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.

A235669 Sum of parts of the form 10...0 and 20...0 with nonnegative number of zeros in ternary representation of n as the corresponding numbers 3^n and 2*3^n.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 6, 3, 4, 9, 4, 5, 4, 3, 4, 7, 4, 5, 18, 7, 8, 5, 4, 5, 8, 5, 6, 27, 10, 11, 6, 5, 6, 9, 6, 7, 10, 5, 6, 5, 4, 5, 8, 5, 6, 19, 8, 9, 6, 5, 6, 9, 6, 7, 54, 19, 20, 9, 8, 9, 12, 9, 10, 11, 6, 7, 6, 5, 6, 9, 6, 7, 20, 9, 10, 7, 6, 7, 10, 7, 8, 81, 28, 29, 12
Offset: 0

Views

Author

Vladimir Shevelev, Jan 13 2014

Keywords

Comments

The number of appearances of k is the number of compositions of k into numbers of the form 3^n and 2*3^n, A235684(k).

Links

Crossrefs

Cf. A124771, A162439, A233249, A233312, A233416, A233420, A233564, A233569, A233655.

Programs

  • Mathematica
    bitPatt[n_,b_]:=Split[IntegerDigits[n,b ],#2==0&]; Map[Plus@@Map[FromDigits[#,3]&,bitPatt[#,3]]&,Range[0,50]] (* Peter J. C. Moses, Jan 13 2014 *)

A331469 a(n) is the greatest value of the form p_1 + ... + p_k where p_1, ..., p_k are powers of primes and such that the concatenation of the binary representations of p_1, ..., p_k equals the binary representation of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 4, 11, 5, 13, 5, 8, 16, 17, 6, 19, 6, 7, 7, 23, 9, 25, 5, 27, 7, 29, 9, 31, 32, 17, 10, 18, 8, 37, 11, 20, 10, 41, 6, 43, 9, 15, 13, 47, 17, 49, 7, 26, 7, 53, 15, 28, 11, 26, 7, 59, 11, 61, 10, 32, 64, 33, 18, 67, 12, 13, 19, 71, 12
Offset: 1

Views

Author

Rémy Sigrist, Jan 17 2020

Keywords

Comments

We can always split the binary representation of a number into powers of 2, so the sequence is well defined.

Examples

			For n = 22:
- the binary representation of 22 is "10110",
- we can split it into "10" and "1" and "10" (2^1 and 2^0 and 2^1),
- or into "101" and "10" (5^2 and 2^1),
- hence a(22) = max(5, 7) = 7.

Links

Crossrefs

Cf. A000961, A162439, A331362.

Programs

  • PARI
    See Links section.

Formula

a(n) >= A162439(n).
a(n) <= n with equality iff n is a power of a prime.

A235590 Sum of parts of the form 10...0 with nonnegative number of zeros in binary representation of c-squarefree numbers (A233564) as the corresponding powers of 2.

Original entry on oeis.org

1, 2, 4, 3, 3, 8, 5, 5, 16, 9, 6, 6, 9, 32, 17, 10, 7, 7, 10, 7, 7, 17, 7, 7, 64, 33, 18, 12, 11, 11, 12, 18, 11, 11, 33, 11, 11, 128, 65, 34, 20, 19, 19, 13, 13, 20, 13, 13, 34, 19, 19, 65, 19, 13, 13, 19, 256, 129, 66, 36, 35
Offset: 1

Views

Author

Vladimir Shevelev, Jan 12 2014

Keywords

Comments

Subsequence of A162439.
The number of times of appearances of number k in the sequence is the number of compositions of k into distinct powers of 2, i.e., it is A000120(k)!

Examples

			Let n=17, A233564(17)=37. In binary a concatenation of parts of the form 10...0 which gives 37 is (100)(10)(1). Thus a(17)= 4+2+1 = 7.

Crossrefs

Cf. A233564, A162439, A000120.

Programs

  • Mathematica
    bitPatt[n_]:=bitPatt[n]=Split[IntegerDigits[n,2],#2==0&]; Map[Plus@@(Map[FromDigits[#,2]&,bitPatt[#]])&,Select[Range[300],#==DeleteDuplicates[#]&[bitPatt[#]]&]] (* Peter J. C. Moses, Jan 15 2014 *)

Formula

Let, for k_1>k_2>...>k_r, A233564(n) = 2^k_1 + 2^k_2 +...+ 2^k_r. Then a(n) = 2^(k_1-k_2-1) + 2^(k_2-k_3-1) + 2^(k_(r-1)-k_r-1) + 2^k_r.
Showing 1-3 of 3 results.

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

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