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A275288 Least k such that there exists a sequence b_1 < b_2 < ... < b_t = k that includes n and has a reciprocal sum of 1. 1
1, 6, 6, 12, 20, 6, 28, 24, 18, 15, 33, 12, 65, 28, 15, 48, 85, 18, 76, 20, 28, 33, 115, 24, 100, 52, 54, 28, 145, 30, 217, 96, 33, 85, 35, 36, 296, 95, 52, 40, 246, 42, 301, 55, 45, 138, 329, 48, 196, 75, 102, 52, 371, 54, 55, 56, 76, 174 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

From Robert Price, Jan 04 2017: (Start)

a(11) = 33       [2,3,11,22,33]

65>= a(13) >26  [2,3,13,26,52,60,65] No better solution with less than 15 terms.

48>= a(16) >32  [2,3,16,18,36,48] No better solution with less than 24 terms.

85>= a(17) >34  [2,3,15,17,34,85] No better solution with less than 12 terms.

76>= a(19) >19  [2,3,12,19,57,76] No better solution with less than 12 terms.

a(20) = 20       [2,4,5,20]

a(21) = 28       [2,4,8,21,24,28]

a(22) = 33       [2,4,11,20,22,30,33]

115>= a(23) >23  [2,3,10,23,69,115] No better solution with less than 11 terms.

a(24) = 24       [2,3,8,24]

100>= a(25) >25  [2,3,10,25,60,100] No better solution with less than 11 terms.

52>= a(26) >26  [2,3,12,26,39,52] No better solution with less than 16 terms.

54>= a(27) >27  [2,3,12,27,36,54] No better solution with less than 9 terms.

a(28) = 28       [2,3,12,21,28]

145>= a(29) >29  [2,4,5,29,116,145] No better solution with less than 9 terms.

a(30) = 30       [2,3,12,20,30]

217>= a(31) >31  [2,3,9,31,93,126,217] No better solution with less than 9 terms.

96>= a(32) >32  [2,3,9,32,72,96] No better solution with less than 11 terms.

a(33) = 33       [2,3,11,22,33]

85>= a(34) >34  [2,3,17,20,34,60,85] No better solution with less than 9 terms.

a(35) = 35       [2,3,14,15,35]

a(36) = 36       [2,3,12,18,36]

296>= a(37) >37  [2,3,8,37,148,222,296] No better solution with less than 8 terms.

95>= a(38) >38  [2,4,5,38,76,95] No better solution with less than 11 terms.

52<= a(39) >39  [2,4,6,26,39,52] No better solution with less than 15 terms.

a(40) = 40       [2,3,10,24,40]

246>= a(41) >41  [2,3,8,41,120,205,246] No better solution with less than 9 terms.

a(42) = 42       [2,3,7,42]

192>= a(64)      [2,3,8,48,64,192] No better solution with less than 9 terms.

162>= a(81)      [2,3,8,72,81,108,162] No better solution with less than 9 terms.

384>= a(128)     [2,3,7,96,128,336,384] No better solution with less than 8 terms.

486>= a(243)     [2,3,7,81,243,336,432,486] No better solution with less than 9 terms.

a(216) = 216     [2,3,8,27,216]

196>= a(49)      [2,3,8,49,98,168,196] No better solution with less than 8 terms.

a(100) = 100     [2,4,5,25,100]

363>= a(121)     [2,3,7,121,176,242,336,363] No better solution with less than 8 terms.

a(144) = 144     [2,3,7,112,126,144]

a(196) = 196     [2 ,3,7,84,147,196]

a(225) = 225     [2,3,9,25,90,225]

a(500) = 500     [2,4,5,25,125,500]

It appears that in most cases a(n) is a small multiple of n. For example: a(8)=3*8, a(11)=3*11, a(35)=1*35.

If not a small multiple of n, then a small rational times n.  For example: a(10)=3/2*10, a(21)=4/3*21, a(22)=3/2*22.

Conjectures:

   a(2^n) = 3*n

   a(3^n) = 2*n

   a(5^n) = 4*n

   a(6^n) = n

   a(7^n) = 4*n

(End)

From Peter Kagey, Jul 20 2017: (Start)

a(n) = n if and only if n is in A092671.

Every term in this sequence is in A092671.

a(a(n)) = a(n); that is, this sequence is idempotent.

(End)

LINKS

Table of n, a(n) for n=1..58.

EXAMPLE

a(1)  = 1  via [1]

a(2)  = 6  via [2, 3, 6]

a(3)  = 6  via [2, 3, 6]

a(4)  = 12 via [2, 4, 6, 12]

a(5)  = 20 via [2, 4, 5, 20]

a(6)  = 6  via [2, 3, 6]

a(7)  = 28 via [2, 4, 7, 14, 28]

a(8)  = 24 via [2, 3, 8, 24]

a(9)  = 18 via [2, 3, 9, 18]

a(10) = 15 via [2, 3, 10, 15]

a(11) > 30

a(12) = 12 via [2, 4, 6, 12]

a(13) > 30

a(14) = 28 via [2, 4, 7, 14, 28]

a(15) = 15 via [2, 3, 10, 15]

a(16) > 30

a(17) > 30

a(18) = 18 via [2, 3, 9, 18]

MATHEMATICA

Table[SelectFirst[Range@ 20, MemberQ[Map[Total, 1/DeleteCases[Rest@ Subsets[Range@ #, #], w_ /; FreeQ[w, n]]], 1] &] /. k_ /; MissingQ@ k -> 0, {n, 12}] (* Michael De Vlieger, Aug 18 2016, Version 10.2, values of a(n) > 20 appear as 0 *)

CROSSREFS

Cf. A006255, A092671, A272036, A272083.

Sequence in context: A315796 A242951 A022089 * A110357 A091827 A315797

Adjacent sequences:  A275285 A275286 A275287 * A275289 A275290 A275291

KEYWORD

nonn

AUTHOR

Peter Kagey, Aug 18 2016

EXTENSIONS

a(11)-a(12) from Robert Price, Jan 07 2017

a(13)-a(58) from David A. Corneth, Jul 20 2017

STATUS

approved

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Last modified April 19 20:04 EDT 2019. Contains 322291 sequences. (Running on oeis4.)