Steven J. Kifowit - cp's OEIS frontend

User: Steven J. Kifowit

Steven J. Kifowit has authored 2 sequences.

A277603 Exceptional Bobo numbers: terms of A242679 that satisfy frac[e*A242679(n)]<(e-1)/2.

36, 9045, 5195512, 5311399545, 8488859795196, 25466579385587, 19542965851120621, 58628897553361862, 61250772004870841520, 183752316014612524559, 250769086731739376780337, 752307260195218130341010, 1299515735021702625544976020, 3898547205065107876634928059

Offset: 1

Steven J. Kifowit, Oct 22 2016

nonn

The exceptional Bobo numbers (EBNs) are very rare relative to the Bobo numbers (A242679).

Exceptional Bobo numbers come in two varieties. Type-1 EBNs are given by the recurrence E(0)=1,E(1)=1,E(k)=(2*k-1)*(2*E(k-1)-1)+E(k-2) for k=3,5,7,... These are derived from the denominators of the odd-indexed convergents of the continued fraction expansion of (e-1)/2 = [0;1,6,10,14,18,...]. The Type-2 EBNs are derived from the Type-1 EBNs. They have the form n*m-(m-1)/2 where n is a Type-1 EBN and m>=3 is an odd integer. However, not every number of this form is an EBN.

S. J. Kifowit, A. Mitchell, and S. Zandi, Exceptional Bobo Numbers, in preparation 2016

Steven J. Kifowit, Table of n, a(n) for n = 1..99
E. R. Bobo, A sequence related to the harmonic series, College Math. J. 26 (1995), 308-310.
D. T. Clancy and S. J. Kifowit, A closer look at Bobo's sequence, College Math. J. 45 (2014), 199-206.
Steve Kifowit, Bobo Numbers, Bobbers, and Bears—Experiences, Undergraduate Research, Preprint, 2016.

Cf. A103762, A242679 (Bobo numbers).

A242679 Bobo numbers: Numbers k with the property that floor(e*k) = least m with Sum_{j=k..m} 1/j > 1.

Original entry on oeis.org

4, 11, 18, 25, 32, 36, 43, 50, 57, 64, 71, 75, 82, 89, 96, 103, 114, 121, 128, 135, 142, 146, 153, 160, 167, 174, 185, 192, 199, 206, 213, 217, 224, 231, 238, 245, 256, 263, 270, 277, 284, 288, 295, 302, 309, 316, 327, 334, 341, 348, 355, 359, 366, 373, 380, 387, 398, 405, 412, 419, 426, 430, 437, 444, 451, 458, 469, 476, 483, 490, 497

Offset: 1

Steven J. Kifowit, May 20 2014

nonn

These are the numbers n for which A103762(n) = floor(e*n).
If frac(e*n) > (e-1)/2, then n is a Bobo number, but not every Bobo number has this property. The exceptions are in A277603.
In Bobo's article (see Bobo link), the Bobo numbers through 2105 are listed. There is a typo: the number 143 is given in place of the correct number 142.
These numbers are mentioned in the comments associated with A103762. Differences between consecutive Bobo numbers are indeed 4, 7, or 11. An elementary proof is given in the Clancy/Kifowit link.

Steven J. Kifowit, Table of n, a(n) for n = 1..10000
E. R. Bobo, A sequence related to the harmonic series, College Math. J. 26 (1995), 308-310.
D. T. Clancy and S. J. Kifowit, A closer look at Bobo's sequence, College Math. J. 45 (2014), 199-206.
Steve Kifowit, Bobo Numbers, Bobbers, and Bears—Experiences, Undergraduate Research, Preprint, 2016.

Cf. A103762, A277603.

is(n)=my(e=exp(1),s); if(frac(e*n)>(e-1)/2, return(1)); s=sum(j=n,e*n\1-1, 1/j); s<=1 && s+e*n\1>1 \\ Charles R Greathouse IV, Sep 17 2016

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User: Steven J. Kifowit

A277603 Exceptional Bobo numbers: terms of A242679 that satisfy frac[e*A242679(n)]<(e-1)/2.

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