A198327 -id:A198327 - cp's OEIS frontend

A200925 Numbers k such that Omega(k) = Omega(k - Omega(k)), where Omega = A001222.

1, 3, 6, 30, 35, 40, 45, 51, 57, 60, 66, 78, 87, 88, 93, 95, 102, 104, 105, 117, 121, 123, 136, 140, 143, 145, 156, 161, 174, 175, 185, 187, 203, 205, 215, 217, 219, 221, 232, 237, 245, 249, 258, 261, 267, 282, 285, 289, 291, 301, 303, 305, 321, 323, 325, 329

Offset: 1

Michel Lagneau, Nov 24 2011

nonn

Omega=A001222: Number of prime divisors counted with multiplicity.

A198327 is a subsequence because, if k and k-2 are semiprimes, Omega(k) = 2, and k - 2 is semiprime, therefore Omega(k-2) = 2.

			a(5) = 35 because Omega(35) = 2 and Omega(35 - 2) = Omega(33) = 2.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A001222.

with(numtheory):
isA200925 := proc(n)
        bigomega(n-bigomega(n)) = bigomega(n) ;
end proc:
for n from 1 to 400 do
        if isA200925(n) then printf("%d,",n) ; end if;
end do: # R. J. Mathar, Nov 28 2011

Select[Range[329], PrimeOmega[#] == PrimeOmega[# - PrimeOmega[#]] &] (* T. D. Noe, Nov 29 2011 *)

A241764 Semiprimes sp such that sp-3 is also semiprime.

Original entry on oeis.org

9, 25, 38, 49, 58, 65, 77, 85, 94, 118, 121, 122, 145, 146, 158, 161, 169, 205, 206, 209, 217, 218, 221, 262, 265, 298, 301, 302, 305, 326, 329, 358, 361, 365, 394, 398, 454, 469, 481, 485, 505, 514, 517, 529, 538, 545, 554, 562, 565, 586, 589, 614

Offset: 1

K. D. Bajpai, Apr 29 2014

nonn

Also semiprimes of the form 2^x - x.

The primes of the form 2^x - x are in A081296.

			a(3)= 38 = 2*19, which is semiprime: 38-3 = 35 = 5*7 is also semiprime.
a(5)= 58 = 2*29, which is semiprime: 58-3 = 55 = 5*11 is also semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..1100

Cf. A001358, A092207, A123017, A198327.

with(numtheory): A241764:= proc(); if bigomega(x)=2 and bigomega(x-3)=2 then RETURN (x); fi; end: seq(A241764 (), x=1..2000);

Flatten[Position[Partition[Table[If[PrimeOmega[n]==2,1,0],{n,700}],4,1], ?(First[ #] ==Last[#]==1&),{1},Heads->False]]+3 (* _Harvey P. Dale, Dec 21 2015 *)

for(k=1, 500, if(bigomega(k)==2 && bigomega(k-3)==2, print1(k, ", "))) \\ Colin Barker, May 07 2014

A241817 Semiprimes sp such that sp-3 is prime.

Original entry on oeis.org

6, 10, 14, 22, 26, 34, 46, 62, 74, 82, 86, 106, 134, 142, 166, 194, 202, 214, 226, 254, 274, 314, 334, 362, 382, 386, 422, 446, 466, 482, 502, 526, 566, 622, 634, 662, 694, 746, 842, 862, 866, 886, 914, 922, 974, 1042, 1094, 1126, 1154, 1174, 1226, 1234, 1262

Offset: 1

K. D. Bajpai, Apr 29 2014

nonn

Even numbers of the form 2p, p prime, that can be expressed as the sum of two primes in at least two ways as 2p = p + p = 3 + (2p-3). For example, 34 is in the sequence because 34 = 2*17 = 17 + 17 = 3 + 31. These are the only numbers that have Goldbach partitions with both a minimum and a maximum possible difference between their prime parts, i.e., |p-p| = 0 and |(2p-3)-3| = 2p-6 respectively. - Wesley Ivan Hurt, Apr 08 2018

			a(2) = 10 = 2*5, which is semiprime and 10-3 = 7 is a prime.
a(6) = 34 = 2*17, which is semiprime and 34-3 = 31 is a prime.

K. D. Bajpai, Table of n, a(n) for n = 1..2500

Cf. A001358, A063908, A092207, A123017, A198327.

with(numtheory): A241817:= proc(); if bigomega(x)=2 and isprime(x-3) then  RETURN (x); fi; end: seq(A241817 (), x=1..3000);

2 Select [Prime[Range[5!]], PrimeQ[2 # - 3] &] (* Vincenzo Librandi, Apr 10 2018 *)
Select[Range[1500],PrimeOmega[#]==2&&PrimeQ[#-3]&] (* Harvey P. Dale, Oct 14 2018 *)

a(n) = 2 * A063908(n). - Wesley Ivan Hurt, Apr 08 2018

A338049 a(n) is the smallest prime that is not less than prime(n) and is such that prime(n)*a(n)+2 is semiprime.

Original entry on oeis.org

2, 11, 11, 7, 11, 19, 17, 29, 29, 29, 37, 37, 43, 43, 61, 53, 61, 71, 89, 79, 73, 79, 83, 103, 97, 103, 107, 109, 113, 127, 131, 151, 137, 151, 157, 197, 167, 167, 173, 181, 211, 199, 191, 197, 199, 211, 227, 257, 257, 241, 233, 251, 257, 251, 263, 263, 269

Offset: 1

N. J. A. Sloane, Oct 08 2020, based on an email from Todor Szimeonov, Oct 07 2020

nonn

Motivated by a question about arranging square tiles in a rectangle.

Cf. A000040, A001358, A092207, A198327, A338048.

cp's OEIS Frontend

A200925 Numbers k such that Omega(k) = Omega(k - Omega(k)), where Omega = A001222.

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A241764 Semiprimes sp such that sp-3 is also semiprime.

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A241817 Semiprimes sp such that sp-3 is prime.

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A338049 a(n) is the smallest prime that is not less than prime(n) and is such that prime(n)*a(n)+2 is semiprime.

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