A063638 -id:A063638 - 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 *)

A092105 Primes p such that both p-2 and p-4 are semiprimes.

Original entry on oeis.org

37, 53, 59, 89, 97, 163, 223, 239, 251, 269, 293, 307, 331, 397, 419, 449, 521, 547, 593, 673, 683, 691, 701, 757, 853, 953, 997, 1061, 1103, 1123, 1151, 1171, 1193, 1259, 1319, 1373, 1511, 1531, 1567, 1693, 1783, 1801, 1823, 1931, 1987, 2053, 2161, 2203

Offset: 1

Zak Seidov, Feb 20 2004

Intersection of A063638 and A092104.

			97 is a member because 95=5*19 and 93=3*31

Cf. A063638, A092104.

isok(n) = (n>5) && isprime(n) && (bigomega(n-2) ==  2) && (bigomega(n-4) == 2); \\ Michel Marcus, Oct 05 2013

More terms from Ray Chandler, Feb 21 2004

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 *)

A267067 Primes p such that mu(p-2) = 1; that is, p-2 is squarefree and has an even number of prime factors, where mu is the Moebius function (A008683).

Original entry on oeis.org

3, 17, 23, 37, 41, 53, 59, 67, 71, 79, 89, 97, 113, 131, 157, 163, 179, 211, 223, 239, 251, 269, 293, 307, 311, 331, 337, 367, 373, 379, 383, 397, 409, 419, 439, 449, 487, 491, 499, 503, 521, 547, 593, 599, 613, 631, 673, 683, 691, 701, 709, 719, 733, 739

Offset: 1

Vincenzo Librandi, Jan 10 2016

nonn

From Robert Israel, Jan 10 2016: (Start)
Includes all members of A063638 except 11.
The first terms not in A063638 are 3 and 1367. (End)

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Moebius Function

Cf. A008683, A063638, A088179.

[n: n in [3..1000] | IsPrime(n) and MoebiusMu(n-2) eq 1];

select(p -> isprime(p) and numtheory:-mobius(p-2)=1, [seq(i,i=3..1000,2)]); # Robert Israel, Jan 10 2016

Select[Prime[Range[200]], MoebiusMu[# - 2] == 1 &]

isok(p) = isprime(p) && (p>2) && (moebius(p-2)==1); \\ Michel Marcus, Mar 08 2023

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

A285915 Integers n such that A112528(n) - A103274(n) = 1.

Original entry on oeis.org

5, 7, 9, 12, 13, 16, 17, 19, 20, 22, 24, 25, 30, 32, 37, 38, 41, 47, 48, 52, 54, 57, 62, 63, 64, 67, 68, 73, 74, 75, 76, 78, 80, 81, 85, 87, 93, 94, 95, 96, 98, 101, 108, 109, 112, 115, 122, 124, 125, 126, 127, 128, 130, 131, 133, 134, 136, 137, 138, 140, 147

Offset: 1

Zak Seidov, Apr 28 2017

nonn

In general, A112528(n) - A103274(n) = 0 or 1.

Also, A000040(a(n)) = A063638(n).

Cf. A000040, A063638, A103274, A112528.

Select[Range@ 150, And[# != 1, PrimeOmega[Prime@ # - 2] == 2] &] (* Michael De Vlieger, May 01 2017 *)

is(n)=n!=1&&bigomega(prime(n)-2)==2 \\ David A. Corneth, Apr 29 2017

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

A092105 Primes p such that both p-2 and p-4 are semiprimes.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

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

A267067 Primes p such that mu(p-2) = 1; that is, p-2 is squarefree and has an even number of prime factors, where mu is the Moebius function (A008683).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A285915 Integers n such that A112528(n) - A103274(n) = 1.

Views

Author