A005360 Flimsy numbers.

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

Author

Jeffrey Shallit

Keywords

• nonn
• nice
• base

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

Examples

			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
		

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.

Crossrefs

Cf. A000120, A125121 (complement).

Programs

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 *)

Extensions

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