A002858 -id:A002858 - cp's OEIS frontend

A199017 Number of partitions of n into distinct terms of (1,2)-Ulam sequence, cf. A002858.

1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9, 10, 11, 12, 13, 14, 14, 16, 16, 17, 19, 20, 22, 23, 25, 26, 27, 29, 30, 31, 34, 35, 38, 40, 41, 45, 45, 48, 51, 52, 57, 60, 62, 66, 68, 71, 75, 78, 83, 86, 93, 97, 100, 107, 109, 115, 120, 124, 132, 138

Offset: 0

Reinhard Zumkeller, Nov 03 2011

nonn

			The first terms of A002858 are 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, ...
a(10) = #{8+2, 6+4, 6+3+1, 4+3+2+1} = 4;
a(11) = #{11, 8+3, 8+2+1, 6+4+1, 6+3+2} = 5;
a(12) = #{11+1, 8+4, 8+3+1, 6+4+2, 6+3+2+1} = 5.

Eric Weisstein's World of Mathematics, Ulam Sequence
Wikipedia, Ulam number
Index entries for Ulam numbers

Cf. A000586; A199016, A199119, A199121, A199123.

a199017 = p a002858_list where
   p _  0 = 1
   p (u:us) m | m < u = 0
              | otherwise = p us (m - u) + p us m

A199016 Number of partitions of n into terms of (1,2)-Ulam sequence, cf. A002858.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 12, 18, 22, 30, 37, 50, 60, 78, 94, 120, 143, 179, 213, 262, 309, 376, 440, 531, 618, 737, 855, 1012, 1167, 1372, 1575, 1840, 2104, 2442, 2783, 3214, 3649, 4193, 4746, 5430, 6126, 6980, 7853, 8914, 10002, 11311, 12660, 14274, 15934

Offset: 0

Reinhard Zumkeller, Nov 03 2011

nonn

			The first terms of A002858 are 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, ...
a(6) = #{6, 4+2, 4+1+1, 3+3, 3+2+1, 3+1+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1, 1+1+1+1+1+1} = 10;
a(7) = #{6+1, 4+3, 4+2+1, 4+1+1+1, 3+3+1, 3+2+2, 3+2+1+1, 3+1+1+1+1, 2+2+2+1, 2+2+1+1+1, 2+1+1+1+1+1, 1+1+1+1+1+1+1} = 12.

Eric Weisstein's World of Mathematics, Ulam Sequence
Wikipedia, Ulam number
Index entries for Ulam numbers

Cf. A000607; A199017, A199118, A199120, A199122.

199016 = p a002858_list where
   p _ 0 = 1
   p us'@(u:us) m | m < u     = 0
                  | otherwise = p us' (m - u) + p us m

A072832 First differences of Ulam's sequence A002858.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 2, 3, 2, 8, 2, 8, 2, 9, 1, 5, 4, 5, 7, 3, 5, 5, 5, 10, 2, 3, 4, 8, 12, 5, 7, 7, 3, 7, 20, 2, 3, 2, 7, 8, 9, 3, 10, 2, 15, 2, 3, 2, 10, 5, 2, 13, 9, 27, 7, 3, 5, 15, 2, 15, 2, 5, 7, 12, 8, 10, 2, 7, 3, 2, 15, 2, 3, 7, 10, 5, 27, 2, 12, 5, 20, 2, 20, 2, 20, 2, 17, 17, 3, 2, 5, 12, 3

Offset: 1

N. J. A. Sloane, Jul 25 2002

No pattern in this sequence has ever been observed, according to Finch.
In the first 49,999,999 terms, 1 occurs four times, 36.9% of the terms are 2, and some terms have not occurred. - Jud McCranie, Mar 02 2012
Even though some values seem to get larger as the sequence grows, it appears that the combined frequency of the values 2 and 3 approaches 50%.  This shows up even in the first 10000 terms (combined frequency: 49.9%). - Enrique Navarrete, May 08 2017

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 145-151.

Jud McCranie, Table of n, a(n) for n = 1..10000
Steven R. Finch, Stolarsky-Harborth Constant [Broken link]
Steven R. Finch, Stolarsky-Harborth Constant [From the Wayback machine]

Cf. A002858, A072540 (conjectured missing terms).

More terms from Matthew Conroy, Aug 24 2002

A080330 Index in A002858 of the smaller of the two Ulam numbers that sum to the n-th Ulam number.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 2, 3, 2, 6, 2, 6, 2, 7, 1, 5, 4, 11, 9, 3, 6, 8, 10, 12, 2, 3, 4, 6, 18, 19, 13, 16, 3, 17, 20, 2, 3, 2, 24, 6, 11, 3, 8, 2, 15, 2, 3, 2, 15, 20, 2, 8, 27, 13, 25, 3, 6, 18, 2, 15, 2, 15, 25, 11, 6, 10, 2, 17, 3, 2, 15, 2, 3, 27, 20, 15, 20, 2, 25, 27, 42, 2, 15

Offset: 1

Jud McCranie, Feb 15 2003

nonn

The first two are zero because the Ulam sequence (A002858) is initialized for those terms. The terms of the sum are in A080328 and A080329 and the indices are in A080330 and A080331.

			The 11th Ulam number (26) is the sum of the 6th Ulam number (8) and the 10th Ulam number (18), so a(11)=6.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A080330

Cf. A002858, A080328, A080329, A080331.

PARI
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See Links section.
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A080331 Index in A002858 of the larger of the two Ulam numbers that sum to the n-th Ulam number.

Original entry on oeis.org

0, 0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 13, 15, 15, 17, 13, 17, 20, 20, 20, 20, 20, 25, 26, 27, 28, 20, 20, 27, 25, 33, 27, 28, 36, 37, 38, 27, 40, 38, 42, 42, 44, 40, 46, 47, 48, 42, 40, 51, 52, 38, 53, 44, 56, 56, 54, 59, 55, 61, 56, 53, 61, 65, 65, 67, 61, 69, 70

Offset: 1

Jud McCranie, Feb 15 2003

nonn

The first two terms in this sequence are zero because the Ulam sequence (A002858) is initialized for those terms. The terms of the sum are in A080328 and A080329 and the indices are in A080330 and A080331.

			The 11th Ulam number (26) is the sum of the 6th Ulam number (8) and the 10th Ulam number (18), so a(11)=10.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A080331

Cf. A002858, A080328, A080329, A080330.

PARI
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See Links section.
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A307328 Indices of prime Ulam numbers: numbers k such that A002858(k) is prime.

Original entry on oeis.org

2, 3, 7, 8, 15, 17, 25, 31, 41, 48, 69, 73, 91, 97, 106, 107, 123, 125, 138, 167, 172, 177, 181, 193, 194, 241, 242, 246, 267, 280, 286, 287, 297, 306, 312, 322, 323, 338, 340, 343, 353, 354, 382, 388, 393, 398, 403, 411, 412, 415, 416, 433, 444, 448, 460

Offset: 1

Amiram Eldar, Apr 02 2019

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002858, A068820.

ulams = {1, 2}; Do[AppendTo[ulams, n = Last[ulams]; While[n++; Length[ DeleteCases[Intersection[ulams, n - ulams], n/2, 1, 1]] != 2]; n], {100}]; Flatten[Position[ulams, ?(PrimeQ[#] &)]] (* after _Jean-François Alcover at A002858 *)

A002858(a(n)) = A068820(n).

A072540 Numbers that are not differences between successive Ulam numbers (A002858).

Original entry on oeis.org

6, 11, 14, 16, 18, 21

Offset: 1

Benoit Cloitre, Aug 04 2002

Unfortunately, it appears that the entries are all conjectural. I am not aware of any proofs that the listed numbers can never appear. The terms shown are those listed by Pickover. - N. J. A. Sloane, Jun 19 2008
However, a gap of 35 doesn't occur until after the 35755308th term (483379914), so these missing gaps could eventually occur. - Jud McCranie, Jun 13 2008
The sequence is conjectured to continue: 23, 26, 28, 31, 33.  These gaps do not appear in the first 158000000 terms, so are candidates for the sequence. [Jud McCranie, Sep 12 2013]
The conjectured list of gaps that don't appear holds through the first 28 billion Ulam numbers. - Jud McCranie, Jan 07 2016

Clifford A. Pickover, "Wonders of Numbers, ...", Oxford University Press, 2000

Philip Gibbs and Judson McCranie, The Ulam Numbers up to One Trillion, (2017).
Shyam Sunder Gupta, Ulam Numbers. In: Exploring the Beauty of Fascinating Numbers. Springer Praxis Books(). Springer, Singapore, (2025).
C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review

Cf. A002858, A072832.

Edited by Max Alekseyev, Dec 19 2011

A285884 For n => 1, the number of distinct summands u and v that can be used in the representation of n as u+v, where u and v are two (possibly equal) Ulam numbers A002858.

Original entry on oeis.org

0, 1, 2, 3, 4, 3, 4, 3, 4, 4, 2, 5, 2, 6, 4, 3, 6, 2, 8, 4, 4, 5, 0, 6, 0, 3, 4, 2, 8, 4, 4, 5, 0, 6, 0, 3, 4, 2, 8, 4, 4, 6, 0, 8, 0, 4, 2, 2, 8, 4, 6, 5, 2, 10, 4, 7, 2, 4, 6, 4, 6, 2, 6, 10, 6, 8, 0, 4, 2, 6, 4, 3, 10, 6, 10, 5, 2, 6, 4, 8, 4, 2, 10, 4, 12

Offset: 1

Enrique Navarrete, Apr 27 2017

nonn

An odd number in the sequence means that there exists the "pseudo-representation" u + u, where u is an Ulam number. For example, a(22)=5 since 22 = 18 + 4 = 16 + 6 = 11 + 11, and the 5 distinct summands 18,4,16,6,11 are Ulam numbers.
Note that both 2 and 3 are values for Ulam numbers since, by the previous comment, a value of 3 means that the Ulam number has the additional "pseudo-representation" u + u (see the Examples).
It seems that all nonnegative integers occur as values of this sequence.

			a(23) = 0 since 23 can't be written as the sum of two distinct Ulam numbers. This type of numbers are in A033629.
a(94) = 1 since 94 = 47 + 47, where 47 is an Ulam number.  This type of numbers are in A287611.
a(11) = 2 since 11 has the unique representation 11 = 8 + 3, where 8,3 are Ulam numbers. If such n is also an Ulam number (such as 11), then it is in A002858.
a(8) = 3 since it has the representation 8 = 6 + 2 and also the additional "pseudo-representation" 8 = 4 + 4, where 6, 2, and 4 are Ulam numbers. If n has such a "pseudo-representation" and is an Ulam number, then it is in A068799.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C program for A285884
Rémy Sigrist, Density plot of the first 2500000 terms

Cf. A002858, A068799, A033629, A287611.

C
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See Links section.
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A307329 Indices of twin Ulam primes: numbers k such that both u(k) and u(k+1) are primes, where u(k) = A002858(k) are the Ulam numbers.

Original entry on oeis.org

2, 7, 106, 193, 241, 286, 322, 353, 411, 415, 753, 858, 859, 1086, 1164, 1305, 1547, 1548, 1625, 1631, 1648, 1678, 1896, 1972, 2007, 2103, 2406, 2503, 2515, 2516, 2530, 2553, 2638, 2714, 3003, 3059, 3060, 3337, 3903, 4012, 4072, 4299, 4386, 4404, 4625, 4698

Offset: 1

Amiram Eldar, Apr 02 2019

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002858, A068820, A307328.

ulams = {1, 2}; Do[AppendTo[ulams, n = Last[ulams]; While[n++; Length[ DeleteCases[Intersection[ulams, n - ulams], n/2, 1, 1]] != 2]; n], {1000}]; p=PrimeQ[ulams]; len=Length[p]; s={}; Do[If[p[[n]]&&p[[n+1]], AppendTo[s,n]], {n,1,len-1}]; s (* after Jean-François Alcover at A002858 *)

A331729 Number of Ulam numbers u (A002858) between powers of 2, 2^n < u <= 2^(n+1).

Original entry on oeis.org

1, 2, 2, 3, 3, 7, 11, 20, 31, 47, 92, 162, 312, 632, 1235, 2460, 4844, 9665, 19335, 38727, 77569, 155729, 310405, 620596, 1240580, 2481645, 4966229, 9926596, 19855760, 39717367, 79428417

Offset: 0

Frank M Jackson, Jan 25 2020

Conjecture 1: For all m > 1 there is always at least one Ulam number u(j) such that m < u(j) < 2m.
Conjecture 2: For all m > 4 there is always at least two Ulam numbers u(j), u(j+1) such that m < u(j) < u(j+1) < 2m.
This sequence illustrates how far these conjectures are oversatisfied.
Conjecture 1 implies that Ulam numbers form a complete sequence because u(1) = 1 and 2u(j) >= u(j+1).
Conjecture 2 implies that three consecutive Ulam numbers satisfies the triangle inequality because 2u(j) > u(j+2) > u(j+1) > u(j) and u(j) + u(j+1) > 2u(j) > u(j+2). It further implies that n consecutive Ulam numbers can always form an n-gon.

			a(6) = 11 because the Ulam numbers between 64 and 128 are (69, 72, 77, 82, 87, 97, 99, 102, 106, 114, 126).

Cf. A002858, A199017, A307331.

ulams = {1, 2}; Do[AppendTo[ulams, n = Last[ulams]; While[n++; Length[DeleteCases[Intersection[ulams, n-ulams], n/2, 1, 1]]!=2]; n], {10^4}]; ulst = ulams; (* Jean-François Alcover, Sep 08 2011 *)
upi[n_] := Module[{p = 1}, While[ulst[[p]] <= n, p++]; p - 1]; Table[upi[2^(n + 1)] - upi[2^n], {n, 0, 16}]

a(20)-a(21) from Sean A. Irvine, Feb 29 2020

a(22)-a(30) from Amiram Eldar, Aug 22 2020

cp's OEIS Frontend

A199017 Number of partitions of n into distinct terms of (1,2)-Ulam sequence, cf. A002858.

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A199016 Number of partitions of n into terms of (1,2)-Ulam sequence, cf. A002858.

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Haskell

A072832 First differences of Ulam's sequence A002858.

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A080330 Index in A002858 of the smaller of the two Ulam numbers that sum to the n-th Ulam number.

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PARI

A080331 Index in A002858 of the larger of the two Ulam numbers that sum to the n-th Ulam number.

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PARI

A307328 Indices of prime Ulam numbers: numbers k such that A002858(k) is prime.

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Mathematica

Formula

A072540 Numbers that are not differences between successive Ulam numbers (A002858).

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Extensions

A285884 For n => 1, the number of distinct summands u and v that can be used in the representation of n as u+v, where u and v are two (possibly equal) Ulam numbers A002858.

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C

A307329 Indices of twin Ulam primes: numbers k such that both u(k) and u(k+1) are primes, where u(k) = A002858(k) are the Ulam numbers.

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