Search: a110529 -id:a110529
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A110562
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Numbers n such that n in binary representation has a block of exactly a nontrivial pentagonal number of zeros.
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+10
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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
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OFFSET
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1,1
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COMMENTS
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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.
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REFERENCES
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J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 157.
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LINKS
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EXAMPLE
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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 sub-blocks of 5 zeros, sub-blocks 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.
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CROSSREFS
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KEYWORD
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base,easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A309092
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Integers whose hexadecimal representation contains a run of zeros of prime length.
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+10
0
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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Select[Range@ 7168, Select[Split@ IntegerDigits[#, 16], #[[1]] == 0 && PrimeQ@ Length@ # &] != {} &] (* Giovanni Resta, Jul 16 2019 *)
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PROG
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(Python)
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
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CROSSREFS
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KEYWORD
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nonn,easy,base
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AUTHOR
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STATUS
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approved
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