primes in arithmetic progression

[Jonathan Post]

The analogue for 3-almost primes (or, as njas prefers, numbers which are the
product of exactly 3 primes) is left as an exercise for the reader...  I've
just submitted the following.

Subject: PRE-NUMBERED NEW SEQUENCE A124570 FROM Jonathan Vos Post


%I A124570
%S A124570 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, 1383, 0, 4, 4, 4, 77, 111, 0, 8129, 0, 4, 4, 14
%N 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.
%C A124570 Semiprime analogue of A124064 Array read by antidiagonals:
T(d,k) (k >= 1, d = 2,4,6,8,...) = smallest prime p of k (not
necessarily consecutive) primes in arithmetic progression with common
difference d. The row d=0 and the column k=1 are degenerate and are filled with
the semiprime 4. Once a 0 occurs in a row, there are no more nonzero
vales in the row. For example, 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).
%e A124570 Array begins:
d.\...k=1.k=2..k=3..k=4..k=5..k=6..k=7..k=8
0..|..4...4....4....4....4....4....4....4
1..|..4...9....33...0....0....0....0....0
2..|..4...4....91...213..1383.8129.3091.0
3..|..4...6....115..0....0....0....0....0
4..|..4...6....6....6....111...
5..|..4...4....77...
6..|..4...4....
7..|..4...14...51...
8..|..4...6....6....69.
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.
%Y A124570 Cf. A000040, A001358, A056809, A070552, A092125, A092126,
A092127, A092128, A092129, A124064.
%O A124570 1,1
%K A124570 ,easy,more,nonn,tabl,
%A A124570 Jonathan Vos Post (jvospost2 at yahoo.com
<http://us.f551.mail.yahoo.com/ym/Compose?To=jvospost2@yahoo.com&YY=99063&y5beta=yes&y5beta=yes&order=down&sort=date&pos=0&view=a&head=b>),
Nov 04 2006
RH
RA 192.20.225.32
RU
RI



On 11/4/06, N. J. A. Sloane <njas at research.att.com> wrote:
>
> Just to get the ball rolling, I am adding this entry:
>
> %I A124064
> %S A124064 2,2,2,2,3,2,2,3,3,2,2,5,3
> %N A124064 Array read by antidiagonals: T(d,k) (k >= 1, d = 2,4,6,8,...) =
> smallest prime p of k (not necessarily consecutive) primes in arithmetic
> progression with common difference d.
> %C A124064 The row d=0 and the column k=1 are degenerate and are filled
> with the prime 2.
> %O A124064 1,1
> %K A124064 nonn,tabl,more
> %e A124064 Array begins:
> %e A124064 d.\...k=1.k=2..k=3..k=4..k=5..k=6
> %e A124064 0..|..2...2....2....2....2....2
> %e A124064 2..|..2...3....3
> %e A124064 4..|..2...3....3
> %e A124064 6..|..2...5....5....5....5
> %e A124064 8..|..2...3....3
> %e A124064 10.|..2...3....3
> %e A124064 12.|..2...5....5....5....5
> %e A124064 14.|..2...3....3.
> %e A124064 16.|..2...3
> %e A124064 18.|..2...5....5....5
> %e A124064 20.|..2...3
> %e A124064 22.|..2...7
> %e A124064 24.|..2...5....5...59
> %e A124064 26.|..2...3
> %e A124064 28.|..2...3....3
> %e A124064 30.|..2...7....7....7.....7....7
> %e A124064 32.|..2...5
> %e A124064 34.|..2...3....3
> %e A124064 36.|..2...5....7...31
> %e A124064 Example for row d=24 and column k=4: the 4 numbers
> 59,59+24,59+2*24 and 59+3*24 are all primes.
> %A A124064 Richard Mathar (mathar(AT)strw.leidenuniv.nl), Nov 04 2006
>
>
>
>
> NJAS
>
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