A092129 -id:A092129 - cp's OEIS frontend

A082919 Numbers k such that k, k+2, k+4, k+6, k+8, k+10, k+12 and k+14 are semiprimes.

8129, 9983, 99443, 132077, 190937, 237449, 401429, 441677, 452639, 604487, 802199, 858179, 991289, 1471727, 1474607, 1963829, 1999937, 2376893, 2714987, 3111977, 3302039, 3869237, 4622087, 4738907, 6156137, 7813559, 8090759

Offset: 1

Hugo Pfoertner, Apr 22 2003

nonn

Start of a cluster of 8 consecutive odd semiprimes. Semiprimes in arithmetic progression. All terms are odd, see also A056809.
Note that there cannot exist 9 consecutive odd semiprimes. Out of any 9 consecutive odd numbers, one of them will be divisible by 9. The only multiple of 9 which is a semiprime is 9 itself and it is easy to see that's not part of a solution. - Jack Brennen, Jan 04 2006
For the first 500 terms, a(n) is roughly 40000*n^1.6, so the sequence appears to be infinite. Note that (a(n)+4)/3 and (a(n)+10)/3 are twin primes. - Don Reble, Jan 05 2006
All terms == 11 (mod 18). - Zak Seidov, Sep 27 2012
There is at least one even semiprime between k and k+14 for 1812 of the first 10000 terms. - Donovan Johnson, Oct 01 2012
All terms == {29,47,83} (mod 90). - Zak Seidov, Sep 13 2014
Among the first 10000 terms, from all 80000 numbers a(n)+m, m=0,2,4,6,8,10,12,14, the only square is a(4637) + 2 = 23538003241 = 153421^2 (153421 is prime, of course). - Zak Seidov, Dec 22 2014

			a(1)=8129 because 8129=11*739, 8131=47*173, 8133=3*2711, 8135=5*1627, 8137=79*103, 8139=3*2713, 8141=7*1163, 8143=17*479 are semiprimes.

Author of this sequence is Jack Brennen, who provided the terms up to 991289 in a posting to the seqfan mailing list on April 5, 2003.

Donovan Johnson, Table of n, a(n) for n = 1..10000 (terms a(1001) to a(2000) from Zak Seidov)
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A001358, A082130, A082131, A056809, A070552, A092207, A092125, A092126, A092127, A092128, A092129, A092209, A217222 (consecutive semiprimes).

PrimeFactorExponentsAdded[n_] := Plus @@ Flatten[Table[ #[[2]], {1}] & /@ FactorInteger[n]]; Select[ Range[3*10^6], PrimeFactorExponentsAdded[ # ] == PrimeFactorExponentsAdded[ # + 2] == PrimeFactorExponentsAdded[ # + 4] == PrimeFactorExponentsAdded[ # + 6] == PrimeFactorExponentsAdded[ # + 8] == PrimeFactorExponentsAdded[ # + 10] == PrimeFactorExponentsAdded[ # + 12] == PrimeFactorExponentsAdded[ # + 14] == 2 &] (* Robert G. Wilson v and Zak Seidov, Feb 24 2004 *)

A092207 Semiprimes k such that k+2 is also a semiprime.

Original entry on oeis.org

4, 33, 49, 55, 85, 91, 93, 119, 121, 141, 143, 159, 183, 185, 201, 203, 213, 215, 217, 219, 235, 247, 265, 287, 289, 299, 301, 303, 319, 321, 327, 339, 391, 393, 411, 413, 415, 445, 451, 469, 471, 515, 517, 527, 533, 535, 543, 551, 579, 581, 589, 633, 667

Offset: 1

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

nonn

Starting with 33 all terms are odd. First squares are 4, 49, 169, 361, 529, 961, 1369, 2209, 2809, 4489, ... - Zak Seidov, Feb 17 2017

Zak Seidov, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Semiprime

Cf. A056809, A070552, A092125, A092126, A092127, A092128, A092129, A082919, A092209.

PrimeFactorExponentsAdded[n_] := Plus @@ Flatten[Table[ #[[2]], {1}] & /@ FactorInteger[n]]; Select[ Range[ 668], PrimeFactorExponentsAdded[ # ] == PrimeFactorExponentsAdded[ # + 2] == 2 &]
Select[Range[700],PrimeOmega[#]==PrimeOmega[#+2]==2&] (* Harvey P. Dale, Aug 20 2011 *)
SequencePosition[Table[If[PrimeOmega[n]==2,1,0],{n,700}],{1,,1}] [[All,1]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, May 29 2017 *)

is(n)=if(n%2==0, return(n==4)); bigomega(n)==2 && bigomega(n+2)==2 \\ Charles R Greathouse IV, Feb 21 2017

from sympy import factorint
from itertools import count, islice
def agen(): # generator of terms
    yield 4
    nxt = 0
    for k in count(5, 2):
        prv, nxt = nxt, sum(factorint(k+2).values())
        if prv == nxt == 2: yield k
print(list(islice(agen(), 53))) # Michael S. Branicky, Nov 26 2022

A092125 Numbers n such that n, n+2, n+4 are semiprimes.

Original entry on oeis.org

91, 119, 141, 183, 201, 213, 215, 217, 287, 299, 301, 319, 391, 411, 413, 469, 515, 533, 579, 667, 685, 695, 789, 813, 1055, 1077, 1133, 1135, 1137, 1145, 1165, 1203, 1253, 1313, 1343, 1345, 1347, 1383, 1385, 1387, 1389, 1401, 1561, 1639, 1685, 1687, 1761

Offset: 1

Zak Seidov, Feb 22 2004

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

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

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

IsSemiprime:=func< n| &+[ k[2]: k in Factorization(n) ] eq 2 >; [ n: n in [2..4300]|IsSemiprime(n) and IsSemiprime(n+2) and IsSemiprime(n+4)] // Vincenzo Librandi, Dec 16 2010

PrimeFactorExponentsAdded[n_] := Plus @@ Flatten[Table[ #[[2]], {1}] & /@ FactorInteger[n]]; Select[ Range[ 1792], PrimeFactorExponentsAdded[ # ] == PrimeFactorExponentsAdded[ # + 2] == PrimeFactorExponentsAdded[ # + 4] == 2 &] (* Robert G. Wilson v, Feb 24 2004 *)
SequencePosition[Table[If[PrimeOmega[n]==2,1,0],{n,2000}],{1,,1,,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 17 2020 *)

A092126 Numbers n such that n, n+2, n+4, n+6 are semiprimes.

Original entry on oeis.org

213, 215, 299, 411, 1133, 1135, 1343, 1345, 1383, 1385, 1387, 1685, 1793, 1835, 1837, 1891, 1937, 1939, 1957, 2045, 2315, 2317, 2513, 2567, 2807, 2809, 2929, 3091, 3093, 3095, 3097, 3147, 3149, 3647, 3957, 3977, 3979, 4115, 4313, 4315, 4411, 4529, 4531

Offset: 1

Zak Seidov, Feb 22 2004

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

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

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

PrimeFactorExponentsAdded[n_] := Plus @@ Flatten[Table[ #[[2]], {1}] & /@ FactorInteger[n]]; Select[ Range[ 4626], PrimeFactorExponentsAdded[ # ] == PrimeFactorExponentsAdded[ # + 2] == PrimeFactorExponentsAdded[ # + 4] == PrimeFactorExponentsAdded[ # + 6] == 2 &] (* Robert G. Wilson v, Feb 24 2004 *)
Transpose[SequencePosition[Table[If[PrimeOmega[n]==2,1,0],{n,5000}],{1,,1,,1,,1}]][[1]] (* The program uses the SequencePosition function from Mathematica version 10 *) (* _Harvey P. Dale, Feb 10 2016 *)

A092128 Numbers n such that n, n+2, n+4, n+6, n+8, n+10 are semiprimes.

Original entry on oeis.org

1383, 3091, 3093, 5609, 8129, 8131, 8133, 9753, 9983, 9985, 9987, 10401, 11013, 12053, 13637, 16499, 22457, 30991, 43339, 45803, 49083, 53761, 55559, 55561, 58277, 63047, 63951, 64829, 69603, 71727, 76803, 80633, 92603, 92605, 98493

Offset: 1

Zak Seidov, Feb 22 2004

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

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

PrimeFactorExponentsAdded[n_] := Plus @@ Flatten[Table[ #[[2]], {1}] & /@ FactorInteger[n]]; Select[ Range[ 99210], PrimeFactorExponentsAdded[ # ] == PrimeFactorExponentsAdded[ # + 2] == PrimeFactorExponentsAdded[ # + 4] == PrimeFactorExponentsAdded[ # + 6] == PrimeFactorExponentsAdded[ # + 8] == PrimeFactorExponentsAdded[ # + 10] == 2 &] (* Robert G. Wilson v, Feb 24 2004 *)
spQ[n_]:=PrimeOmega[n]==2; Select[Range[100000],AllTrue[#+{0,2,4,6,8,10},spQ]&] (* Harvey P. Dale, Dec 19 2021 *)

More terms from Don Reble, Feb 23 2004

More terms from Robert G. Wilson v, Feb 24 2004

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

Original entry on oeis.org

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 *)

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

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

A091016 a(n) is the least x such that the n values x+0, x+6, x+12, ..., x+6*(n-1) are all products of exactly two primes (i.e., semiprimes).

Original entry on oeis.org

4, 4, 9, 203, 1333, 1333, 1333, 2159, 8309, 18799, 60499, 60499, 186779, 186779, 186779, 12015573923, 12015573923, 6763513182767

Offset: 1

Don Reble, Feb 25 2004

a(19) > 1.e14 if it exists. - Hugo van der Sanden, Aug 06 2021

			a(4)=203 because 203 = 7*29, 209 = 11*19, 215 = 5*43, 221 = 13*17.

Cf. A082919, A092129. Row 6 of A124570.

a(8) and a(9) corrected and a(18) from Donovan Johnson, Oct 01 2010

cp's OEIS Frontend

A082919 Numbers k such that k, k+2, k+4, k+6, k+8, k+10, k+12 and k+14 are semiprimes.

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A092207 Semiprimes k such that k+2 is also a semiprime.

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A092125 Numbers n such that n, n+2, n+4 are semiprimes.

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Magma

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A092126 Numbers n such that n, n+2, n+4, n+6 are semiprimes.

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A092128 Numbers n such that n, n+2, n+4, n+6, n+8, n+10 are semiprimes.

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Extensions

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

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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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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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