A123017 -id:A123017 - cp's OEIS frontend

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.

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

A241764 Semiprimes sp such that sp-3 is also semiprime.

Original entry on oeis.org

9, 25, 38, 49, 58, 65, 77, 85, 94, 118, 121, 122, 145, 146, 158, 161, 169, 205, 206, 209, 217, 218, 221, 262, 265, 298, 301, 302, 305, 326, 329, 358, 361, 365, 394, 398, 454, 469, 481, 485, 505, 514, 517, 529, 538, 545, 554, 562, 565, 586, 589, 614

Offset: 1

K. D. Bajpai, Apr 29 2014

nonn

Also semiprimes of the form 2^x - x.

The primes of the form 2^x - x are in A081296.

			a(3)= 38 = 2*19, which is semiprime: 38-3 = 35 = 5*7 is also semiprime.
a(5)= 58 = 2*29, which is semiprime: 58-3 = 55 = 5*11 is also semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..1100

Cf. A001358, A092207, A123017, A198327.

with(numtheory): A241764:= proc(); if bigomega(x)=2 and bigomega(x-3)=2 then RETURN (x); fi; end: seq(A241764 (), x=1..2000);

Flatten[Position[Partition[Table[If[PrimeOmega[n]==2,1,0],{n,700}],4,1], ?(First[ #] ==Last[#]==1&),{1},Heads->False]]+3 (* _Harvey P. Dale, Dec 21 2015 *)

for(k=1, 500, if(bigomega(k)==2 && bigomega(k-3)==2, print1(k, ", "))) \\ Colin Barker, May 07 2014

A241817 Semiprimes sp such that sp-3 is prime.

Original entry on oeis.org

6, 10, 14, 22, 26, 34, 46, 62, 74, 82, 86, 106, 134, 142, 166, 194, 202, 214, 226, 254, 274, 314, 334, 362, 382, 386, 422, 446, 466, 482, 502, 526, 566, 622, 634, 662, 694, 746, 842, 862, 866, 886, 914, 922, 974, 1042, 1094, 1126, 1154, 1174, 1226, 1234, 1262

Offset: 1

K. D. Bajpai, Apr 29 2014

nonn

Even numbers of the form 2p, p prime, that can be expressed as the sum of two primes in at least two ways as 2p = p + p = 3 + (2p-3). For example, 34 is in the sequence because 34 = 2*17 = 17 + 17 = 3 + 31. These are the only numbers that have Goldbach partitions with both a minimum and a maximum possible difference between their prime parts, i.e., |p-p| = 0 and |(2p-3)-3| = 2p-6 respectively. - Wesley Ivan Hurt, Apr 08 2018

			a(2) = 10 = 2*5, which is semiprime and 10-3 = 7 is a prime.
a(6) = 34 = 2*17, which is semiprime and 34-3 = 31 is a prime.

K. D. Bajpai, Table of n, a(n) for n = 1..2500

Cf. A001358, A063908, A092207, A123017, A198327.

with(numtheory): A241817:= proc(); if bigomega(x)=2 and isprime(x-3) then  RETURN (x); fi; end: seq(A241817 (), x=1..3000);

2 Select [Prime[Range[5!]], PrimeQ[2 # - 3] &] (* Vincenzo Librandi, Apr 10 2018 *)
Select[Range[1500],PrimeOmega[#]==2&&PrimeQ[#-3]&] (* Harvey P. Dale, Oct 14 2018 *)

a(n) = 2 * A063908(n). - Wesley Ivan Hurt, Apr 08 2018

cp's OEIS Frontend

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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A241764 Semiprimes sp such that sp-3 is also semiprime.

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Maple

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A241817 Semiprimes sp such that sp-3 is prime.

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Maple

Mathematica

Formula