A005243 -id:A005243 - cp's OEIS frontend

A118164 Number of representations of A005243(n) as sum of consecutive earlier terms in A005243.

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

Offset: 1

Reinhard Zumkeller, Apr 13 2006

nonn

A118165(n) = Min{m: a(m) = n};

for n>2: a(n) > 0 by definition of A005243.

			A005243(33) = 54 = 29+25 = Sum(A005243[17:18]) =
19+18+17 = Sum(A005243[11:13]) = 14+11+10+8+6+5 =
Sum(A005243[4:9]), therefore a(33) = 3.

Eric Weisstein's World of Mathematics, Hofstadter Sequences

A048973 Complement of A005243.

Original entry on oeis.org

4, 7, 9, 12, 13, 15, 20, 23, 26, 27, 28, 31, 36, 38, 39, 42, 44, 48, 50, 52, 53, 55, 56, 61, 63, 64, 66, 74, 79, 83, 85, 89, 98, 101, 103, 107, 109, 114, 120, 123, 125, 128, 131, 133, 136, 144, 152, 157, 159, 160, 165, 168, 182, 184, 190, 192, 198, 203, 208, 212

Offset: 1

D. R. Hofstadter

The sequence appears to be growing slightly faster than quadratically. - T. D. Noe, Nov 04 2007

T. D. Noe, Table of n, a(n) for n=1..6564 (terms < 10^8)

import Data.List.Ordered (minus)
a048973 n = a048973_list !! (n-1)
a048973_list = [1..] `minus` a005243_list
-- Reinhard Zumkeller, Dec 17 2015

nmax = 250; 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]]]; A048973 = Complement[Range[nmax], s] (* Jean-François Alcover, Oct 21 2016 *)

More terms from Larry Reeves (larryr(AT)acm.org), Oct 02 2000

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

A085921 Duplicate of A005243.

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

Offset: 1

dead

A118065 a(1) = a(2) = 1, a(n) = smallest number greater than a(n-1) that can be written as sum of consecutive earlier terms.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78

Offset: 1

Reinhard Zumkeller, Apr 11 2006

nonn

Complement of A118066.

			a(20) = a(11) + a(10) = 11 + 10 = 21;
a(21) = a(9) + a(8) + a(7) = 9 + 7 + 6 = 22;
a(22) = a(12) + a(11) = 12 + 11 = 23;
a(23) = a(13) + a(12) = 13 + 12 = 25;
a(24) = a(10) + a(9) + a(8) = 10 + 9 + 7 = 26.

Eric Weisstein's World of Mathematics, Hofstadter Sequences

Cf. A005243, A118066.

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

Original entry on oeis.org

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

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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A080338 Product of consecutive previous terms (starting with 2,3).

Original entry on oeis.org

2, 3, 6, 18, 36, 108, 324, 648, 3888, 11664, 23328, 34992, 69984, 209952, 419904, 1259712, 2519424, 22674816, 45349632, 136048896, 272097792, 408146688, 816293376, 2448880128, 14693280768, 29386561536, 88159684608, 264479053824

Offset: 1

Benoit Cloitre, Mar 19 2003

nonn

			Sequence begins : 2,3,6,18 and 2*3*6=36 is the smallest product of consecutive terms greater than 18, hence a(7)=36.

Cf. A005243 (sum of consecutive previous terms).

More terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net), Jul 22 2005

cp's OEIS Frontend

A118164 Number of representations of A005243(n) as sum of consecutive earlier terms in A005243.

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A048973 Complement of A005243.

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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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A085921 Duplicate of A005243.

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A118065 a(1) = a(2) = 1, a(n) = smallest number greater than a(n-1) that can be written as sum of consecutive earlier terms.

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

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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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Author

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Programs

PARI

A080338 Product of consecutive previous terms (starting with 2,3).

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Keywords

Examples

Crossrefs

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