cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A387413 The length of binary expansion of n minus 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).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Sep 01 2025

Keywords

Links

Crossrefs

Cf. A000523, A003961, A387412, A387414 (positions of 0's).
Cf. also A387423.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A387413(n) = { my(a=binary(n), b=binary(A003961(n)), i=1); while(i<=#a,if(a[i]!=b[i],return(#a-(i-1))); i++); (0); };

Formula

a(n) = (1+A000523(n)) - A387412(n).

A387414 Numbers k such that the binary expansion of k is a prefix of the binary expansion of A003961(k), where A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

1, 4, 10, 18, 57, 348, 1054, 2626, 60625, 68727, 129260, 192276, 675348, 960320, 5368464, 12371554, 30078308, 356311953, 1158654378, 1673018314
Offset: 1

Views

Author

Antti Karttunen, Sep 01 2025

Keywords

Comments

Numbers k such that A003961(k) = 2^e * k + r, for some k >= 1, e >= 0, 0 <= r < 2^e.

Examples

			A007088(4) = 100, and A007088(A003961(4)) = A007088(9) = 1001 begins with the same binary string, therefore 4 is included.
A007088(18) = 10010, and A007088(A003961(18)) = A007088(75) = 1001011 begins with the same binary string, therefore 18 is included as a term. Also, 75 = 2^2 * 18 + 3.

Links

Crossrefs

Positions of 0's in A387413.
Cf. A000523, A003961, A007088, A387412.
Subsequences: A348514 (which is also a subsequence of A387411).

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
is_A387414(n) = { my(s=A003961(n)); while(s>n, s >>= 1); (s==n); };

A387422 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

1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 2, 2, 2, 1, 3, 4, 2, 4, 2, 2, 1, 2, 2, 1, 1, 5, 3, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 3, 4, 1, 3, 3, 2, 2, 1, 2, 2, 1, 1, 3, 4, 2, 1, 3, 1, 1, 3, 1, 4, 2, 2, 1, 2, 3, 4, 1, 1, 3, 3, 1, 5, 1, 1, 3, 1, 2, 2, 3, 1, 1, 4, 1, 1, 2, 1, 5, 5, 1, 1, 3, 2, 2, 1, 2, 5, 1, 1, 3, 5, 3, 3, 6, 3
Offset: 1

Views

Author

Antti Karttunen, Sep 01 2025

Keywords

Links

Crossrefs

Cf. A000203, A000523, A387423.
Cf. also A347380, A387412.

Programs

  • PARI
    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); };

Formula

a(n) = (1+A000523(n)) - A387423(n).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.