A387408 -id:A387408 - cp's OEIS frontend

A387407 Partial sums of A387422, where A387422 is the length of the maximal common prefix of the binary expansions of n and sigma(n).

1, 2, 3, 4, 5, 8, 9, 10, 11, 13, 14, 16, 18, 20, 22, 23, 26, 30, 32, 36, 38, 40, 41, 43, 45, 46, 47, 52, 55, 56, 57, 58, 59, 60, 61, 63, 67, 68, 69, 72, 76, 77, 80, 83, 85, 87, 88, 90, 92, 93, 94, 97, 101, 103, 104, 107, 108, 109, 112, 113, 117, 119, 121, 122, 124, 127, 131, 132, 133, 136, 139, 140, 145, 146, 147

Offset: 1

Antti Karttunen, Sep 03 2025

Cf. A003961, A387422, A387408, A387409.

Cf. also A387424.

up_to = 65537;
A387422(n) = { my(a=binary(n), b=binary(sigma(n)), i=1); while(i<=#a,if(a[i]!=b[i],return(i-1)); i++); (#a); };
A387407list(up_to) = { my(v=vector(up_to)); v[1] = A387422(1); for(n=2,up_to,v[n] = v[n-1]+A387422(n)); (v); };
v387407 = A387407list(up_to);
A387407(n) = v387407[n];

a(1) = 1; and for n > 1, a(n) = a(n-1) + A387422(n).

a(n) = A387408(n) + A387409(n).

A387409 Partial sums of A387422 minus partial sums of A387412.

Original entry on oeis.org

0, 0, 0, -2, -2, -1, -1, -1, -1, -3, -3, -2, -1, 0, 1, 0, 0, -1, -1, 2, 3, 3, 3, 4, 2, 0, -1, 2, 2, 1, 1, 1, -3, -3, -5, -4, -2, -4, -5, -5, -5, -8, -8, -6, -7, -8, -8, -10, -10, -10, -10, -8, -6, -5, -5, -3, -8, -8, -8, -8, -5, -5, -4, -5, -4, -2, -2, -3, -5, -3, -3, -4, -3, -3, -3, -1, -3, -2, -2, -1, -2, -3, -2

Offset: 1

Antti Karttunen, Sep 03 2025

Also partial sums of A387413 minus partial sums of A387423.
This sequence gives some measure of how much longer the common prefix of the binary expansions of n and sigma(n) is - on average - than the common prefix of the binary expansions of n and A003961(n).
Question: Does the sequence eventually grow without limit and is it just because A003961(n) >= A000203(n)? Does ratio a(n)/n converge to any limit?

Cf. A000203, A003961, A387407, A387408, A387412, A387413, A387422, A387423, A387424, A387425.

PARI
```
A387409(n) = (A387407(n) - A387408(n));
```

a(n) = A387407(n) - A387408(n).

a(n) = A387425(n) - A387424(n).

cp's OEIS Frontend

A387407 Partial sums of A387422, where A387422 is the length of the maximal common prefix of the binary expansions of n and sigma(n).

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