

A005360


Flimsy numbers.
(Formerly M4771)


6



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
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OFFSET

1,1


COMMENTS

Definition: n is flimsy if and only if 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, Jul 07 2008
A143073(n) gives the least k for each n in this sequence.  T. D. Noe, Jul 22 2008
If k is in this sequence then so is 2*k.  David A. Corneth, Oct 01 2016


REFERENCES

Bojan Basic, The existence of nflimsy numbers in a given base, The Ramanujan Journal, March 7, 2016, pages 111. DOI 10.1007/s1113901597687.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

T. D. Noe, Table of n, a(n) for n=1..1000
Trevor Clokie et al., Computational Aspects of Sturdy and Flimsy Numbers, arxiv preprint arXiv:2002.02731 [cs.DS], February 7 2020.
R. J. Mathar, Examples of a(n), k and the two binary representations of a(n) and a(n)*k
Tony D. Noe, Odd sturdy numbers, Integer Sequences #S000848.
K. B. Stolarsky, Integers whose multiples have anomalous digital frequencies, Acta Arith. 38 (1980) 117128.


EXAMPLE

11 is flimsy because A000120(3*11) = 2 < A000120(11) = 3.
107 is flimsy because A000120(3*107) = 3 < A000120(107) = 5.
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, 137*2^j with k = 125400505, any j >= 0.  R. J. Mathar, Jul 14 2008


MATHEMATICA

nmax = 121; kmax = 200; nn = {37, 67, 81, 83, 97, 101, 113}; flimsyQ[n_ /; MemberQ[nn, n]  MatchQ[FactorInteger[n], {{2, _} , {Alternatives @@ nn, 1}}]] = True; flimsyQ[n_] := For[k = 2, True, k++, Which[DigitCount[k * n, 2, 1] < DigitCount[n, 2, 1], Return[True], k > kmax, Return[False]]]; Reap[Do[If[flimsyQ[n], Sow[n]], {n, 2, nmax}]][[2, 1]] (* JeanFrançois Alcover, May 23 2012, after R. J. Mathar *)
nmax = 200; Bits[n_Integer] := Count[IntegerDigits[n, 2], 1]; FlimsyQ[n_Integer] := FlimsyQ[n] = Module[{res, b = Bits[n], k}, If[b <= 2, False, If[EvenQ[n], FlimsyQ[n/2], res = Union[Mod[2^Range[n], n]]; If[Length[res] == n  1, True, k = 2; While[k < b && ! MemberQ[Union[Mod[Plus @@@ Subsets[res, {k}], n]], 0], k++]; k < b]]]]; Select[Range[nmax], FlimsyQ] (* JeanFrançois Alcover, Feb 11 2016, this code is due to T. D. Noe *)


PROG

(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, Jul 14 2008 */


CROSSREFS

Cf. A000120, A125121 (complement).
Sequence in context: A357074 A268487 A216687 * A269806 A062019 A057891
Adjacent sequences: A005357 A005358 A005359 * A005361 A005362 A005363


KEYWORD

nonn,nice,base


AUTHOR

Jeffrey Shallit


EXTENSIONS

More terms from R. J. Mathar, Jul 14 2008


STATUS

approved



