpai, May 11 2014

nonn

The n-th triangular number T(n) = n*(n+1)/2.

Triangular numbers of the form p*q - 2 and r*s + 2 where p, q, r and s are primes.

			a(1) = 36 = 8*(8+1)/2 = 36 + 2 = 38 = 2 * 19 and  36 - 2 = 34 = 2 * 17 both are semiprimes.
a(2) = 120 = 15*(15+1)/2 = 120 + 2 = 122 = 2 * 61 and 120 - 2 = 118 = 2 * 59 both are semiprimes.

K. D. Bajpai, Table of n, a(n) for n = 1..6565

Cf. A001358, A000217, A063637, A063638.

with(numtheory): A242356:= proc()local t; t:=x*(x+1)/2; if bigomega(t+2)=2 and bigomega(t-2)=2 then  RETURN (t); fi;end: seq(A242356 (),x=1..500);

Select[Table[n*(n + 1)/2, {n, 500}], PrimeOmega[# + 2] == 2 && PrimeOmega[# - 2] == 2 &]
Select[Accumulate[Range[300]],PrimeOmega[#+{2,-2}]=={2,2}&] (* Harvey P. Dale, Apr 21 2016 *)

A277993 Sophie Germain primes p such that p + 2 and p - 2 are semiprimes.

Original entry on oeis.org

23, 53, 89, 113, 131, 251, 293, 491, 683, 719, 953, 1439, 1499, 1511, 1733, 2393, 3491, 3779, 5171, 7043, 7151, 7433, 7649, 7901, 8069, 8663, 9689, 10781, 12011, 12653, 13049, 13229, 13451, 13553, 14669, 15569, 16001, 16253, 18899, 19709, 20411, 22469, 22751, 23099

Offset: 1

K. D. Bajpai, Nov 07 2016

nonn

Intersection of A005384 and A063643.

			a(1) = 23 is Sophie Germain prime because 2*23 + 1 = 47 is prime. Also, 23 + 2 = 25 =  5*5; 23 - 2 = 21 = 7*3; are both semiprime.
a(2) = 53 is Sophie Germain prime because 2*53 + 1 = 107 is prime. Also, 53 + 2 = 55 =  11*5; 23 - 2 = 51 = 17*3; are both semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..4000

Cf. A005384, A063637, A063638, A063643.

Select[Select[Prime[Range[10000]], PrimeQ[2 # + 1] &], PrimeOmega[# - 2] == 2 && PrimeOmega[# + 2] == 2 &]
Select[Prime[Range[3000]],PrimeQ[2#+1]&&PrimeOmega[#+{2,-2}]=={2,2}&] (* Harvey P. Dale, Dec 16 2017 *)

is(n) = ispseudoprime(n) && ispseudoprime(2*n+1) && bigomega(n+2)==2 && bigomega(n-2)==2 \\ Felix Fröhlich, Nov 07 2016

A111067 Number of odd primes p < 10^n such that p+2 = product of 2 primes (no twin Chen primes).

Original entry on oeis.org

1, 11, 79, 427, 3009, 21779, 166649, 1322266, 10752066, 89305602, 754868608, 6472917998

Offset: 1

Pierre CAMI, Oct 08 2005

A006880(n) = number of primes < 10^n, A007508(n) = number of twin primes < 10^n. Let F(n) = A006880(n)/A007508(n). For n > 3, we find that F(n) is ~ 0.762373*log(10^n) - 0.968855.

Let FF(n) = A006880(n)/a(n). For n>3, we find that FF(n) is ~ 0.163128*log(10^n) + 1.349255. a(n)/A007508(n) is then ~ 0.762373*log((10^n) - 0.968855 / ( 0.163128*log(10^n) + 1.349255, as n tends to infinity a(n) / A007508(n) needs to tend to 0.762373 / 0.163128 = 4.673465.

			7 is the only prime < 10 with 7+2 = 3*3 = product of 2 odd primes so a(1) = 1.

Cf. A006880, A007508, A063637.

a[n_] := Count[Prime[Range[2, PrimePi[10^n]]], ?(PrimeOmega[# + 2] == 2 &)]; Array[a, 6] (* _Amiram Eldar, Jul 25 2025 *)

list(len) = {my(c = 0, k = 0, pow = 10); forprime(p = 3, , if(p > pow, print1(c, ", "); k++; if(k == len, break); pow *= 10); if(bigomega(p+2) == 2, c++));} \\ Amiram Eldar, Jul 25 2025

a(8) corrected and a(9) computed by Robert G. Wilson v, Oct 10 2005

a(10)-a(12) from Amiram Eldar, Jul 25 2025

cp's OEIS Frontend

A241732 Primes p such that p^3 + 2 and p^3 - 2 are semiprime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A242343 Triangular numbers T such that (T+2) is semiprime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A289250 Primes p such that p + 4 is a semiprime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A104485 Primes p = p(k) such that prime(k) + 2 and prime(k+1) + 2 are both semiprimes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A106667 a(n) = 1 if prime(n) + 2 is a prime, a(n) = -1 if prime(n) + 2 is a semiprime, otherwise 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Extensions

A242344 Triangular numbers T such that T-2 is semiprime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A242356 Triangular numbers T such that both (T+2) and (T-2) are semiprimes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A277993 Sophie Germain primes p such that p + 2 and p - 2 are semiprimes.

Views

Author

Keywords

Comments