A056809 -id:A056809 - cp's OEIS frontend

A092127 Numbers n such that n, n+2, n+4, n+6, n+8 are semiprimes.

213, 1133, 1343, 1383, 1385, 1835, 1937, 2315, 2807, 3091, 3093, 3095, 3147, 3977, 4313, 4529, 4835, 5089, 5609, 5611, 6185, 6533, 7141, 8129, 8131, 8133, 8135, 9753, 9755, 9983, 9985, 9987, 9989, 10401, 10403, 11013, 11015, 11099, 11663, 12053

Offset: 1

Zak Seidov, Feb 22 2004

Semiprimes in arithmetic progression. All terms are odd, see also A056809.

Cf. A056809, A070552, A092125, A092126, A092128, A092129.

PrimeFactorExponentsAdded[n_] := Plus @@ Flatten[Table[ #[[2]], {1}] & /@ FactorInteger[n]]; Select[ Range[ 12054], PrimeFactorExponentsAdded[ # ] == PrimeFactorExponentsAdded[ # + 2] == PrimeFactorExponentsAdded[ # + 4] == PrimeFactorExponentsAdded[ # + 6] == PrimeFactorExponentsAdded[ # + 8] == 2 &] (* Robert G. Wilson v, Feb 24 2004 *)

A109373 Semiprimes of the form semiprime + 1.

Original entry on oeis.org

10, 15, 22, 26, 34, 35, 39, 58, 86, 87, 94, 95, 119, 122, 123, 134, 142, 143, 146, 159, 178, 202, 203, 206, 214, 215, 218, 219, 254, 299, 302, 303, 327, 335, 362, 382, 394, 395, 446, 447, 454, 482, 502, 515, 527, 538, 543, 554, 566, 623, 634, 635, 695, 698

Offset: 1

Jonathan Vos Post, Aug 24 2005

			a(1) = 10 because (3*3+1)=(2*5) = 10.
a(2) = 15 because (2*7+1)=(3*5) = 15.
a(3) = 22 because (3*7+1)=(2*11) = 22.
a(4) = 26 because (5*5+1)=(2*13) = 26.
a(5) = 34 because (3*11+1)=(2*17) = 34.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Primes are in A000040. Semiprimes are in A001358.
Primes of the form semiprime + 1 are in A005385 (safe primes).
Semiprimes of the form semiprime + 1 are in this sequence.
3-almost primes of the form semiprime + 1 are in A109067.
4-almost primes of the form semiprime + 1 are in A109287.
5-almost primes of the form semiprime + 1 are in A109383.
Least n-almost prime of the form semiprime + 1 are in A128665.
Cf. A056809, A077065, A079148, A086005, A086006, A092307.
Subsequence of A088707; A064911.

a109373 n = a109373_list !! (n-1)
a109373_list = filter ((== 1) . a064911) a088707_list
-- Reinhard Zumkeller, Feb 20 2012

fQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; Select[ Range[ 700], fQ[ # - 1] && fQ[ # ] &] (* Robert G. Wilson v *)
With[{sps=Select[Range[700],PrimeOmega[#]==2&]},Transpose[Select[ Partition[ sps,2,1],#[[2]]-#[[1]]==1&]][[2]]] (* Harvey P. Dale, Sep 05 2012 *)

is(n)=bigomega(n)==2 && bigomega(n-1)==2 \\ Charles R Greathouse IV, Jan 31 2017

a(n) is in this sequence iff a(n) is in A001358 and (a(n)-1) is in A001358.

a(n) = A070552(n) + 1.

Extended by Ray Chandler and Robert G. Wilson v, Aug 25 2005

Edited by Ray Chandler, Mar 20 2007

A113789 Numbers n such that n, n+1 and n+2 are products of exactly 3 primes.

Original entry on oeis.org

170, 244, 284, 428, 434, 506, 602, 603, 604, 637, 962, 1074, 1083, 1084, 1130, 1244, 1309, 1412, 1434, 1490, 1532, 1556, 1586, 1604, 1634, 1675, 1771, 1885, 1946, 2012, 2013, 2035, 2084, 2091, 2092, 2162, 2396, 2404, 2522, 2523, 2524, 2525, 2634, 2635

Offset: 1

Jonathan Vos Post, Jan 21 2006

3-almost prime analog of A056809.
This sequence consists of the least of 3 consecutive 3-almost primes, or 4 or more consecutive 3-almost primes (i.e. n, n+1 and n+2 but not excluding n+3 also 3-almost prime). A067813 has some runs of up to 7 consecutive 3-almost primes (i.e. starting 211673). But there cannot be 8 consecutive 3-almost primes, as every run of 8 consecutive positive integers contains exactly one multiple of 8 = 2^3 and only 8 of all positive multiples of 8 is a 3-almost prime (i.e., all larger multiples have at least 4 prime factors, with multiplicity).
Primes counted with multiplicity. - Harvey P. Dale, Sep 04 2019

			a(1) = 170 because 170 = 2 * 5 * 17 and 171 = 3^2 * 19 and 172 = 2^2 * 43 are all 3-almost primes.
a(2) = 244 because 244 = 2^2 * 61 and 245 = 5 * 7^2 and 246 = 2 * 3 * 41 are all 3-almost primes.
a(3) = 284 because 284 = 2^2 * 71 and 285 = 3 * 5 * 19 and 286 = 2 * 11 * 13 are all 3-almost primes.
a(4) = 428 because 428 = 2^2 * 107 and 429 = 3 * 11 * 13 and 430 = 2 * 5 * 43 are all 3-almost primes.
a(5) = 434 because 434 = 2 * 7 * 31 and 435 = 3 * 5 * 29 and 436 = 2^2 * 109 are all 3-almost primes.
a(6) = 506 because 506 = 2 * 11 * 23 and 507 = 3 * 13^2 and 508 = 2^2 * 127 all 3-almost primes.
a(7), a(8), a(9) = 602, 603, 604 because of the record-setting 5 consecutive 3-almost primes: 602 = 2 * 7 * 43; 603 = 3^2 * 67; 604 = 2^2 * 151; 605 = 5 * 11^2; 606 = 2 * 3 * 101.

D. W. Wilson, Table of n, a(n) for n = 1..10000

Cf. A014612, A056809, A067813.

Subsequence of A180117.

fQ[n_] := Plus @@ Last /@ FactorInteger@n == 3; Select[ Range@2664, fQ@# && fQ[ # + 1] && fQ[ # + 2] &] (* Robert G. Wilson v, Jan 21 2006 *)
SequencePosition[Table[If[PrimeOmega[n]==3,1,0],{n,3000}],{1,1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 04 2019 *)

is(n)=bigomega(n)==3 && bigomega(n+1)==3 && bigomega(n+2)==3 \\ Charles R Greathouse IV, Feb 05 2017

n, n+1 and n+2 are all elements of A014612.

Edited, corrected and extended by Robert G. Wilson v, Jan 21 2006

A092209 Smallest number k such that k, k+2, k+4, ..., k+2n are semiprimes.

Original entry on oeis.org

4, 4, 91, 213, 213, 1383, 3091, 8129

Offset: 0

Robert G. Wilson v and Zak Seidov, Feb 24 2004

Semiprimes in arithmetic progression. All terms are odd, except for the first two. See also A056809.

Eric Weisstein's World of Mathematics, Semiprime.

First entry in A001358, A092207, A092125, A092126, A092127, A092128, A092129, A082919 respectively.

A123017 Semiprimes k such that k+3 is also a semiprime.

Original entry on oeis.org

6, 22, 35, 46, 55, 62, 74, 82, 91, 115, 118, 119, 142, 143, 155, 158, 166, 202, 203, 206, 214, 215, 218, 259, 262, 295, 298, 299, 302, 323, 326, 355, 358, 362, 391, 395, 451, 466, 478, 482, 502, 511, 514, 526, 535, 542, 551, 559, 562, 583, 586, 611, 623, 626

Offset: 1

Jonathan Vos Post, Nov 04 2006

When a(n+1) = a(n) + 3 we have that a(n) is a semiprime such that a(n) and a(n)+3 and a(n) + 3 + 3 are all semiprimes, hence at least 3 semiprimes in arithmetic progression with common difference 3. This subsequence begins 115, 155. There cannot be 4 semiprimes in arithmetic progression with common difference 3, starting with k, because modulo 4 we have {k, k+3, k+6, k+9} == {k+0, k+3, k+2, k+1} and one of these must be divisible by 4, hence a nonsemiprime (eliminating k = 4 by inspection).

			a(1) = 6 because 6 = 2 * 3 is semiprime and 6 + 3 = 9 = 3^2 is semiprime.
a(2) = 22 because 22 = 2 * 11 and 22 + 3 = 25 = 5^2.
a(3) = 35 because 35 = 5 * 7  and 35 + 3 = 38 = 2 * 19.
a(4) = 46 because 46 = 2 * 23 and 46 + 3 = 49 = 7^2.
a(5) = 55 because 55 = 5 * 11 and 55 + 3 = 58 = 2 * 29.

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

Cf. A001358, A056809, A070552, A092125, A092126, A092127, A092128, A092129.

semiprimeQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; Select[ Range@ 670, semiprimeQ[ # ] && semiprimeQ[ # + 3] &] (* Robert G. Wilson v, Aug 31 2007 *)
SequencePosition[Table[If[PrimeOmega[n]==2,1,0],{n,700}],{1,,,1}][[All, 1]] (* Requires Mathematica version 10 or later *)  (* Harvey P. Dale, Mar 03 2017 *)

{a(n)} = {k such that k is in A001358 and k+3 is in A001358}.

More terms from Robert G. Wilson v, Aug 31 2007

A124057 Numbers n such that n, n+1, n+2 and n+3 are products of exactly 3 primes.

Original entry on oeis.org

602, 603, 1083, 2012, 2091, 2522, 2523, 2524, 2634, 2763, 3243, 3355, 4202, 4203, 4921, 4922, 4923, 5034, 5035, 5132, 5203, 5282, 5283, 5785, 5882, 5954, 5972, 6092, 6212, 6476, 6962, 6985, 7730, 7731, 7945, 8393, 8825, 8956, 8972, 9188, 9482, 10011

Offset: 1

Jonathan Vos Post, Nov 03 2006

n such that n, n+1, n+2 and n+3 are 3-almost primes. Subset of A113789 Numbers n such that n, n+1 and n+2 are products of exactly 3 primes. A067813 has some runs of up to 7 consecutive 3-almost primes (i.e. starting 211673). But there cannot be 8 consecutive 3-almost primes, as every run of 8 consecutive positive integers contains exactly one multiple of 8 = 2^3 and only 8 of all positive multiples of 8 is a 3-almost prime (i.e. all larger multiples have at least 4 prime factors, with multiplicity).

A subset of A045940. - Zak Seidov, Nov 05 2006

			a(1) = 602 because 602 = 2 * 7 * 43 and 603 = 3^2 * 67 and 604 = 2^2 * 151 and 605 = 5 * 11^2 are all 3-almost primes.
a(2) = 603 because 603 = 3^2 * 67 and 604 = 2^2 * 151 and 605 = 5 * 11^2 and 606 = 2 * 3 * 101 are all 3-almost primes.
a(3) = 1083 because 1083 = 3 * 19^2 and 1084 = 2^2 * 271 and 1085 = 5 * 7 * 31 and 1086 = 2 * 3 * 181 are all 3-almost primes.
a(4) = 2012 because 2012 = 2^2 * 503, 2013 = 3 * 11 * 61, 2014 = 2 * 19 * 53, 2015 = 5 * 13 * 31.
a(5) = 2091 because 2091 = 3 * 17 * 41, 2092 = 2^2 * 523, 2093 = 7 * 13 * 23, 2094 = 2 * 3 * 349.

D. W. Wilson, Table of n, a(n) for n = 1..10000

Cf. A000040, A014612, A056809, A067813, A113789, A045940.

with(numtheory): a:=proc(n) if bigomega(n)=3 and bigomega(n+1)=3 and bigomega(n+2)=3 and bigomega(n+3)=3 then n else fi end: seq(a(n),n=1..15000); # Emeric Deutsch, Nov 07 2006

okQ[{a_,b_,c_,d_}]:=Union[{a,b,c,d}]=={3}; Flatten[Position[Partition[ PrimeOmega[ Range[11000]],4,1],?(okQ)]] (* _Harvey P. Dale, Sep 23 2012 *)

is(n)=if(!isprime((n+3)\4), return(0)); for(k=n,n+3, if(bigomega(k)!=3, return(0))); 1 \\ Charles R Greathouse IV, Feb 05 2017

list(lim)=my(v=List(),u=v,t); forprime(p=2,lim\4, forprime(q=2,min(lim\(2*p),p), t=p*q; forprime(r=2,min(lim\t,q), listput(u,t*r)))); u=Set(u); for(i=4,#u, if(u[i]-u[i-3]==3, listput(v,u[i-3]))); Vec(v) \\ Charles R Greathouse IV, Feb 05 2017

n, n+1, n+2 and n+3 are all elements of A014612. n and n+1 are elements of A113789.

More terms from Zak Seidov, Nov 05 2006

More terms from Emeric Deutsch, Nov 07 2006

A124570 Array read by antidiagonals: T(d,k) (k >= 1, d = 1,2,3,4,5,6,...) = smallest semiprime s of k (not necessarily consecutive) semiprimes in arithmetic progression with common difference d, or 0 if there is no such arithmetic progression.

Original entry on oeis.org

4, 4, 4, 4, 9, 4, 4, 4, 33, 4, 4, 6, 91, 0, 4, 4, 6, 115, 213, 0, 4, 4, 4, 6, 0, 213, 0, 4, 4, 4, 4, 111, 0, 1383, 0, 4, 4, 14, 9, 0, 201, 0, 3091, 0, 4, 4, 6, 51, 203, 0, 201, 0, 8129, 0, 4, 4, 6, 6, 0, 1333, 0, 481, 0, 0, 0, 4, 4, 4, 77, 69, 0, 1333, 0, 5989, 0, 0, 0, 4

Offset: 1

Jonathan Vos Post, Nov 04 2006

Comment from Hugo van der Sanden Aug 14 2021: (Start)
Row d=12 starts 4 9 9 10 10 469 3937 7343 7343 44719 78937 78937 78937 78937 55952333 233761133 597191343199.
Row d=18 starts 4 4 15 15 15 695 695 1727 7711 13951 13951 46159 400847 400847 400847 65737811 13388955301 934046384293.
Row d=24 starts 4 9 9 10 10 793 4819 6415 7271 14069 14069 14069 31589 67344271 616851797 48299373047 48299373047 20302675273219.
Row d=30 starts 4 4 9 25 25 2779 2779 6347 6347 6347 10811 10811 87109 87109 87109 1513723 15009191 15009191 316612697 316612697 1275591688621.
Row d=36 starts 4 10 10 10 15 1333 3161 4997 6865 34885 142171 834863 1327447 35528747 720945097 63389173477 63389173477 16074207679897 41728758250241.
Row d=42 starts 4 4 9 35 35 2701 2987 2987 7729 26995 26995 185795 307553 708385 708385 708385 1090198367 1819546069 20263042201 5672249016001.
Later terms in these rows are always >10^14. (End)
If p is the least prime that does not divide d, then T(d,k) <= p^2 if k >= p^2 (i.e. any a.p. of length >= p^2 with difference d contains a term divisible by p^2, and the only semiprime divisible by p^2 is p^2). Thus every row is eventually 0. - Robert Israel, Aug 11 2024

			Array begins:
d.\...k=1.k=2.k=3.k=4.k=5..k=6..k=7..k=8....k=9..k=10.k=11..k=12.
0..|..4...4...4...4...4....4....4....4......4....4.....4.....4...
1..|..4...9...33..0...0....0....0....0......0....0.....0.....0....
2..|..4...4...91..213.213..1383.3091.8129...0....0.....0.....0.....
3..|..4...6...115.0...0....0....0....0......0....0.....0.....0.....
4..|..4...6...6...111.201..201..481..5989...0....0.....0.....0....
5..|..4...4...4...0...0....0....0....0......0....0.....0.....0.....
6..|..4...4...9...203.1333.1333.1333.2159...8309.18799.60499.60499
7..|..4...14..51..0...0....0....0....0......0....0.....0.....0.....
8..|..4...6...6...69..473..511..511..112697.0....0.....0.....0.....
9..|..4...6...77..0...0....0....0....0......0....0.....0.....0.....
10.|..4...4...15..289.289..289..1631.13501..0....0.....0.....0.....
11.|..4...4...4...0...0....0....0....0......0....0.....0.....0.....
Example for row 3: 115 = 5 * 23 is semiprime, 115+3 = 118 = 2 * 59 is semiprime and 115+3+3 = 121 = 11^2 is semiprime, so T(3,3) = 115.

R. J. Mathar, Table for d <= 999 (PDF)

Semiprime analog of A124064.

Cf. A125025 (row lengths), A001358, A056809, A070552, A092125, A092126, A092127, A092128, A092129, A124064, A092209 (row d=2), A091016 (row d=6).

T(1,2)=A070552(1). T(1,3)=A056809(1). T(2,4)=A092126(1). T(2,5)=A092127(1). T(2,6)=A092128(1). T(2,7)=A092129(1). T(2,8)=A082919(1). T(3,2)=A123017(1). T(d,1)=A001358(1). - R. J. Mathar, Aug 05 2021

Corrected and extended by R. J. Mathar, Nov 06 2006

Definition clarified by Robert Israel, Aug 11 2024

A123255 Numbers k such that 4k+1, 4k+2, and 4k+3 are all semiprimes.

Original entry on oeis.org

8, 21, 23, 30, 35, 50, 53, 54, 75, 98, 111, 158, 174, 210, 230, 260, 284, 315, 336, 350, 410, 440, 459, 473, 485, 495, 525, 545, 554, 576, 590, 608, 615, 629, 660, 680, 683, 774, 846, 900, 923, 966, 975, 989, 1071, 1103, 1133, 1148, 1220, 1400, 1430, 1463, 1499

Offset: 1

Jonathan Vos Post, Oct 09 2006

4k+4 = 4*(k+1) = 2*2*(k+1) cannot be semiprime as well, as it has at least 3 prime factors with multiplicity. Thus there are no four consecutive semiprimes.

			a(1) = 8 because 4*8+1 = 33 = 3*11 is semiprime and 4*8+2 = 34 = 2*17 is semiprime and 4*8+3 = 35 = 3*5 is semiprime.

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

Cf. A001358, A056809, A070552.

IsSemiprime:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [2..1500] | IsSemiprime(4*n+1) and IsSemiprime(4*n+2) and IsSemiprime(4*n+3) ]; // Vincenzo Librandi, Dec 22 2010

Select[Range[1100],Union[PrimeOmega[4#+{1,2,3}]]=={2}&] (* Harvey P. Dale, Feb 02 2015 *)

from sympy import factorint, isprime
def issemiprime(n):
    return sum(factorint(n).values()) == 2 if n&1 else isprime(n//2)
def ok(n): return all(issemiprime(4*n+i) for i in (2, 1, 3))
print([k for k in range(1500) if ok(k)]) # Michael S. Branicky, Nov 26 2022

{k: 4k+1 is in A001358 AND 4k+2 is in A001358 AND 4k+3 is in A001358}.
{k: 4k+1 is in A070552 AND 4k+2 is in A070552}.
{(A056809(i)-1)/4}.

336 and 680 added by Vincenzo Librandi, Dec 22 2010

A115393 Numbers n such that n, n-1 and n-2 are semiprimes.

Original entry on oeis.org

35, 87, 95, 123, 143, 203, 215, 219, 303, 395, 447, 635, 699, 843, 923, 1043, 1139, 1263, 1347, 1403, 1643, 1763, 1839, 1895, 1943, 1983, 2103, 2183, 2219, 2307, 2363, 2435, 2463, 2519, 2643, 2723, 2735, 3099, 3387, 3603, 3695, 3867, 3903, 3959, 4287

Offset: 1

Zak Seidov, Mar 08 2006

nonn

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

Cf. A056809, A086005, A086006.

filter:= proc(n) local k;
   k:= n mod 3;
   if k = 0 then isprime(n/3) and isprime((n-1)/2) and numtheory:-bigomega(n-2)=2
   elif k= 1 then false
   else isprime((n-2)/3) and isprime((n-1)/2) and numtheory:-bigomega(n)=2
   fi
end proc:
select(filter, [seq(i,i=3..10000,4)]); # Robert Israel, Jun 11 2020

upto=5000;p=Prime[Range[PrimePi[upto/2]]];lim=Floor[Sqrt[upto]]; sp={};k=0; While[k++;p[[k]]<=lim,sp=Join[sp,p[[k]] *Take[p,{k,PrimePi[upto/p[[k]]]}]]]; sp=Sort[sp];Transpose[Select [Partition[sp,3,1], Last[#]-#[[2]]==#[[2]]-First[#]==1&]][[3]] (* Harvey P. Dale, Mar 21 2011 -- semiprime generating portion of program from A001358 *)

a(n)=A056809(n)+2=A086005(n)+1=2*A086006(n)+1.

A179502 Numbers k with the property that k^2, k^2+1 and k^2+2 are all semiprimes.

Original entry on oeis.org

11, 29, 79, 271, 379, 461, 521, 631, 739, 881, 929, 1459, 1531, 1709, 2161, 2239, 2341, 2729, 3049, 3491, 3709, 4021, 4349, 4561, 4691, 5021, 5281, 5851, 5879, 6301, 6329, 6829, 7559, 8009, 9151, 10069, 10099, 10151, 10529, 10891, 11719, 11959, 11969, 13799, 14051, 14159

Offset: 1

Zak Seidov, Jan 08 2011

nonn

From the first 10^6 primes, 6680 are terms of the sequence.
Also, all numbers k^2+1 are twice prime, and k^2+2 are thrice prime.
The number of terms less than 10^m beginning with m = 1: 0, 3, 11, 35, 160, 759, 4668, 30319, 204439, ..., .
The number of terms less than the (10^m)-th prime beginning with m = 1: 2, 7, 33, 165, 941, 6680, 48977, 373627, ..., .

Zak Seidov, Table of n, a(n) for n = 1..6680

n^2 are squares in A070552, which is a subsequence of A056809 (m and m+1 are semiprimes) and A001358 (semiprimes).

The sequence is a subsequence of A048161.

fQ[n_] := PrimeQ[(n^2 + 1)/2] && PrimeQ[(n^2 + 2)/3]; Select[ Prime@ Range@ 1667, fQ] (* Robert G. Wilson v, Feb 26 2011 *)
Select[Range[15000],PrimeOmega[#^2+{0,1,2}]=={2,2,2}&] (* Harvey P. Dale, May 12 2025 *)

{n=10;for(i=1,10^4,n=nextprime(n+1);n2=n^2;if(2==bigomega(n2+1)&&2==bigomega(n2+2),print1(n,",")))}

cp's OEIS Frontend

A092127 Numbers n such that n, n+2, n+4, n+6, n+8 are semiprimes.

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A109373 Semiprimes of the form semiprime + 1.

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A113789 Numbers n such that n, n+1 and n+2 are products of exactly 3 primes.

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A092209 Smallest number k such that k, k+2, k+4, ..., k+2n are semiprimes.

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A123017 Semiprimes k such that k+3 is also a semiprime.

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A124057 Numbers n such that n, n+1, n+2 and n+3 are products of exactly 3 primes.

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A124570 Array read by antidiagonals: T(d,k) (k >= 1, d = 1,2,3,4,5,6,...) = smallest semiprime s of k (not necessarily consecutive) semiprimes in arithmetic progression with common difference d, or 0 if there is no such arithmetic progression.

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A123255 Numbers k such that 4k+1, 4k+2, and 4k+3 are all semiprimes.