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A387423

The length of binary expansion of n minus the length of the maximal common prefix of the binary expansions of n and sigma(n), where sigma is the sum of divisors function.

0, 1, 1, 2, 2, 0, 2, 3, 3, 2, 3, 2, 2, 2, 2, 4, 2, 1, 3, 1, 3, 3, 4, 3, 3, 4, 4, 0, 2, 4, 4, 5, 5, 5, 5, 4, 2, 5, 5, 3, 2, 5, 3, 3, 4, 4, 5, 4, 4, 5, 5, 3, 2, 4, 5, 3, 5, 5, 3, 5, 2, 4, 4, 6, 5, 4, 3, 6, 6, 4, 4, 6, 2, 6, 6, 4, 6, 5, 5, 4, 6, 6, 3, 6, 6, 5, 6, 2, 2, 6, 6, 4, 5, 5, 6, 5, 2, 6, 6, 4, 2, 4, 4, 1, 4

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OFFSET

1,4

COMMENTS

Positions of 0's in this sequence is given by such numbers n that sigma(n) = 2^k * n + r, for some n >= 1, k >= 0, 0 <= r < 2^k. These would include also quasi-perfect numbers and their generalizations, numbers n such that sigma(n) = 2^k * n + 2^k - 1, for some n > 1, k > 0 (see comments in A332223), if such numbers exist. However, it is conjectured that there are no other zeros than those given by A336702.

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 sigma(n).

FORMULA

a(n) = (1+A000523(n)) - A387422(n).

MATHEMATICA

A387423[n_] := BitLength[n] - LengthWhile[Transpose[IntegerDigits[{n, DivisorSigma[1, n]}, 2][[All, ;; BitLength[n]]]], Equal @@ # &];

Array[A387423, 100] (* Paolo Xausa, Sep 03 2025 *)

PROG

(PARI) A387423(n) = { my(a=binary(n), b=binary(sigma(n)), i=1); while(i<=#a, if(a[i]!=b[i], return(#a-(i-1))); i++); (0); };

(Python)