A118164 -id:A118164 - cp's OEIS frontend

A118165 Smallest number m such that A118164(m) = n.

1, 3, 8, 27, 23, 112, 236, 1050, 2210, 2561, 4016, 5504, 9325, 18328, 13506, 40150, 38789, 60881, 44541, 74499, 106766, 53339, 158352, 238255, 196734, 224410, 136750, 620509, 372124, 632976, 810969, 411364, 1034731, 1117857, 882335, 1672055

Offset: 0

Reinhard Zumkeller, Apr 13 2006

nonn

A118164(a(n)) = n; A118166(n) = A005243(a(n)).

a(15)-a(35) from Donovan Johnson, Feb 16 2011

A005243 A self-generating sequence: start with 1 and 2, take all sums of any number of successive previous elements and adjoin them to the sequence. Repeat!

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 10, 11, 14, 16, 17, 18, 19, 21, 22, 24, 25, 29, 30, 32, 33, 34, 35, 37, 40, 41, 43, 45, 46, 47, 49, 51, 54, 57, 58, 59, 60, 62, 65, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 80, 81, 82, 84, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 97, 99, 100

Offset: 1

D. R. Hofstadter, Jul 15 1977

Most of the natural numbers are members. Conjecture: there are infinitely many nonmembers. Is there an estimate for a(k)/k ?

A118164(n) = number of representations of a(n) as sum of consecutive earlier terms. - Reinhard Zumkeller, Apr 13 2006

			After 1,2,3,5,6 you can adjoin 8 = 3+5, 10 = 2+3+5, etc.
12 is not a term since it is not the sum of any set of consecutive previous terms.

R. K. Guy, Unsolved Problems in Number Theory, E31.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..1000
Thomas Bloom, Problem 423, Erdős Problems.
D. R. Hofstadter, Eta-Lore [Cached copy, with permission]
D. R. Hofstadter, Pi-Mu Sequences [Cached copy, with permission]
D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991
Eric Weisstein's World of Mathematics, Hofstadter Sequences.

Complement of A048973.

Cf. A118065, A118166.

import Data.Set (singleton, deleteFindMin, fromList, union, IntSet)
a005243 n = a005243_list !! (n-1)
a005243_list = 1 : h [1] (singleton 2) where
   h xs s = m : h (m:xs) (union s' $ fromList $ map (+ m) $ scanl1 (+) xs)
     where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Dec 17 2015, Jun 22 2011

nmax = 200; For[ s = {1, 2}; n = 3, n <= nmax, n++, ls = Length[s]; tt = Total /@ Flatten[Table[s[[i ;; j]], {i, 1, ls-1}, {j, i+1, ls}], 1]; If[MemberQ[tt, n], AppendTo[s, n]]]; A005243 = s (* Jean-François Alcover, Oct 21 2016 *)

More terms from Jud McCranie

A118166 Smallest term in the Hofstadter sequence A005243 having exactly n representations as sum of consecutive earlier terms.

Original entry on oeis.org

1, 3, 11, 43, 35, 162, 311, 1203, 2405, 2769, 4257, 5772, 9639, 18711, 13860, 40635, 39270, 61425, 45045, 75075, 107415, 53865, 159075, 239085, 197505, 225225, 137445, 621621, 373065, 634095, 812175, 412335, 1036035, 1119195, 883575, 1673595

Offset: 0

Reinhard Zumkeller, Apr 13 2006

nonn

a(n) = A005243(A118165(n)).

			a(8) = A005243(A118165(8)) = A005243(2210) = 2405:
#1: 1203 + 1202 = Sum(A005243[1049:1050]) = 2405,
#2: 803 + 802 + 800 = Sum(A005243[671:673]) = 2405,
#3: 483 + 482 + 481 + 480 + 479 = Sum(A005243[382:386]),
#4: 306 + 304 + 302 + 301 + 300 + 299 + 297 + 296 =
Sum(A005243[224:231]) = 2405,
#5: 224 + 223 + 222 + 221 + 220 + 219 + 218 + 216 + 215 +
214 + 213 = Sum(A005243[153:163]) = 2405,
#6: 145 + 143 + 142 + 141 + 140 + 139 + 138 + 137 + 135 +
134 + 132 + 130 + 129 + 127 + 126 + 124 + 122 + 121 =
Sum(A005243[82:99]) = 2405,
#7: 129 + 127 + 126 + 124 + 122 + 121 + 119 + 118 + 117 +
116 + 115 + 113 + 112 + 111 + 110 + 108 + 106 + 105 +
104 + 102 + 100 = Sum(A005243[67:87]) = 2405,
#8: 95 + 94 + 93 + 92 + 91 + 90 + 88 + 87 + 86 + 84 + 82 +
81 + 80 + 78 + 77 + 76 + 75 + 73 + 72 + 71 + 70 + 69 + 68 +
67 + 65 + 62 + 60 + 59 + 58 + 57 + 54 + 51 =
Sum(A005243[32:63]) = 2405.

Eric Weisstein's World of Mathematics, Hofstadter Sequences

Cf. A118164.

a(15)-a(35) from Donovan Johnson, Feb 16 2011

A124145 a(1)=1, a(2)=2, a(n)=smallest number greater than a(n-1) that can be written as sum of consecutive earlier terms in exactly one way.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 10, 16, 17, 18, 19, 22, 25, 26, 29, 32, 33, 37, 40, 41, 43, 45, 47, 48, 50, 54, 55, 57, 59, 62, 66, 67, 68, 69, 73, 75, 76, 77, 81, 83, 85, 86, 87, 95, 98, 99, 101, 105, 109, 117, 118, 120, 126, 128, 129, 131, 133, 134, 137, 139, 140, 141, 143, 146, 148

Offset: 1

Tobias Baumann (baumtobi(AT)students.uni-mainz.de), Dec 01 2006

This sequence is similar to the Hofstadter sequence A005243 except the decomposition into summands has to be unique.

This sequence has similarities with Ulam numbers (A002858); here we consider unique sums of consecutive terms, there unique sums of two distinct terms. - Rémy Sigrist, Jan 02 2022

			a(7)=10 because 2+3+5=10 is the only way to sum up consecutive terms. 11 is not contained in the sequence because 11=5+6=1+2+3+5 has got more than one decompositions.

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000
Rémy Sigrist, PARI program for A124145

Cf. A002858, A005243, A118065, A118164, A118166.

PARI
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See Links section.
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A117118 Number of partitions of n into consecutive parts containing no binary powers.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 2, 1, 1, 0, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 0, 2, 1, 2, 2, 2, 1, 3, 1, 2, 3, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 3, 3, 1, 3, 1, 2, 3, 2, 1, 2, 0, 2, 2, 2, 1, 4, 1, 2, 2, 2, 2, 4, 1, 2, 3, 2, 1, 3, 2, 2, 3, 2, 2, 3, 1, 2, 3, 2, 1, 2, 2, 3, 1, 2, 2, 2, 2, 2, 3, 2, 1, 4

Offset: 1

Reinhard Zumkeller, Apr 17 2006

nonn

a(n) <= A001227(n).

			a(60) = #{10+11+12+13+14, 19+20+21, 60} = 3, 4+5+6+7+8+9+10+11=60 doesn't count because 4=2^2 or 8=2^3.

Cf. A001227, A118164.

a[n_] := a[n] = Count[IntegerPartitions[n, All, Cases[Range[3, n], k_Integer /; Total[IntegerDigits[k, 2]] > 1]], q_List /; Length[q] == Length[Union[q]] && Length[q] == First[q] - Last[q] + 1];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 105}] (* Jean-François Alcover, Oct 06 2021 *)

cp's OEIS Frontend

A118165 Smallest number m such that A118164(m) = n.

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A005243 A self-generating sequence: start with 1 and 2, take all sums of any number of successive previous elements and adjoin them to the sequence. Repeat!

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A118166 Smallest term in the Hofstadter sequence A005243 having exactly n representations as sum of consecutive earlier terms.

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Extensions

A124145 a(1)=1, a(2)=2, a(n)=smallest number greater than a(n-1) that can be written as sum of consecutive earlier terms in exactly one way.

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PARI

A117118 Number of partitions of n into consecutive parts containing no binary powers.

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Mathematica