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A348040 Square array A(n,k) = the length of the common prefix in binary expansions of A156552(n) and A156552(k), read by antidiagonals.

%I A348040 #7 Oct 12 2021 21:56:32
%S A348040 0,0,0,0,1,0,0,1,1,0,0,1,2,1,0,0,1,1,1,1,0,0,1,2,2,2,1,0,0,1,2,1,1,2,
%T A348040 1,0,0,1,2,1,3,1,2,1,0,0,1,1,1,2,2,1,1,1,0,0,1,1,2,3,3,3,2,1,1,0,0,1,
%U A348040 2,2,1,2,2,1,2,2,1,0,0,1,2,1,1,1,4,1,1,1,2,1,0,0,1,2,1,3,1,1,1,1,3,1,2,1,0
%N A348040 Square array A(n,k) = the length of the common prefix in binary expansions of A156552(n) and A156552(k), read by antidiagonals.
%H A348040 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%e A348040 The top left 17x17 corner of the array:
%e A348040   n/k | 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
%e A348040 ------+----------------------------------------------------
%e A348040    1  | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
%e A348040    2  | 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
%e A348040    3  | 0, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2,
%e A348040    4  | 0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1,
%e A348040    5  | 0, 1, 2, 1, 3, 2, 3, 1, 1, 3, 3, 2, 3, 3, 2, 1, 3,
%e A348040    6  | 0, 1, 2, 1, 2, 3, 2, 1, 1, 2, 2, 3, 2, 2, 3, 1, 2,
%e A348040    7  | 0, 1, 2, 1, 3, 2, 4, 1, 1, 3, 4, 2, 4, 4, 2, 1, 4,
%e A348040    8  | 0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 3, 1,
%e A348040    9  | 0, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 2, 1,
%e A348040   10  | 0, 1, 2, 1, 3, 2, 3, 1, 1, 4, 3, 2, 3, 3, 2, 1, 3,
%e A348040   11  | 0, 1, 2, 1, 3, 2, 4, 1, 1, 3, 5, 2, 5, 4, 2, 1, 5,
%e A348040   12  | 0, 1, 2, 1, 2, 3, 2, 1, 1, 2, 2, 4, 2, 2, 3, 1, 2,
%e A348040   13  | 0, 1, 2, 1, 3, 2, 4, 1, 1, 3, 5, 2, 6, 4, 2, 1, 6,
%e A348040   14  | 0, 1, 2, 1, 3, 2, 4, 1, 1, 3, 4, 2, 4, 5, 2, 1, 4,
%e A348040   15  | 0, 1, 2, 1, 2, 3, 2, 1, 1, 2, 2, 3, 2, 2, 4, 1, 2,
%e A348040   16  | 0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1,
%e A348040   17  | 0, 1, 2, 1, 3, 2, 4, 1, 1, 3, 5, 2, 6, 4, 2, 1, 7,
%o A348040 (PARI)
%o A348040 up_to = 105;
%o A348040 Abincompreflen(n, m) = { my(x=binary(n),y=binary(m),u=min(#x,#y)); for(i=1,u,if(x[i]!=y[i],return(i-1))); (u);};
%o A348040 A156552(n) = {my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552
%o A348040 A348040sq(x,y) = Abincompreflen(A156552(x), A156552(y));
%o A348040 A348040list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A348040sq(col,(a-(col-1))))); (v); };
%o A348040 v348040 = A348040list(up_to);
%o A348040 A348040(n) = v348040[n];
%Y A348040 Cf. A252464 (main diagonal).
%Y A348040 Cf. A005940, A156552, A348041.
%Y A348040 Cf. also A347380, A347381.
%K A348040 nonn,tabl
%O A348040 1,13
%A A348040 _Antti Karttunen_, Sep 27 2021