A062721 -id:A062721 - 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

A157931 Numbers that are both the sum and the product of two primes.

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 38, 39, 46, 49, 55, 58, 62, 69, 74, 82, 85, 86, 91, 94, 106, 111, 115, 118, 122, 129, 133, 134, 141, 142, 146, 158, 159, 166, 169, 178, 183, 194, 201, 202, 206, 213, 214, 218, 226, 235, 253, 254, 259, 262, 265, 274, 278

Offset: 1

William Weeks (dach(AT)kuci.org), Mar 09 2009

Assuming the Goldbach conjecture, this is A001358 intersect (A005843 union A052147), since an odd number n is the sum of two primes iff n-2 is prime. - N. J. A. Sloane, Mar 14 2009

The first few terms of A001358: Semiprimes, not members of A157931 are: 35, 51, 57, 65, 77, 87, 93, 95, ..., . - Robert G. Wilson v, Mar 15 2009

			For the numbers up to 100, the solutions are 4 = (2+2) = (2*2); 6 = (3+3) = (2*3); 9 = (2+7) = (3*3); 10 = (3+7) = (2*5); 14 = (3+11) = (2*7); 15 = (2+13) = (3*5); 21 = (2+19) = (3*7); 22 = (3+19) = (2*11); 25 = (2+23) = (5*5); 26 = (3+23) = (2*13); 33 = (2+31) = (3*11); 34 = (3+31) = (2*17); 38 = (7+31) = (2*19); 39 = (2+37) = (3*13); 46 = (3+43) = (2*23); 49 = (2+47) = (7*7); 55 = (2+53) = (5*11); 58 = (5+53) = (2*29); 62 = (3+59) = (2*31); 69 = (2+67) = (3*23); 74 = (3+71) = (2*37); 82 = (3+79) = (2*41); 85 = (2+83) = (5*17); 86 = (3+83) = (2*43); 91 = (2+89) = (7*13); 94 = (5+89) = (2*47).

Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 1096 terms from Robert G. Wilson v)

Cf. A001358, A005843, A052147, A062721.
Cf. A043326 Numbers n such that n is a product of two different primes and n - 2 is prime, A062721 Numbers n such that n is a product of two primes and n - 2 is prime. - Zak Seidov, Mar 15 2009
Cf. A014091, A064911, A100962.

a157931 n = a157931_list !! (n-1)
a157931_list = filter ((== 1) . a064911) a014091_list
-- Reinhard Zumkeller, Oct 15 2014

isA014091 := proc(n) for i from 1 do p := ithprime(i) ; if p > n/2 then RETURN(false); fi; if isprime(n-p) then RETURN(true) ; fi; od: end: isA001358 := proc(n) RETURN(numtheory[bigomega](n) = 2) ; end: for n from 4 to 500 do if isA001358(n) and isA014091(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Mar 15 2009

fQ[n_] := Block[{k = 2}, While[k < n, If[ PrimeQ[n - k], Break[]]; k = NextPrime@k]; k + 1 < n]; semiPrimeQ[n_] := Plus @@ Last /@ FactorInteger@n == 2; Select[ Range@ 295, fQ@# && semiPrimeQ@# &] (* Robert G. Wilson v, Mar 15 2009 *)
Select[Union[Flatten[Table[Prime[i] + Prime[j], {i, 50}, {j, 50}]]], PrimeOmega[#] == 2 &] (* Alonso del Arte, Feb 08 2013 *)
Union[Select[Total/@Tuples[Prime[Range[60]],2],PrimeOmega[#]==2&]] (* Harvey P. Dale, Jul 27 2015 *)

A014091 INTERSECT A001358. - R. J. Mathar, Mar 15 2009

Edited by N. J. A. Sloane, Mar 14 2009

Extended by R. J. Mathar and Robert G. Wilson v, Mar 15 2009

A158318 Primes p such that 5p-2 is prime.

Original entry on oeis.org

3, 5, 11, 17, 23, 47, 53, 59, 71, 89, 101, 113, 131, 137, 149, 173, 191, 197, 233, 239, 257, 311, 317, 347, 383, 401, 431, 443, 449, 467, 479, 509, 569, 593, 617, 641, 683, 719, 761, 773, 827, 857, 929, 941, 947, 1031, 1061, 1097, 1163, 1181, 1223, 1229

Offset: 1

Zak Seidov, Mar 16 2009

nonn

Hence 5p are terms in A157931, A062721 and (except of 25) in A043326.

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

Cf. A043326 Numbers n such that n is a product of two different primes and n-2 is prime, A062721 Numbers n such that n is a product of two primes and n-2 is prime, A157931 Numbers that are both the sum and the product of two primes.

Select[Prime[Range[300]],PrimeQ[5#-2]&] (* Harvey P. Dale, Apr 10 2015 *)

isok(p) = isprime(p) && isprime(5*p-2);  \\ Michel Marcus, Oct 16 2013

A107986 Composite numbers of the form p+2 where p is prime.

Original entry on oeis.org

4, 9, 15, 21, 25, 33, 39, 45, 49, 55, 63, 69, 75, 81, 85, 91, 99, 105, 111, 115, 129, 133, 141, 153, 159, 165, 169, 175, 183, 195, 201, 213, 225, 231, 235, 243, 253, 259, 265, 273, 279, 285, 295, 309, 315, 319, 333, 339, 351, 355, 361, 369, 375, 381, 385, 391

Offset: 1

Cino Hilliard, Jun 13 2005

This sequence is analogous to the sequence formed by the Goldbach-Euler conjecture that every even number greater than 2 is the sum of 2 primes. If p + 2 is prime then p and p + 2 are twin primes. The number of terms in this sequence is infinite. This follows immediately from the proof that the number of primes p is infinite. Conjecture: The ratio of the number of terms in this sequence to Pi(n) tends to a limit < 1.

The first term in this sequence that is not also in A062721 is 45 = 3^2 * 5. - Alonso del Arte, May 03 2014

Cf. A062721, A067774, A107987.

Select[Range[4, 399], Not[PrimeQ[#]] && PrimeQ[# - 2] &] (* Alonso del Arte, May 03 2014 *)

a(n) = A067774(n) + 2. - Amiram Eldar, Jul 05 2024

A043326 Numbers k such that k is a product of two different primes and k-2 is prime.

Original entry on oeis.org

15, 21, 33, 39, 55, 69, 85, 91, 111, 115, 129, 133, 141, 159, 183, 201, 213, 235, 253, 259, 265, 295, 309, 319, 339, 355, 381, 391, 403, 411, 445, 451, 469, 481, 489, 493, 501, 505, 511, 543, 559, 565, 573, 579, 589, 633, 649, 655, 679, 685

Offset: 1

Olivier Gérard, Jul 19 2001

nonn

Zak Seidov and Michael De Vlieger, Table of n, a(n) for n = 1..21728 (first 10000 terms from Zak Seidov)

Cf. A006512, A062721.

Subsequence of A006881.

Select[Range[700],PrimeOmega[#]==PrimeNu[#]==2&&PrimeQ[#-2]&] (* Harvey P. Dale, Feb 01 2014 *)
Select[Prime[Range[200]]+2,PrimeOmega[#]==PrimeNu[#]==2&] (* Harvey P. Dale, Oct 13 2018 *)

is(n)=isprime(n-2) && bigomega(n)==2 && !issquare(n) \\ Charles R Greathouse IV, Jul 20 2014

A089268 Odd semiprimes m such that m-2 is composite.

Original entry on oeis.org

35, 51, 57, 65, 77, 87, 93, 95, 119, 121, 123, 143, 145, 155, 161, 177, 185, 187, 203, 205, 209, 215, 217, 219, 221, 237, 247, 249, 267, 287, 289, 291, 299, 301, 303, 305, 321, 323, 327, 329, 335, 341, 365, 371, 377, 393, 395, 407, 413, 415, 417, 427, 437

Offset: 1

Reinhard Zumkeller, Oct 28 2003

nonn

A087942(a(n)) = 0.

Assuming Goldbach's conjecture that every even number greater than 2 is the sum of two primes, these are the numbers that are the product of two primes but not the sum of two primes. - Michael B. Porter, Feb 08 2013

Bill McEachen, Table of n, a(n) for n = 1..10000

Cf. A062721, A046315, A001358.

Take[Select[Union[Flatten[Table[Prime[i] Prime[j], {i, 2, 25}, {j, 2, 25}]]], Not[PrimeQ[# - 2]] &], 50] (* Alonso del Arte, Feb 08 2013 *)

isok(m) = (m%2) && (bigomega(m)==2) && !isprime(m-2); \\ Michel Marcus, Oct 19 2021

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.

A121885 Excess of n-th prime over previous semiprime.

Original entry on oeis.org

1, 1, 1, 3, 2, 4, 1, 3, 5, 2, 2, 4, 1, 2, 1, 3, 2, 2, 4, 2, 1, 2, 2, 6, 8, 1, 3, 2, 4, 2, 3, 5, 3, 5, 2, 2, 1, 4, 1, 3, 4, 6, 3, 5, 2, 2, 1, 3, 7, 2, 4, 2, 3, 1, 2, 4, 3, 3, 5, 2, 2, 2, 4, 3, 2, 2, 1, 3, 7, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 4, 4, 6, 2, 6, 2, 3, 3

Offset: 3

Jonathan Vos Post, Aug 31 2006

See: A102415 Greatest semiprime less than n-th prime. See: A102414 Smallest semiprime greater than n-th prime.

T. D. Noe, Table of n, a(n) for n = 3..10000

Cf. A000040, A001358, A062721, A077068, A102414, A102415.

SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])]; Table[i = Prime[n] - 1; While[! SemiPrimeQ[i], i--]; Prime[n] - i, {n, 3, 100}] (* T. D. Noe, Oct 08 2012 *)
eps[n_]:=Module[{c=n-1},While[PrimeOmega[c]!=2,c--];n-c]; Table[eps[n],{n,Prime[Range[3,90]]}] (* Harvey P. Dale, Aug 12 2014 *)

dsemi(n)= { local(k=0); if(isprime(n),k=0;while(bigomega(n-k)<>2&&kAntonio Roldán, Oct 08 2012

a(n) = Min{A000040(n)-s for s < A000040(n) and s in A001358(k)}. a(n) = A000040(n) - A102415(n).

Extended by T. D. Noe, Oct 08 2012

A121884 Excess of n-th semiprime over previous prime.

Original entry on oeis.org

1, 1, 2, 3, 1, 2, 2, 3, 2, 3, 2, 3, 4, 1, 2, 3, 2, 4, 2, 4, 5, 1, 4, 2, 1, 4, 3, 2, 3, 4, 2, 4, 5, 6, 3, 2, 2, 5, 6, 8, 9, 10, 2, 2, 3, 2, 3, 4, 6, 7, 4, 1, 2, 4, 3, 2, 4, 5, 2, 4, 6, 1, 2, 3, 4, 6, 7, 10, 2, 3, 4, 6, 7, 8, 10, 3, 2, 4, 6, 8, 2, 3, 2, 5, 2, 4, 3, 1, 4, 6, 8, 2, 5, 6, 8, 9, 10, 12, 2, 1, 2, 4, 6

Offset: 1

Jonathan Vos Post, Aug 31 2006

a(n) = 1 iff n is in A077068. a(n) = 2 iff n is in A062721.

Robert G. Wilson v, Table of n, a(n) for n=1..1000

Cf. A000040, A001358, A062721, A077068.

semiPrimeQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; lst1 = Select[ Range@ 325, semiPrimeQ@# &]; lst = Select[ Range@ 500, semiPrimeQ@# &]; lst - (NextPrime[ #, -1] & /@ lst) (* Robert G. Wilson v, Mar 16 2009 *)
#-NextPrime[#,-1]&/@Select[Range[400],PrimeOmega[#]==2&] (* Harvey P. Dale, Feb 15 2018 *)

a(n) = Min{A001358(n)-p for p < A001358(n) and p in A000040(k)}.

a(31)-a(103) from Robert G. Wilson v, Mar 16 2009

cp's OEIS Frontend

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

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A157931 Numbers that are both the sum and the product of two primes.

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A158318 Primes p such that 5p-2 is prime.

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A107986 Composite numbers of the form p+2 where p is prime.

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A043326 Numbers k such that k is a product of two different primes and k-2 is prime.

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A089268 Odd semiprimes m such that m-2 is composite.

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

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