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A113460 Triangle read by rows: n-th row is the lexicographically earliest arithmetic progression of n numbers all having the same prime signature.

%I A113460 #24 Oct 25 2024 09:32:07
%S A113460 1,2,3,3,5,7,5,11,17,23,5,11,17,23,29,7,37,67,97,127,157,7,157,307,
%T A113460 457,607,757,907,11,1210241,2420471,3630701,4840931,6051161,7261391,
%U A113460 8471621,11,32671181,65342351,98013521,130684691,163355861,196027031,228698201,261369371
%N A113460 Triangle read by rows: n-th row is the lexicographically earliest arithmetic progression of n numbers all having the same prime signature.
%C A113460 Presumably this triangle will differ from that in A130791 after some point. - _N. J. A. Sloane_, Sep 22 2007
%C A113460 Apart from the initial term, this sequence coincides with A130791 for at least the first 210 rows. - _David W. Wilson_, Sep 22 2007
%H A113460 <a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>
%e A113460 Triangle begins:
%e A113460   1;
%e A113460   2,   3;
%e A113460   3,   5,   7;
%e A113460   5,  11,  17,  23;
%e A113460   5,  11,  17,  23,  29;
%e A113460   7,  37,  67,  97, 127, 157;
%e A113460   ...
%e A113460 From _M. F. Hasler_, Oct 10 2024: (Start)
%e A113460 For row 1, we can take 1, which is the only integer to have prime signature {}.
%e A113460 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.
%e A113460 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.
%e A113460   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.
%e A113460 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.
%e A113460 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.
%e A113460 (* 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.)
%e A113460 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)
%Y A113460 Cf. A113459 (leading terms).
%Y A113460 Cf. A007918, A086786, A113459, A113461, A130791, A127781.
%K A113460 nonn,tabl
%O A113460 1,2
%A A113460 _David Wasserman_, Jan 08 2006
%E A113460 Erroneous commas in sequence deleted by _N. J. A. Sloane_, Sep 22 2007