A175612 -id:A175612 - cp's OEIS frontend

A175648 Semiprimes m such that m+4 is also semiprime.

6, 10, 21, 22, 34, 35, 51, 58, 65, 82, 87, 91, 111, 115, 118, 119, 129, 141, 142, 155, 183, 201, 202, 205, 209, 213, 214, 215, 217, 249, 274, 287, 291, 295, 298, 299, 301, 305, 319, 323, 335, 358, 361, 377, 382, 391, 394, 403, 407, 411, 413, 447, 454, 469, 478

Offset: 1

Juri-Stepan Gerasimov, Aug 02 2010

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A001358, A001359, A175612.

select(t -> numtheory:-bigomega(t)=2 and numtheory:-bigomega(t+4)=2, [$1..1000]); # Robert Israel, Feb 12 2015

Select[Range@ 480, And[PrimeOmega@ # == 2, PrimeOmega[# + 4] == 2] &] (* Michael De Vlieger, Dec 09 2015 *)

isok(n) = (bigomega(n) == 2) && (bigomega(n+4) == 2); \\ Michel Marcus, Feb 12 2015

Corrected (309 removed) by R. J. Mathar, Aug 03 2010

Definition corrected by Zak Seidov, Feb 12 2015

A175634 Chen semiprimes: semiprimes m such that m+4 is either a prime or a semiprime.

Original entry on oeis.org

6, 9, 10, 15, 21, 22, 25, 33, 34, 35, 39, 49, 51, 55, 57, 58, 65, 69, 82, 85, 87, 91, 93, 111, 115, 118, 119, 123, 129, 133, 141, 142, 145, 155, 159, 169, 177, 183, 187, 201, 202, 205, 209, 213, 214, 215, 217, 219, 235, 237, 247, 249, 253, 259, 265, 267, 274, 287

Offset: 1

Juri-Stepan Gerasimov, Aug 01 2010

nonn

Cf. A109611, A175612.

from sympy.ntheory.factor_ import primeomega, isprime
def issemiprime(n): return primeomega(n) == 2
def ok(n): return issemiprime(n) and (issemiprime(n+4) or isprime(n+4))
print(list(filter(ok, range(1, 288)))) # Michael S. Branicky, Apr 14 2021

Corrected (86 replaced by 85) by R. J. Mathar, Aug 03 2010

A175663 Maximal run length of primes of the form n, n+2, n+23, n+23*5,..

Original entry on oeis.org

0, 1, 2, 0, 3, 0, 1, 0, 0, 0, 4, 0, 1, 0, 0, 0, 5, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 9, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 6, 0, 1, 0, 0

Offset: 1

Vladislav-Stepan Malakovsky & Juri-Stepan Gerasimov, Aug 04 2010

nonn

			a(107)=8 because 107=prime, 107+2=109=prime, 107+2*3=113=prime, 107+2*3*5=137=prime, 107+2*3*5*7=317=prime, 107+2*3*5*7*11=2417=prime, 107+2*3*5*7*11*13=30137=prime, 107+2*3*5*7*11*13*17=510617=prime.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A006512 (greater of twin primes), A175612 (list of twin semiprimes), A175648 (lesser of twin semiprimes).

Cf. also A175682.

A002110 := proc(n) option remember; mul(ithprime(i),i=1..n) ; end proc:
A175663 := proc(n) if isprime(n) then for p from 1 do if not isprime(n+A002110(p)) then return p ; end if; end do: else return 0 ; end if; end proc:
seq(A175663(n),n=1..120) ; # R. J. Mathar, Aug 07 2010

Array[If[PrimeQ@ #, Block[{s = {1}}, While[PrimeQ[# + Times @@ Prime@ s], AppendTo[s, s[[-1]] + 1]]; Last@ s], 0] &, 105] (* Michael De Vlieger, Jan 03 2019 *)

A175663(n) = if(!isprime(n),0,my(pr=2); for(k=1, oo, if(!isprime(pr+n), return(k)); pr *= prime(1+k))); \\ Antti Karttunen, Jan 03 2019

a(n) <= A175682(n). - Antti Karttunen, Jan 03 2019

A175664 Greater of twin semiprimes.

Original entry on oeis.org

10, 14, 25, 26, 38, 39, 55, 62, 69, 86, 91, 95, 115, 119, 122, 123, 133, 145, 146, 159, 187, 205, 206, 209, 213, 217, 218, 219, 221, 253, 278, 291, 295, 299, 302, 303, 305, 309, 323, 327, 339, 362, 365, 381, 386, 395, 398, 407, 411, 415, 417, 451, 458, 473

Offset: 1

Juri-Stepan Gerasimov, Aug 04 2010

nonn

Semiprimes m such that m-4 is also semiprime.

			a(1)=10 because 10 (semiprime) - 4 = 6 (semiprime);
a(2)=14 because 14 (semiprime) - 4 = 10 (semiprime).

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

Cf. A006512 (greater of twin primes), A175612 (list of twin semiprimes), A175648 (lesser of twin semiprimes).

A175664 := proc(n) option remember; if n = 1 then 10; else for a from procname(n-1)+1 do if numtheory[bigomega](a) = 2 and numtheory[bigomega](a-4) = 2 then return a; end if; end do: end if; end proc: seq(A175664(n),n=1..100) ; # R. J. Mathar, Aug 07 2010

SequencePosition[Table[If[PrimeOmega[n]==2,1,0],{n,500}],{1,,,_,1}][[All,2]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 16 2017 *)

a(n) = 4 + A175648(n). - R. J. Mathar, Aug 07 2010

Corrected (313 removed) by R. J. Mathar, Aug 07 2010

A175668 First differences of A175648.

Original entry on oeis.org

4, 11, 1, 12, 1, 16, 7, 7, 17, 5, 4, 20, 4, 3, 1, 10, 12, 1, 13, 28, 18, 1, 3, 4, 4, 1, 1, 2, 32, 25, 13, 4, 4, 3, 1, 2, 4, 14, 4, 12, 23, 3, 16, 5, 9, 3, 9, 4, 4, 2, 34, 7, 15, 9, 3, 4, 4, 4, 4, 4, 10, 4, 14, 4, 5, 24, 17, 43, 7, 38, 14, 4, 9, 1, 4, 4, 10, 4, 28, 4, 14, 4, 14, 4, 4, 10, 4, 10

Offset: 1

Juri-Stepan Gerasimov, Aug 05 2010

nonn

Distance between twin semiprime pairs.

Cf. A053319, A175612, A175648.

A175648 := proc(n) option remember; if n = 1 then 6; else for a from procname(n-1)+1 do if numtheory[bigomega](a) = 2 and numtheory[bigomega](a+4) = 2 then return a; end if; end do: end if; end proc:
A175668 := proc(n) A175648(n+1)-A175648(n) ; end proc:
seq(A175668(n),n=1..100) ; # R. J. Mathar, Aug 07 2010

Terms from a(33) on corrected by R. J. Mathar, Aug 07 2010

A175710 Numbers k such that k-4, k and k+4 are all semiprimes.

Original entry on oeis.org

10, 91, 115, 119, 205, 209, 213, 217, 291, 295, 299, 305, 323, 407, 411, 485, 489, 493, 497, 501, 515, 533, 685, 699, 703, 717, 749, 767, 785, 789, 803, 917, 955, 989, 1007, 1077, 1115, 1137, 1141, 1145, 1195, 1199, 1203, 1207, 1257, 1267, 1333, 1343, 1347

Offset: 1

Juri-Stepan Gerasimov, Aug 12 2010

nonn

If k is a term, one of k-4, k and k+4 is 3 times a prime. - Robert Israel, Mar 25 2025

			a(1)=10 because 10-4=6, 10 and 10+4=14 are all semiprimes.

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

Cf. A001358, A175612, A175648, A175664.

SP:= select(t -> numtheory:-bigomega(t) = 2, {$1..10000}):
sort(convert(SP intersect (SP +~ 4) intersect (SP -~ 4),list)); # Robert Israel, Mar 25 2025

Corrected (299, 411 etc inserted) by R. J. Mathar, Aug 13 2010

Name edited by Robert Israel, Mar 25 2025

A175765 Pairs (t,t+8) of 3-almost primes t separated by 8.

Original entry on oeis.org

12, 20, 20, 28, 42, 50, 44, 52, 68, 76, 70, 78, 102, 110, 116, 124, 117, 125, 130, 138, 164, 172, 174, 182, 182, 190, 222, 230, 230, 238, 236, 244, 238, 246, 258, 266, 282, 290, 284, 292, 310, 318, 325, 333, 366, 374, 402, 410, 404, 412, 410, 418, 418, 426, 426, 434

Offset: 1

Juri-Stepan Gerasimov, Aug 30 2010

Pairs (12,20), (20,28), (42,50) etc, sorted by the smaller member, such that both numbers are members of A014612 and their difference is 8.

Cf. A014612, A077800, A175612.

Corrected (236 replaced by 238, 258, 266 inserted etc.) by R. J. Mathar, Sep 01 2010

A175766 Greater of "twin numbers": Omega(n) = Omega(n - 2^Omega(n)), where Omega is A001222.

Original entry on oeis.org

5, 7, 10, 13, 14, 19, 20, 25, 26, 28, 31, 38, 39, 40, 43, 50, 52, 55, 56, 61, 62, 69, 73, 76, 78, 80, 86, 91, 95, 100, 103, 104, 109, 110, 112, 115, 119, 122, 123, 124, 125, 133, 138, 139, 145, 146, 151, 152, 156, 159, 160, 172, 181, 182, 187, 190, 193, 199, 200, 205, 206, 208

Offset: 1

Juri-Stepan Gerasimov, Sep 01 2010

nonn

Generalizes the twin primes. Superset of A006512.

			"Twin numbers" are (3, 5), (5, 7), (6, 10), (11, 13), (10, 14), (17, 19), (12, 20), (21, 25), (22, 26), (20, 28), (29, 31), (34, 38), (35, 39), (24, 40), (41, 43), (42, 50), (44, 52), (51, 55), (40, 56), (59, 61), (58, 62), (65, 69), (71, 73), ...

Cf. A001222, A006512, A175612.

for(n=1,1e3,o=bigomega(n);trap(, ,if(o==bigomega(n-2^o),print1(n","))))

Entry rewritten by Charles R Greathouse IV, Sep 01 2010

A175636 Maximal (nonzeros) run length of semiprimes of the form k+4, k+8, k+12,.., where k is a non-semiprime.

Original entry on oeis.org

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

Offset: 1

Vladislav-Stepan Malakovsky & Juri-Stepan Gerasimov, Aug 01 2010

nonn

			a(1)=1 because k=0=non-semiprime and 0+4=4=semiprime; a(2)=3 because k=2=non-semiprime and 2+4=6=semiprime, 2+8=10=semiprime, 2+12=14=semiprime; a(3)=1 because k=5=non-semiprime and 5+4=9=semiprime.

Cf. A175612, A175632.

A175767 List of pairs of "twin numbers": Omega(n) = Omega(n - 2^Omega(n)), where Omega is A001222.

Original entry on oeis.org

3, 5, 5, 7, 6, 10, 11, 13, 10, 14, 17, 19, 12, 20, 21, 25, 22, 26, 20, 28, 29, 31, 34, 38, 35, 39, 24, 40, 41, 43, 42, 50, 44, 52, 51, 55, 40, 56, 59, 61, 58, 62, 65, 69, 71, 73, 68, 76, 70, 78, 48, 80, 82, 86, 87, 91, 91, 95, 84, 100, 101, 103, 88, 104, 107, 109, 102, 110, 80, 112, 111

Offset: 1

Juri-Stepan Gerasimov, Sep 01 2010

nonn

The k-primes = numbers that are divisible by exactly k primes (counted with multiplicity).

			"Twin numbers" are (3, 5), (5, 7), (6, 10), (11, 13), (10, 14), (17, 19), (12, 20), (21, 25), (22, 26), (20, 28), (29, 31), (34, 38), (35, 39), (24, 40), (41, 43), (42, 50), (44, 52), (51, 55), (40, 56), (59, 61), (58, 62), (65, 69), (71, 73), ...

Cf. A175766, A001222, A077800, A175612, A175765.

Flatten[{#[[3]],#[[1]]}&/@Select[Table[{n,PrimeOmega[n],n-2^PrimeOmega[n],PrimeOmega[ n-2^PrimeOmega[n]]},{n,150}],#[[2]]==#[[4]]&]] (* Harvey P. Dale, Mar 23 2024 *)

a(2n+1) = A175766(n) = a(2n) + 2^A001222(a(2n+1)).

Rewritten by Charles R Greathouse IV, Sep 03 2010

cp's OEIS Frontend

A175648 Semiprimes m such that m+4 is also semiprime.

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A175634 Chen semiprimes: semiprimes m such that m+4 is either a prime or a semiprime.

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A175663 Maximal run length of primes of the form n, n+2, n+2*3, n+2*3*5,..

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A175664 Greater of twin semiprimes.

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A175668 First differences of A175648.

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A175710 Numbers k such that k-4, k and k+4 are all semiprimes.

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Maple

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A175765 Pairs (t,t+8) of 3-almost primes t separated by 8.

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Extensions

A175766 Greater of "twin numbers": Omega(n) = Omega(n - 2^Omega(n)), where Omega is A001222.

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A175663 Maximal run length of primes of the form n, n+2, n+23, n+23*5,..