A110529 -id:A110529 - cp's OEIS frontend

A110562 Numbers n such that n in binary representation has a block of exactly a nontrivial pentagonal number of zeros.

32, 65, 96, 130, 131, 160, 193, 224, 260, 261, 262, 263, 288, 321, 352, 386, 387, 416, 449, 480, 520, 521, 522, 523, 524, 525, 526, 527, 544, 577, 608, 642, 643, 672, 705, 736, 772, 773, 774, 775, 800, 833, 864, 898, 899, 928, 961, 992, 1040, 1041, 1042

Offset: 1

Jonathan Vos Post, Sep 12 2005

a(n) is the index of zeros in the complement of the pentagonal number analog of the Baum-Sweet sequence, which is b(n) = 1 if the binary representation of n contains no block of consecutive zeros of exactly a nontrivial pentagonal number length A000326(i) for i>1; otherwise b(n) = 0.

			a(1) = 32 because 32 (base 2) = 100000, which has a block of 5 = A000326(2) zeros.
a(2) = 65 because 65 (base 2) = 1000001, which has a block of 5 zeros.
64 is not in this sequence because, though 64 (base 2) = 1000000 has a block of 6 zeros, which has subblocks of 5 zeros, subblocks do not count.
2080 is in this sequence because 2080 (base 2) = 100000100000 has 2 blocks of 5 zeros, but we do not require only one such 5-zero block.
4096 is in this sequence because 4096 (base 2) = 1000000000000, which has a block of 12 = A000326(3) zeros, as do 8193 and many more.
4194304 is in this sequence because 4194304 (base 2) = 10000000000000000000000, which has a block of 22 = A000326(4) zeros.

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 157.

J.-P. Allouche, Finite Automata and Arithmetic, Séminaire Lotharingien de Combinatoire, B30c (1993), 23 pp.

Cf. A000326, A037011, A086747, A110471, A110472, A110474, A110502, A110529.

Corrected by Ray Chandler, Sep 17 2005

A309092 Integers whose hexadecimal representation contains a run of zeros of prime length.

Original entry on oeis.org

256, 512, 768, 1024, 1280, 1536, 1792, 2048, 2304, 2560, 2816, 3072, 3328, 3584, 3840, 4096, 4097, 4098, 4099, 4100, 4101, 4102, 4103, 4104, 4105, 4106, 4107, 4108, 4109, 4110, 4111, 4352, 4608, 4864, 5120, 5376, 5632, 5888, 6144, 6400, 6656, 6912, 7168

Offset: 1

W. Zane Billings, Jul 11 2019

			256 = 100_(16) is a term because 100 has a run of two zeros, and two is prime. 258 = 102_(16) is not a term, because its only run of zeros is of length 1, which is not prime.

Cf. A110529, A319302.

Select[Range@ 7168, Select[Split@ IntegerDigits[#, 16], #[[1]] == 0 && PrimeQ@ Length@ # &] != {} &] (* Giovanni Resta, Jul 16 2019 *)

from re import split
from sympy import isprime
seq_list, n = [],1
while len(seq_list) < 10000:
    for d in split('[1-9]+|[a-f]+', format(n,'x')):
        if isprime(len(d)):
            seq_list.append(n)
    n += 1

cp's OEIS Frontend

A110562 Numbers n such that n in binary representation has a block of exactly a nontrivial pentagonal number of zeros.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Extensions

A309092 Integers whose hexadecimal representation contains a run of zeros of prime length.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Python