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A376115 Least common differences in the arithmetic progressions corresponding to A376109.

%I A376115 #6 Sep 11 2024 00:42:51
%S A376115 1,1,1,1,2,2,2,1,3,1,4,4,2,4,1,1,6,6,6,2,6,8,6,8,3,11,7,8,6,2,12,1,12,
%T A376115 12,1,12,6,12,1,4,12,12,12,16,1,12,18,16,14,5,13,22,12,14,17,16,11,12,
%U A376115 6,4,24,24,18,1,7,24,24,24,18,2,12,24,6,35,5,13,19,33,6,8,21,24,12,24,8,24
%N A376115 Least common differences in the arithmetic progressions corresponding to A376109.
%C A376115 a(n) is the least d >= 1 such that A001222(n-i*d) = A001222(n) for 0 <= i < A376109(n).
%H A376115 Robert Israel, <a href="/A376115/b376115.txt">Table of n, a(n) for n = 1..10000</a>
%e A376115 a(7) = 2 because the arithmetic progression 3, 5, 7 of A376109(7) = 3 primes ending in 7 has common difference of 5 - 3 = 7 - 5 = 2.
%e A376115 There are two arithmetic progressions of semiprimes of A376109(14) = 3 ending in 14, namely 6, 10, 14 with common difference 4 and 4, 9, 14 with common difference 5, so a(14) = 4.
%p A376115 M:= Array(1..10):
%p A376115 for n from 2 to 100 do
%p A376115   v:= numtheory:-bigomega(n);
%p A376115   if M[v] = 0 then M[v]:= n else M[v]:= M[v], n fi;
%p A376115 od:
%p A376115 for i from 1 to 10 do M[i]:= [M[i]] od:
%p A376115 f:= proc(s) local n,i,m,d,v,j,dm;
%p A376115    m:= 1; dm:= 1;
%p A376115    v:= numtheory:-bigomega(s);
%p A376115    member(s,M[v],n);
%p A376115    for i from n-1 to 1 by -1 do
%p A376115      d:= s - M[v][i];
%p A376115      if s - m*d < M[v][1] then return dm fi;
%p A376115      for j from 2 while ListTools:-BinarySearch(M[v],s-j*d) <> 0 do od:
%p A376115      if j > m then m:= j; dm:= d fi;
%p A376115    od;
%p A376115   dm;
%p A376115 end proc:
%p A376115 f(1):= 1:
%p A376115 map(f, [$1..100]);
%Y A376115 Cf. A001222, A375386, A375511, A376109.
%K A376115 nonn
%O A376115 1,5
%A A376115 _Robert Israel_, Sep 10 2024