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).

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, 46, 48, 50, 53, 55, 56, 57, 62, 63, 66, 67, 69, 72, 74, 77, 81, 85, 88, 89, 92, 95, 96, 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

Offset: 1

Author

Antti Karttunen, Sep 03 2025

Keywords

• nonn
• base
• new

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences related to binary expansion of n.
• Index entries for sequences related to prime indices in the factorization of n.

Crossrefs

Cf. A000203, A387412, A387407, A387409.

Cf. also A387425.

Programs

PARI

up_to = 65537;
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
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); };
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); };
v387408 = A387408list(up_to);
A387408(n) = v387408[n];

Formula

a(1) = 1; and for n > 1, a(n) = a(n-1) + A387412(n).

a(n) = A387407(n) - A387409(n).