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A387408 Partial sums of A387412, where A387412(n) is the length of the maximal common prefix of the binary expansions of n and A003961(n).

%I A387408 #7 Sep 03 2025 13:11:16
%S A387408 1,2,3,6,7,9,10,11,12,16,17,18,19,20,21,23,26,31,33,34,35,37,38,39,43,
%T A387408 46,48,50,53,55,56,57,62,63,66,67,69,72,74,77,81,85,88,89,92,95,96,
%U A387408 100,102,103,104,105,107,108,109,110,116,117,120,121,122,124,125,127,128,129,133,135,138,139,142,144,148,149
%N A387408 Partial sums of A387412, where A387412(n) is the length of the maximal common prefix of the binary expansions of n and A003961(n).
%H A387408 Antti Karttunen, <a href="/A387408/b387408.txt">Table of n, a(n) for n = 1..65537</a>
%H A387408 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>.
%H A387408 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%F A387408 a(1) = 1; and for n > 1, a(n) = a(n-1) + A387412(n).
%F A387408 a(n) = A387407(n) - A387409(n).
%o A387408 (PARI)
%o A387408 up_to = 65537;
%o A387408 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A387408 A387412(n) = { my(a=binary(n), b=binary(A003961(n)), i=1); while(i<=#a,if(a[i]!=b[i],return(i-1)); i++); (#a); };
%o A387408 A387408list(up_to) = { my(v=vector(up_to)); v[1] = A387412(1); for(n=2,up_to,v[n] = v[n-1]+A387412(n)); (v); };
%o A387408 v387408 = A387408list(up_to);
%o A387408 A387408(n) = v387408[n];
%Y A387408 Cf. A000203, A387412, A387407, A387409.
%Y A387408 Cf. also A387425.
%K A387408 nonn,base,new
%O A387408 1,2
%A A387408 _Antti Karttunen_, Sep 03 2025