A092209 -id:A092209 - 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

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

A113457 Least number that begins an n-term arithmetic progression with common difference 2 in which all terms have the same prime signature.

Original entry on oeis.org

1, 3, 3, 213, 213, 1383, 3091, 8129

Offset: 1

David Wasserman, Jan 08 2006

Second column of A113456. a(9) >= 37887000000.

Cf. A092209 A113456 A113466.

A217222 Initial terms of sets of 8 consecutive semiprimes with gap 2.

Original entry on oeis.org

8129, 237449, 401429, 452639, 604487, 858179, 1471727, 1999937, 2376893, 2714987, 3111977, 3302039, 3869237, 4622087, 7813559, 9795449, 10587899, 10630739, 11389349, 14186387, 14924153, 15142547, 15757337, 18017687, 18271829, 19732979, 22715057, 25402907

Offset: 1

Zak Seidov, Sep 28 2012

nonn

All terms == 11 (mod 18).
Also all terms of sets of 8 consecutive semiprimes are odd, e.g., {8129, 8131, 8133, 8135, 8137, 8139, 8141, 8143} is the smallest set of 8 consecutive semiprimes.
Note that in all cases "9th term" (in this case 8143+2=8145) is divisible by 9 and hence is not semiprime.
Also note that all seven "intermediate" even integers (in this case {8130, 8132, 8134, 8136, 8138, 8140, 8142}) have at least three prime factors counting with multiplicity. Up to n = 40*10^9 there are 5570 terms of this sequence.

Zak Seidov, Table of n, a(n) for n = 1..5570

Subsequence of A082919.

Cf. A056809, A070552, A092125-A092129, A092207-A092209.

Transpose[Select[Partition[Select[Range[26*10^6],PrimeOmega[#] == 2&],8,1], Union[ Differences[#]]=={2}&]][[1]] (* Harvey P. Dale, Sep 02 2015 *)

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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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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A113457 Least number that begins an n-term arithmetic progression with common difference 2 in which all terms have the same prime signature.

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A217222 Initial terms of sets of 8 consecutive semiprimes with gap 2.

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