A061558
The smallest difference of an increasing arithmetic progression of n primes with the minimal possible first term (A007918(n)).
Original entry on oeis.org
0, 1, 2, 6, 6, 30, 150, 1210230, 32671170, 224494620, 1536160080, 1482708889200, 9918821194590, 266029822978920, 266029822978920, 11358256064006271420, 341976204789992332560, 128642760444772214170530, 2166703103992332274919550
Offset: 1
For n = 10, the smallest difference a(10) = 224494620 with the first term 11 (= A007918(10)) producing an arithmetic progression of 10 primes.
- Jens Kruse Andersen, Records for primes in arithmetic progression
- Jaroslaw Wroblewski, Re: AP19 starting with 19, Yahoo group "primenumbers", Apr 10 2013.
- Jaroslaw Wroblewski, Mike Oakes, Jens Kruse Andersen, AP19 starting with 19, digest of 6 messages in primenumbers Yahoo group, Feb 25 - Apr 10, 2013.
- Index entries for sequences related to primes in arithmetic progressions
A113460
Triangle read by rows: n-th row is the lexicographically earliest arithmetic progression of n numbers all having the same prime signature.
Original entry on oeis.org
1, 2, 3, 3, 5, 7, 5, 11, 17, 23, 5, 11, 17, 23, 29, 7, 37, 67, 97, 127, 157, 7, 157, 307, 457, 607, 757, 907, 11, 1210241, 2420471, 3630701, 4840931, 6051161, 7261391, 8471621, 11, 32671181, 65342351, 98013521, 130684691, 163355861, 196027031, 228698201, 261369371
Offset: 1
Triangle begins:
1;
2, 3;
3, 5, 7;
5, 11, 17, 23;
5, 11, 17, 23, 29;
7, 37, 67, 97, 127, 157;
...
From _M. F. Hasler_, Oct 10 2024: (Start)
For row 1, we can take 1, which is the only integer to have prime signature {}.
For row 2, we can't use 1 (no two integers with that prime signature), but primes 2 & 3 are a valid and then also minimal choice.
For row 3, primes {3, 5, 7} are a valid choice and also smallest: we can't use 1, nor 2, for reasons of parity: the next prime would be odd but the third term of the arithmetic progression would then again be even and not prime.
The same reasoning also excludes any higher power 2^m as starting term, which would require the same (m-th) power of odd primes as subsequent terms.
For rows 4 and 5, we can't start with the prime 3, because the 4th term of any arithmetic progression starting with 3 is again divisible by 3. Also 4 = 2^2 is excluded, see above. Thus, 5 is the smallest possible starting term for n = 4 and 5.
For row 6 and 7, we again can't start with a prime < nextprime(6) = 7, because there can't be more than 5 primes in AP starting with 5: the sixth term would always be divisible by 5 again. To start with the even semiprime 6 = 2*3 would require an AP of even semiprimes. Dividing by 2, we would have an AP of 6 primes starting with 3, which is impossible.(*) So, 7 is the smallest possibility.
(* This actually excludes all even semiprimes 2*p between prime(k-1) and prime(k) from being a starting term of a row in that range, because that would yield an AP of >= prime(k-1) primes starting with p < prime(k)/2 < prime(k-1), impossible.)
Rows 8 through 11 can't start with a prime < nextprime(8) = 11, as before. We have also excluded any 2^m and 2*3 as starting value. Starting with 9 = 3^2 would require an AP of squares of primes, but all larger squares of primes have a difference (6k +- 1)^2 - (6m +- 1)^2 divisible by 12, which is not the case for the difference with 3^2 = 9. The even semiprime 10 = 2*5 was also excluded above (*). Therefore, 11 is the smallest possible initial term. And so on. (End)
A130791
Triangle read by rows: n-th row is the lexicographically earliest arithmetic progression of n primes beginning with A007918(n).
Original entry on oeis.org
2, 2, 3, 3, 5, 7, 5, 11, 17, 23, 5, 11, 17, 23, 29, 7, 37, 67, 97, 127, 157, 7, 157, 307, 457, 607, 757, 907, 11, 1210241, 2420471, 3630701, 4840931, 6051161, 7261391, 8471621, 11, 32671181, 65342351, 98013521, 130684691, 163355861, 196027031, 228698201, 261369371
Offset: 1
Triangle begins:
2
2 3
3 5 7
5 11 17 23
5 11 17 23 29
7 37 67 97 127 157
7 157 307 457 607 757 907
11 1210241 2420471 3630701 4840931 6051161 7261391 8471621
11 32671181 65342351 98013521 130684691 163355861 196027031 228698201 261369371
For common differences see
A061558.
A113459
Least number that begins an arithmetic progression of n numbers with the same prime signature.
Original entry on oeis.org
1, 2, 3, 5, 5, 7, 7, 11, 11, 11, 11, 13
Offset: 1
Showing 1-4 of 4 results.
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