A387422 -id:A387422 - cp's OEIS frontend

A387412 The length of the maximal common prefix of the binary expansions of n and A003961(n), where A003961 is fully multiplicative with a(p) = nextprime(p).

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

Offset: 1

Antti Karttunen, Sep 01 2025

Cf. A000523, A003961, A387413.

Cf. also A387422.

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); };

a(n) = (1+A000523(n)) - A387413(n).

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.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 01 2025

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.

Cf. A000203, A000523, A332223, A336700, A336701, A336702 (conjectured positions of 0's), A387422.

Cf. also A347381, A387413.

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); };

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

cp's OEIS Frontend

A387412 The length of the maximal common prefix of the binary expansions of n and A003961(n), where A003961 is fully multiplicative with a(p) = nextprime(p).

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