A063638 -id:A063638 - cp's OEIS frontend

A063637 Primes p such that p+2 is a semiprime.

2, 7, 13, 19, 23, 31, 37, 47, 53, 67, 83, 89, 109, 113, 127, 131, 139, 157, 167, 181, 199, 211, 233, 251, 257, 263, 293, 307, 317, 337, 353, 359, 379, 389, 401, 409, 443, 449, 467, 479, 487, 491, 499, 503, 509, 541, 557, 563, 571, 577, 587, 631, 647, 653, 677

Offset: 1

Reinhard Zumkeller, Jul 21 2001

nonn

Primes of the form p*q - 2, where p and q are primes.

Union of A049002 and A115093. - T. D. Noe, Mar 01 2006

			From _K. D. Bajpai_, Sep 06 2014: (Start)
a(3) = 13, which is prime, and 13 + 2 = 15 = 3 * 5, which is a semiprime.
a(4) = 19, which is prime, and 19 + 2 = 21 = 3 * 7, which is a semiprime.
(End)

J.-R. Chen, On the representation of a large even integer as the sum of a prime and a product of at most two primes, Sci. Sinica 16 (1973), 157-176.

K. D. Bajpai, Table of n, a(n) for n = 1..14190 (first 1000 terms from T. D. Noe)
P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 146. [?Broken link]
P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 146.
T. Tao, Obstructions to uniformity and arithmetic patterns in the primes, arXiv:math/0505402 [math.NT], 2005.

Cf. A005383, A001358, A063638.

Cf. A109611 (Chen primes).

a063637 n = a063637_list !!(n-1)
a063637_list = filter ((== 1) . a064911 . (+ 2)) a000040_list
-- Reinhard Zumkeller, Nov 15 2011

select(t -> isprime(t) and numtheory:-bigomega(t+2)=2, [2, seq(2*i+1,i=1..500)]); # Robert Israel, Sep 07 2014

f[n_] := Plus @@ Flatten[ Table[ # [[2]], {1}] & /@ FactorInteger[ n]]; Select[ Prime[ Range[ 123]], f[ # + 2] == 2 &] (* Robert G. Wilson v, Apr 30 2005 *)
Select[Prime[Range[500]],PrimeOmega[#+2]==2&]  (* K. D. Bajpai, Sep 06 2014 *)

{ n=0; for (m=1, 10^9, p=prime(m); if (bigomega(p + 2) == 2, write("b063637.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 26 2009

a(n) = A062721(n) - 2.

A010051(a(n)) * A064911(a(n) + 2) = 1. - Reinhard Zumkeller, Nov 15 2011

A063643 Primes with 2 representations: pq - 2 = uv + 2 where p, q, u and v are primes.

Original entry on oeis.org

23, 37, 53, 67, 89, 113, 131, 157, 211, 251, 293, 307, 337, 379, 409, 449, 487, 491, 499, 503, 631, 683, 701, 719, 751, 769, 787, 919, 941, 953, 991, 1009, 1039, 1117, 1193, 1201, 1259, 1381, 1399, 1439, 1459, 1471, 1499, 1511, 1567, 1709, 1733, 1759, 1801

Offset: 1

Reinhard Zumkeller, Jul 21 2001

nonn

Or, primes p such that p+/-2 are semiprimes. - Zak Seidov, Mar 08 2006

			A063643(25) = 751: 751 = A063637(60)= 753 - 2 = 3*251 - 2, 751 = A063638(55)= 749 + 2 = 7*107 + 2.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 term from T. D. Noe)

Cf. A063637, A063638, A001358.

Cf. A109611 (Chen primes).

q:= p-> isprime(p) and map(numtheory[bigomega], {p-2, p+2})={2}:
select(q, [$2..2000])[];  # Alois P. Heinz, Apr 01 2024

Select[Prime[Range[300]], PrimeOmega[#+2] == PrimeOmega[#-2] == 2&] (* Jean-François Alcover, Mar 02 2019 *)

{ n=0; for (m=2, 10^9, p=prime(m); if (bigomega(p + 2) == 2 && bigomega(p - 2) == 2, write("b063643.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 26 2009

Intersection of A063637 and A063638. - Zak Seidov, Mar 14 2011

A189827 a(n) = d(n-1) + d(n+1), where d(k) is the number of divisors of k.

Original entry on oeis.org

3, 5, 4, 7, 4, 8, 5, 8, 5, 10, 4, 10, 6, 9, 6, 11, 4, 12, 6, 10, 6, 12, 5, 12, 7, 10, 6, 14, 4, 14, 6, 10, 8, 13, 6, 13, 6, 12, 6, 16, 4, 14, 8, 10, 8, 14, 5, 16, 7, 12, 6, 14, 6, 16, 8, 12, 6, 16, 4, 16, 8, 11, 10, 15, 6, 14, 6, 14, 6, 20, 4, 16, 8, 10, 10

Offset: 2

T. D. Noe, Apr 28 2011

nonn

d(n-1) + d(n+1) is a measure of the compositeness of the numbers next to n. Sequence A189825 lists the first occurrence of each number.

It is conjectured that every number greater than 3 occurs an infinite number of times. Note that an infinite number of 4's is equivalent to there being an infinite number of twin primes (A001097). An infinite number of 5's is equivalent to there being an infinite number of primes of the form p^2-2 (A028871) or p^2+2 (A056899) for prime p. An infinite number of 6's is equivalent to there being an infinite number of primes of the form p^3-2 (A066878), p^3+2 (A048636), p*q-2 (A063637), or p*q+2 (A063638), where p and q are distinct primes.

			a(5) = d(4) + d(6) = 3 + 4 = 7.

T. D. Noe, Table of n, a(n) for n = 2..10000

Cf. A175144, A189825.

Table[DivisorSigma[0,n-1] + DivisorSigma[0,n+1], {n, 2, 100}]
First[#]+Last[#]&/@Partition[DivisorSigma[0,Range[80]],3,1] (* Harvey P. Dale, May 27 2013 *)

A176229 The smaller members p of cousin prime pairs (p,p+4) with a semiprime arithmetic mean p+2.

Original entry on oeis.org

7, 13, 19, 37, 67, 109, 127, 307, 379, 487, 499, 769, 877, 937, 1009, 1297, 1567, 2269, 2389, 2659, 2857, 3037, 3187, 3457, 3847, 3907, 3919, 4447, 4789, 4969, 4999, 5077, 5167, 5347, 5737, 6007, 6997, 7039, 7669, 8689, 8779, 9199, 10597, 11467, 11827

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 12 2010

nonn

By definition a subsequence of A063637 and of A023200.
The associated p+4 are members of A063638.
Because all members of A023200 are == 1 (mod 3), the semiprimes p+2 are all == 0 (mod 3), so one of their two factors is 3.
The least-significant digit (LSD) of p > 13 in A023200 is always 3, 7 or 9, but those with LSD equal to 3 demand p+2 to have LSD 5 and therefore divisor 5 which contradicts the semiprime property above, so 13 is the only member of the sequence with LSD equal to 3.

			7 = prime(4), 11 = prime(5), (7+11)/2 = 3^2 = semiprime(3), so 7 is in the sequence.
13 = prime(6), 17 = prime(7), (13+17)/3 = 3 * 5 = semiprime(6), so 13 is in the sequence.
19 = prime(8), 23 = prime(9), (19+23)/3 = 3 * 7 = semiprime(7), so 19 is in the sequence.

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

Cf. A001358, A023200, A046132, A063637, A063638.

aQ[n_] := PrimeQ[n] && PrimeOmega[n + 2] == 2 && PrimeQ[n + 4]; Select[Range[12000], aQ] (* Amiram Eldar, Sep 08 2019 *)
Select[Prime[Range[1500]],PrimeQ[#+4]&&PrimeOmega[#+2]==2&] (* Harvey P. Dale, May 15 2023 *)

A241659 Primes p such that p^3 + 2 is semiprime.

Original entry on oeis.org

2, 11, 13, 17, 19, 23, 31, 41, 53, 59, 89, 101, 131, 137, 149, 193, 211, 223, 227, 229, 233, 239, 251, 271, 293, 317, 331, 359, 401, 449, 461, 557, 563, 571, 593, 599, 619, 641, 659, 677, 691, 719, 739, 751, 809, 821, 853, 929, 971, 991, 1009, 1013, 1039, 1051

Offset: 1

K. D. Bajpai, Apr 26 2014

nonn

			11 is prime and appears in the sequence because 11^3 + 2 = 1333 = 31 * 43, which is a semiprime.
17 is prime and appears in the sequence because 17^3 + 2 = 4915 =  5 * 983, which is a semiprime.
37 is prime but does not appear in the sequence because 37^3 + 2 = 50655 =  3 * 5 * 11 * 983, which is not a semiprime.

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

Cf. A001358, A063637, A063638, A228183, A237627, A241483, A241493.

with(numtheory): KD:= proc() local a, b, k; k:=ithprime(n); a:=bigomega(k^3+2); if a=2 then RETURN (k); fi; end: seq(KD(), n=1..500);

A241659 = {}; Do[t = Prime[n]; If[PrimeOmega[t^3 + 2] == 2, AppendTo[A241659, t]], {n, 500}]; A241659
(*For the b-file*) c = 0; Do[t = Prime[n]; If[PrimeOmega[t^3 + 2] == 2, c++; Print[c, "  ", t]], {n, 1,6*10^4}];
Select[Prime[Range[200]],PrimeOmega[#^3+2]==2&] (* Harvey P. Dale, Feb 05 2025 *)

s=[]; forprime(p=2, 1200, if(bigomega(p^3+2)==2, s=concat(s, p))); s \\ Colin Barker, Apr 27 2014

A241809 Semiprimes sp such that sp+2 is a prime.

Original entry on oeis.org

9, 15, 21, 35, 39, 51, 57, 65, 69, 77, 87, 95, 111, 129, 155, 161, 177, 209, 221, 237, 249, 267, 291, 305, 309, 329, 335, 365, 371, 377, 381, 395, 407, 417, 437, 447, 485, 489, 497, 501, 519, 545, 591, 597, 611, 629, 671, 681, 689, 699, 707, 717, 731, 737, 749

Offset: 1

K. D. Bajpai, Apr 29 2014

			a(2) = 15 = 3*5, which is semiprime and 15+2 = 17 is a prime.
a(6) = 51 = 3*17, which is semiprime and 51+2 = 53 is a prime.

K. D. Bajpai, Table of n, a(n) for n = 1..1370

Cf. A001358, A062721, A063638, A107986.

with(numtheory): A241809:= proc(); if bigomega(x)=2 and isprime(x+2)then  RETURN (x); fi; end: seq(A241809 (), x=1..2000);

A241809={};Do[If[PrimeOmega[n]==2&&PrimeQ[n+2],AppendTo[A241809,n]],{n,1000}];A241809
Select[Prime[Range[200]]-2,PrimeOmega[#]==2&] (* Harvey P. Dale, Aug 06 2015 *)
SequencePosition[Table[Which[PrimeQ[n],1,PrimeOmega[n]==2,2,True,0],{n,800}],{2,,1}][[;;,1]] (* _Harvey P. Dale, Oct 05 2023 *)

for(k=1, 1000, if(bigomega(k)==2 && isprime(k+2), print1(k, ", "))) \\ Colin Barker, May 07 2014

a(n) = A063638(n) - 2.

A092104 Primes of form p*q + 4, with prime p and q.

Original entry on oeis.org

13, 19, 29, 37, 43, 53, 59, 61, 73, 89, 97, 127, 137, 149, 163, 173, 181, 191, 223, 239, 241, 251, 257, 263, 269, 271, 293, 307, 313, 331, 359, 397, 419, 421, 431, 449, 457, 509, 521, 523, 541, 547, 557, 563, 569, 577, 587, 593, 601, 653, 659, 673, 683, 691

Offset: 1

Zak Seidov, Feb 20 2004

Primes of form p*q + 2, A063638. Primes common in A063638 and A092104, A092105. Primes of form p*p + 4, A045637.

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

Cf. A045637, A063638, A092105.

With[{upto=700},Select[Times@@#+4&/@Tuples[Prime[Range[PrimePi[upto/2]]], 2], PrimeQ[#]&&#<+upto&]]//Union (* Harvey P. Dale, Jul 23 2016 *)

list(lim)=my(v=List(),t); forprime(p=3,(lim-4)\3, forprime(q=3,min((lim-4)\p, p), t=p*q+4; if(isprime(t), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Aug 05 2016

A207526 Primes p such that p-2 or p+2 are semiprimes.

Original entry on oeis.org

2, 7, 11, 13, 17, 19, 23, 31, 37, 41, 47, 53, 59, 67, 71, 79, 83, 89, 97, 109, 113, 127, 131, 139, 157, 163, 167, 179, 181, 199, 211, 223, 233, 239, 251, 257, 263, 269, 293, 307, 311, 317, 331, 337, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 439

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 18 2012

nonn

			2+2=2^2(semiprime), 7+2=3^2(semiprime), 11-2=3^2(semiprime), 13+2=3*5(semiprime), 17-2=3*5(semiprime)

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A063637, A063638, A063643.

Select[Table[Prime[n], {n,200}], (Last /@ FactorInteger[#-2] == {1,1} || Last /@ FactorInteger[#-2] == {2}) || (Last /@ FactorInteger[#+2] == {1,1} || Last /@ FactorInteger[#+2] == {2}) &]

A217195 Primes p such that p-2 is the greatest semiprime less than p.

Original entry on oeis.org

17, 37, 41, 53, 67, 71, 79, 89, 97, 113, 131, 157, 163, 211, 223, 239, 251, 269, 293, 307, 311, 331, 337, 367, 373, 379, 397, 409, 419, 439, 449, 487, 491, 499, 521, 547, 593, 599, 613, 631, 673, 683, 691, 701, 709, 733, 739, 751, 757, 769, 773, 787, 809

Offset: 1

Antonio Roldán, Sep 27 2012

nonn

This is a subsequence of A063638.

			487 is prime, 486 = 2*3^5 is not semiprime and 485 = 5*97 is semiprime.

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

SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])]; Select[Prime[Range[200]], ! SemiPrimeQ[# - 1] && SemiPrimeQ[# - 2] &] (* T. D. Noe, Sep 27 2012 *)

forprime(p=3, 9999, bigomega(p-2)==2 && bigomega(p-1)!=2 & print1(p", "))

A241716 Primes p such that p^3 - 2 is semiprime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 41, 43, 47, 61, 79, 89, 101, 107, 139, 157, 181, 199, 239, 271, 307, 311, 331, 337, 347, 349, 379, 397, 409, 421, 431, 479, 487, 499, 521, 523, 541, 571, 607, 613, 641, 643, 661, 673, 701, 719, 761, 769, 811, 823, 829, 839, 877, 881, 883

Offset: 1

K. D. Bajpai, Apr 27 2014

nonn

			11 is prime and appears in the sequence because 11^3 - 2 = 1329 = 3 * 443, which is a semiprime.
17 is prime and appears in the sequence because 17^3 - 2 = 4911 = 3 * 1637, which is a semiprime.
23 is prime but does not appear in the sequence because 23^3 - 2 = 12165 =  3 * 5 * 811, which is not a semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..9380

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

with(numtheory):A241716:= proc() local k; k:=ithprime(x); if bigomega(k^3-2)=2 then RETURN (k); fi; end: seq(A241716(), x=1..500);

A241716 = {}; Do[t = Prime[n]; If[PrimeOmega[t^3 - 2] == 2, AppendTo[A241716, t]], {n, 500}]; A241716
Select[Prime[Range[200]],PrimeOmega[#^3-2]==2&] (* Harvey P. Dale, Dec 09 2018 *)

cp's OEIS Frontend

A063637 Primes p such that p+2 is a semiprime.

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A063643 Primes with 2 representations: p*q - 2 = u*v + 2 where p, q, u and v are primes.

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A189827 a(n) = d(n-1) + d(n+1), where d(k) is the number of divisors of k.

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A176229 The smaller members p of cousin prime pairs (p,p+4) with a semiprime arithmetic mean p+2.

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A241659 Primes p such that p^3 + 2 is semiprime.

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A241809 Semiprimes sp such that sp+2 is a prime.

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A092104 Primes of form p*q + 4, with prime p and q.

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A063643 Primes with 2 representations: pq - 2 = uv + 2 where p, q, u and v are primes.