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
If k is in this sequence then so is 2*k. - David A. Corneth, Oct 01 2016
REFERENCES
Bojan Basic, The existence of n-flimsy numbers in a given base, The Ramanujan Journal, March 7, 2016, pages 1-11. DOI 10.1007/s11139-015-9768-7.
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.
Tony D. Noe, Odd sturdy numbers, Integer Sequences #S000848.
K. B. Stolarsky, Integers whose multiples have anomalous digital frequencies, Acta Arith. 38 (1980) 117-128.
EXAMPLE
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]] (* Jean-François 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] (* Jean-François 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
KEYWORD
nonn,nice,base
AUTHOR
EXTENSIONS
More terms from R. J. Mathar, Jul 14 2008
STATUS
approved