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Search: id:A005360
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| A005360 |
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Flimsy numbers.
(Formerly M4771)
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+0
5
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| 11, 13, 19, 22, 23, 25, 26, 27, 29, 37, 38, 39, 41, 43, 44, 46, 47, 50, 52, 53, 54, 55, 57, 58, 59, 61, 67, 71, 74, 76, 77, 78, 79, 81, 82, 83, 86, 87, 88, 91, 92, 94, 95, 97, 99, 100, 101, 103, 104, 106, 107, 108, 109, 110, 111, 113, 114, 115, 116, 117, 118, 119, 121
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Definition: a(n) is flimsy iff there exists a k such that A000120(k*n) < A000120(n). That is, some multiple of n has fewer ones in its binary expansion than does n. What are the associated k for each n? What is the smallest n for each k? Stolarsky says "at least half the primes are flimsy." - Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 07 2008
A143073(n) gives the least k for each n in this sequence. -T. D. Noe (noe(AT)sspectra.com), Jul 22 2008
Stolarsky says "at least half the primes are flimsy". (Jonathan Vos Post, Jul 14 2008)
The list has been created assuming k<=140000000; terms which require larger k may still be missing. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 14 2008
The numbers 37*2^j are flimsy with k=7085. The numbers 67*2^j are flimsy with k=128207979, 81*2^j are flimsy with k=1657009, 83*2^j are flimsy with k=395, 97*2^j with k=172961, 101*2^j with k=365, 113*2^j with k=145, 134*2^j with k=128207979, 137*2^j with k=125400505, any j>=0. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 14 2008
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
K. B. Stolarsky, Integers whose multiples have anomalous digital frequencies, Acta Arithmetica, 38 (1980), 117-128.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
R. J. Mathar, Examples of a(n), k and the two binary representations of a(n) and a(n)*k
K. B. Stolarsky, Integers whose multiples have anomalous digital frequencies, Acta Arith. 38 (1981) 117-128.
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EXAMPLE
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11 is flimsy because A000120(3*11) = 2 < A000120(11) = 3.
107 is flimsy because A000120(3*107) = 3 < A000120(107) = 5.
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PROGRAM
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(C++) #include <iostream> #include <cstdlib> int A000120(unsigned long long n) { int b=0 ; while(n>0) { b += n & 1 ; n >>= 1 ; } return b; } using namespace std ; int main(int argc, char *argv[]) { unsigned long long kmax=atoi(argv[1]) ; for(unsigned long long n=1; ; n++) { const int n120=A000120(n) ; for(unsigned long long k=3; k < kmax ; k+= 2) if ( A000120(k*n) < n120) { cout << n << " " << k << endl ; break ; } } } /* R. J. Mathar, mathar(AT)strw.leidenuniv.nl), Jul 14 2008 */ - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 14 2008
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CROSSREFS
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Cf. A000120, A125121 (complement).
Sequence in context: A107932 A090137 A136491 this_sequence A062019 A057891 A164708
Adjacent sequences: A005357 A005358 A005359 this_sequence A005361 A005362 A005363
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KEYWORD
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nonn,nice,base
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AUTHOR
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Jeffrey Shallit
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EXTENSIONS
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More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 14 2008
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