A005360 Flimsy numbers. (Formerly M4771)

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

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 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.

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) 117-128.

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]] (* 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

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