A002858 -id:A002858 - cp's OEIS frontend

A183528 An Ulam-type sequence: a(n) = n if n<=5; for n>5, a(n) = least number > a(n-1) which is a unique sum of 5 distinct earlier terms.

1, 2, 3, 4, 5, 15, 25, 26, 27, 28, 29, 35, 43, 45, 165, 171, 172, 174, 180, 181, 328, 333, 338, 339, 340, 341, 493, 499, 500, 647, 652, 657, 658, 659, 660, 661, 662, 663, 815, 818, 819, 971, 1127, 1137, 1138, 1139, 1140, 1141, 1142

Offset: 1

Jonathan Vos Post and Alois P. Heinz, Jan 05 2011

nonn

An Ulam-type sequence - see A002858 for further information.

			a(6) = 15 = 1 + 2 + 3 + 4 + 5 = 5*6/2, because it is the least number >5 with a unique sum of 5 distinct earlier terms.
a(7) = 25 = 1 + 2 + 3 + 4 + 15 = 5^2, because it is the least number >15 with a unique sum of 5 distinct earlier terms.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for Ulam numbers

Column k=5 of A183534.

Cf. A002858, A007086, A183527-A183533, A135737.

Maple
```
# see A183534 for programs.
```

A274522 Index of the smaller Ulam number of record gaps (A080287).

Original entry on oeis.org

1, 4, 6, 10, 14, 24, 29, 35, 54, 107, 122, 150, 181, 207, 231, 439, 486, 1509, 2929, 4952, 18857, 69180, 21431879, 24576523, 49395953, 51744075, 71482877, 16614427386, 23647775833

Offset: 1

Jud McCranie, Jun 26 2016

nonn

			There is a gap of 8 between the Ulam numbers 18 and 26, this is the 4th term in A080287 and 18 is the 10th Ulam number, so a(4)=10.

Cf. A080287, A080288, A002858.

A287612 Ulam numbers k such that 4*k is also an Ulam number.

Original entry on oeis.org

1, 2, 4, 18, 114, 131, 180, 258, 324, 412, 431, 605, 646, 864, 1037, 1081, 1101, 1167, 1296, 1313, 1404, 1470, 1765, 1856, 1858, 1900, 1944, 2032, 2117, 2327, 2572, 2630, 2635, 2674, 2787, 2809, 2814, 2897, 3068, 3214, 3219, 3261, 3393, 3451, 3591, 3622, 3630

Offset: 1

Enrique Navarrete, May 27 2017

nonn

Cf. A002858, A068791, A285885.

(* First run one of the programs for A002858 to populate the ulams list *) A287612 = ulams; curr = 1; While[curr < Length[ulams] + 1, A287612[[curr]] = ulams[[curr]] * Boole[MemberQ[ulams, 4ulams[[curr]]]]; curr++]; DeleteCases[A287612, 0] (* Alonso del Arte, May 28 2017 *)

A336528 a(1) = 1; a(2) = 2; for n > 2, a(n) is the least number > a(n-1) whose decimal representation is uniquely the concatenation of the decimal representations of two distinct earlier terms.

Original entry on oeis.org

1, 2, 12, 21, 112, 122, 211, 221, 1112, 1121, 1211, 1222, 2111, 2122, 2212, 2221, 11112, 11122, 11221, 11222, 12211, 12222, 21111, 21122, 22111, 22112, 22211, 22221, 111112, 111121, 111212, 112112, 112121, 112122, 112212, 121111, 121122, 121211, 121222, 122122

Offset: 1

Rémy Sigrist, Jul 24 2020

This sequence is inspired by Ulam sequence (A002858).
All terms belong to A007931.
Applying the mapping 1 -> 0, 2 -> 1 to the decimal representations of the terms of this sequence gives the sequence U({0, 1}) described in the article by Bade et al. in Links section. - Rémy Sigrist, Aug 08 2020

			The first terms, alongside A007931 and the corresponding concatenations, are:
  n   a(n)  A007931  concatenations
  --  ----  -------  --------------
   1     1        1
   2     2        2
                 11
   3    12       12  1|2
   4    21       21  2|1
                 22
                111  1|11, 11|1
   5   112      112  1|12
                121  1|21, 12|1
   6   122      122  12|2
   7   211      211  21|1
                212  2|12, 21|2
   8   221      221  2|21
                222
               1111
   9  1112     1112  1|112
  10  1121     1121  112|1

Rémy Sigrist, Table of n, a(n) for n = 1..15616 (terms < 10^15)
Tej Bade, Kelly Cui, Antoine Labelle, and Deyuan Li, Ulam Sets in New Settings, arXiv:2008.02762 [math.CO], 2020. See also Integers (2020) Vol. 20, #A102.
Rémy Sigrist, PARI program for A336528

Cf. A002858, A007931, A336527 (binary variant).

PARI
```
See Links section.
```

A349462 The a_i coefficients of the standard form decomposition of U(1,X), the Ulam sequence starting with 1, X in the ordered abelian group of linear integer polynomials in X, where the ordering is lexigraphical.

Original entry on oeis.org

0, 1, 2, 4, 4, 5, 7, 10, 11, 13, 16, 17, 19, 20, 22, 23, 25, 26, 28, 31, 34, 38, 40, 40, 43, 44, 46, 49, 52, 55, 59, 62, 64, 68, 70, 76, 79, 85, 88, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 137, 139, 142, 145, 148, 151, 154, 157, 163, 166, 172, 176, 178, 181, 184, 187

Offset: 0

Arseniy (Senia) Sheydvasser, Nov 19 2021

nonn

The set U(1,X) can be defined as follows: it is the unique set of linear integer polynomials such that 1,X are the two smallest elements; for any interval [P,Q] where P,Q are linear integer polynomials, U(1,X) intersect [P,Q] has a minimum and a maximum; and finally, an element u > X is in U(1,X) if and only if it is the smallest element larger than the maximum of U(1,X) intersect [1,u - 1] and which can be written as a sum of two distinct elements in U(1,X) in exactly one way.
U(1,X) can be written uniquely as a union of intervals [a_i X + b_i, c_i X + d_i], where c_i X + d_i + 1 < a_{i + 1} X + b_{i + 1} for all indices i. Here, we give just the coefficients a_i.
This set is related to Ulam sequences in an odd way. Let U(1,n) be the sequence of integers starting with 1,n such that every subsequent term is the next smallest element that can be written as the sum of two distinct prior terms in exactly one way. Then, for all integers k, for all sufficiently large n, U(1,n) intersect [1,c_k n + d_k + 1] = eval_n(U(1,X) intersect [1,c_k X + d_k + 1]), where eval_n is the evaluation map sending X to n.
It is conjectured that for all k and n > 3, U(1,n) intersect [1,c_k n + d_k + 1] = eval_n(U(1,X) intersect [1,c_k X + d_k + 1]). At the time of writing, this had been checked up to k = 217529.
U(1,X) can also be defined model-theoretically, as follows. Let *Z denote the hyper-integers. The sequence of sets U(1,n) indexed over positive integers has a unique extension to a sequence of sets indexed over positive hyper-integers. Take any positive non-standard integer H, and consider the corresponding set U(1,H) in this sequence. Then the intersection of U(1,H) with the set of integer polynomials in H will be isomorphic to U(1,X).

			The first four intervals of U(1,X) are [1,1], [X,2X], [2X + 2, 2X + 2], [4X, 4X] hence the corresponding a_i coefficients are 0,1,2,4.

J. Hinman, B. Kuca, A. Schlesinger, and A. Sheydvasser, The Unreasonable Rigidity of Ulam Sequences, J. Number Theory, 194 (2019), 409-425.
A. Sheydvasser, The Ulam Sequence of the Integer Polynomial Ring, J. Integer Seq., accepted.

J. Hinman, B. Kuca, A. Schlesinger, and A. Sheydvasser, The Unreasonable Rigidity of Ulam Sequences, arXiv:1711.00145 [math.NT], 2017.

Cf. A349463, A349464, A349465 for the other coefficients. The original Ulam sequence U(1,2) is A002858.

A349463 The b_i coefficients of the standard form decomposition of U(1,X), the Ulam sequence starting with 1, X in the ordered abelian group of linear integer polynomials in X, where the ordering is lexigraphical.

Original entry on oeis.org

1, 0, 2, 0, 2, 1, 3, 2, 2, 4, 2, 2, 3, 2, 3, 4, 4, 3, 4, 5, 5, 6, 5, 8, 7, 6, 7, 8, 8, 9, 8, 8, 10, 9, 10, 11, 12, 12, 13, 14, 15, 15, 16, 16, 16, 17, 17, 18, 18, 19, 21, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 26, 26, 27, 27, 27, 28, 28, 29, 29, 29, 31, 31, 32, 33, 33, 34, 34, 35, 35

Offset: 0

Arseniy (Senia) Sheydvasser, Nov 19 2021

nonn

The set U(1,X) can be defined as follows: it is the unique set of linear integer polynomials such that 1,X are the two smallest elements; for any interval [P,Q] where P,Q are linear integer polynomials, U(1,X) intersect [P,Q] has a minimum and a maximum; and finally, an element u > X is in U(1,X) if and only if it is the smallest element larger than the maximum of U(1,X) intersect [1,u - 1] and which can be written as a sum of two distinct elements in U(1,X) in exactly one way.
U(1,X) can be written uniquely as a union of intervals [a_i X + b_i, c_i X + d_i], where c_i X + d_i + 1 < a_{i + 1} X + b_{i + 1} for all indices i. Here, we give just the coefficients b_i.
This set is related to Ulam sequences in an odd way. Let U(1,n) be the sequence of integers starting with 1,n such that every subsequent term is the next smallest element that can be written as the sum of two distinct prior terms in exactly one way. Then, for all integers k, for all sufficiently large n, U(1,n) intersect [1,c_k n + d_k + 1] = eval_n(U(1,X) intersect [1,c_k X + d_k + 1]), where eval_n is the evaluation map sending X to n.
It is conjectured that for all k and n > 3, U(1,n) intersect [1,c_k n + d_k + 1] = eval_n(U(1,X) intersect [1,c_k X + d_k + 1]). At the time of writing, this had been checked up to k = 217529.
U(1,X) can also be defined model-theoretically, as follows. Let *Z denote the hyper-integers. The sequence of sets U(1,n) indexed over positive integers has a unique extension to a sequence of sets indexed over positive hyper-integers. Take any positive non-standard integer H, and consider the corresponding set U(1,H) in this sequence. Then the intersection of U(1,H) with the set of integer polynomials in H will be isomorphic to U(1,X).

			The first four intervals of U(1,X) are [1,1], [X,2X], [2X + 2, 2X + 2], [4X, 4X] hence the corresponding b_i coefficients are 1,0,2,0.

J. Hinman, B. Kuca, A. Schlesinger, and A. Sheydvasser, The Unreasonable Rigidity of Ulam Sequences, J. Number Theory, 194 (2019), 409-425.
A. Sheydvasser, The Ulam Sequence of the Integer Polynomial Ring, J. Integer Seq., accepted.

J. Hinman, B. Kuca, A. Schlesinger, and A. Sheydvasser, The Unreasonable Rigidity of Ulam Sequences, arXiv:1711.00145 [math.NT], 2017.

Cf. A349462, A349464, A349465 for the other coefficients. The original Ulam sequence U(1,2) is A002858.

A349464 The c_i coefficients of the standard form decomposition of U(1,X), the Ulam sequence starting with 1, X in the ordered abelian group of linear integer polynomials in X, where the ordering is lexigraphical.

Original entry on oeis.org

0, 2, 2, 4, 5, 5, 8, 10, 11, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 32, 34, 38, 40, 41, 44, 44, 46, 50, 53, 56, 59, 62, 65, 68, 70, 77, 80, 86, 89, 98, 101, 104, 107, 110, 113, 116, 119, 122, 125, 137, 140, 143, 146, 149, 152, 155, 158, 164, 167, 173, 176, 179, 182, 185, 188, 191, 194

Offset: 0

Arseniy (Senia) Sheydvasser, Dec 27 2021

nonn

The set U(1,X) can be defined as follows: it is the unique set of linear integer polynomials such that 1,X are the two smallest elements; for any interval [P,Q] where P,Q are linear integer polynomials, U(1,X) intersect [P,Q] has a minimum and a maximum; and finally, an element u > X is in U(1,X) if and only if it is the smallest element larger than the maximum of U(1,X) intersect [1,u - 1] and which can be written as a sum of two distinct elements in U(1,X) in exactly one way.
U(1,X) can be written uniquely as a union of intervals [a_i X + b_i, c_i X + d_i], where c_i X + d_i + 1 < a_{i + 1} X + b_{i + 1} for all indices i. Here, we give just the coefficients c_i.
This set is related to Ulam sequences in an odd way. Let U(1,n) be the sequence of integers starting with 1,n such that every subsequent term is the next smallest element that can be written as the sum of two distinct prior terms in exactly one way. Then, for all integers k, for all sufficiently large n, U(1,n) intersect [1,c_k n + d_k + 1] = eval_n(U(1,X) intersect [1,c_k X + d_k + 1]), where eval_n is the evaluation map sending X to n.
It is conjectured that for all k and n > 3, U(1,n) intersect [1,c_k n + d_k + 1] = eval_n(U(1,X) intersect [1,c_k X + d_k + 1]). At the time of writing, this had been checked up to k = 217529.
U(1,X) can also be defined model-theoretically, as follows. Let *Z denote the hyper-integers. The sequence of sets U(1,n) indexed over positive integers has a unique extension to a sequence of sets indexed over positive hyper-integers. Take any positive non-standard integer H, and consider the corresponding set U(1,H) in this sequence. Then the intersection of U(1,H) with the set of integer polynomials in H will be isomorphic to U(1,X).

			The first four intervals of U(1,X) are [1,1], [X,2X], [2X + 2, 2X + 2], [4X, 4X] hence the corresponding c_i coefficients are 0,2,2,4.

J. Hinman, B. Kuca, A. Schlesinger, and A. Sheydvasser, The Unreasonable Rigidity of Ulam Sequences, J. Number Theory, 194 (2019), 409-425.
A. Sheydvasser, The Ulam Sequence of the Integer Polynomial Ring, J. Integer Seq., accepted.

J. Hinman, B. Kuca, A. Schlesinger, and A. Sheydvasser, The Unreasonable Rigidity of Ulam Sequences, arXiv:1711.00145 [math.NT], 2017.

Cf. A349462, A349463, A349465 for the other coefficients. The original Ulam sequence U(1,2) is A002858.

A349465 The d_i coefficients of the standard form decomposition of U(1,X), the Ulam sequence starting with 1, X in the ordered abelian group of linear integer polynomials in X, where the ordering is lexigraphical.

Original entry on oeis.org

1, 0, 2, 0, -1, 1, 1, 2, 2, 1, 2, 2, 3, 2, 3, 4, 5, 3, 4, 3, 5, 6, 5, 4, 4, 6, 7, 6, 7, 6, 8, 8, 8, 9, 10, 10, 10, 11, 11, 13, 13, 14, 14, 14, 15, 15, 16, 16, 16, 19, 19, 19, 20, 20, 20, 21, 21, 22, 22, 23, 24, 24, 25, 25, 25, 26, 26, 27, 27, 27, 29, 30, 31, 31, 31, 32, 32, 33, 33

Offset: 0

Arseniy (Senia) Sheydvasser, Nov 19 2021

sign

The set U(1,X) can be defined as follows: it is the unique set of linear integer polynomials such that 1,X are the two smallest elements; for any interval [P,Q] where P,Q are linear integer polynomials, U(1,X) intersect [P,Q] has a minimum and a maximum; and finally, an element u > X is in U(1,X) if and only if it is the smallest element larger than the maximum of U(1,X) intersect [1,u - 1] and which can be written as a sum of two distinct elements in U(1,X) in exactly one way.
U(1,X) can be written uniquely as a union of intervals [a_i X + b_i, c_i X + d_i], where c_i X + d_i + 1 < a_{i + 1} X + b_{i + 1} for all indices i. Here, we give just the coefficients d_i.
This set is related to Ulam sequences in an odd way. Let U(1,n) be the sequence of integers starting with 1,n such that every subsequent term is the next smallest element that can be written as the sum of two distinct prior terms in exactly one way. Then, for all integers k, for all sufficiently large n, U(1,n) intersect [1,c_k n + d_k + 1] = eval_n(U(1,X) intersect [1,c_k X + d_k + 1]), where eval_n is the evaluation map sending X to n.
It is conjectured that for all k and n > 3, U(1,n) intersect [1,c_k n + d_k + 1] = eval_n(U(1,X) intersect [1,c_k X + d_k + 1]). At the time of writing, this had been checked up to k = 217529.
U(1,X) can also be defined model-theoretically, as follows. Let *Z denote the hyper-integers. The sequence of sets U(1,n) indexed over positive integers has a unique extension to a sequence of sets indexed over positive hyper-integers. Take any positive non-standard integer H, and consider the corresponding set U(1,H) in this sequence. Then the intersection of U(1,H) with the set of integer polynomials in H will be isomorphic to U(1,X).

			The first four intervals of U(1,X) are [1,1], [X,2X], [2X + 2, 2X + 2], [4X, 4X] hence the corresponding d_i coefficients are 1,0,2,0.

J. Hinman, B. Kuca, A. Schlesinger, and A. Sheydvasser, The Unreasonable Rigidity of Ulam Sequences, J. Number Theory, 194 (2019), 409-425.
A. Sheydvasser, The Ulam Sequence of the Integer Polynomial Ring, J. Integer Seq., accepted.

J. Hinman, B. Kuca, A. Schlesinger, and A. Sheydvasser, The Unreasonable Rigidity of Ulam Sequences, arXiv:1711.00145 [math.NT], 2017.

Cf. A349462, A349463, A349464 for the other coefficients. The original Ulam sequence U(1,2) is A002858.

A379162 Ulam numbers that are sphenics.

Original entry on oeis.org

102, 114, 138, 182, 238, 258, 273, 282, 370, 402, 429, 434, 483, 602, 627, 646, 861, 986, 1023, 1030, 1311, 1335, 1338, 1406, 1462, 1790, 1834, 1902, 1946, 2054, 2093, 2134, 2247, 2330, 2354, 2445, 2486, 2613, 2630, 2635, 2674, 2919, 2985, 3070, 3219, 3395

Offset: 1

Massimo Kofler, Dec 17 2024

nonn

Ulam numbers: a(1) = 1; a(2) = 2; for n>2, a(n) = least number > a(n-1) which is a unique sum of two distinct earlier terms.

			102 is a term because 102=2*3*17 is the product of 3 distinct primes and 102 is an Ulam number.
114 is a term because 114=2*3*19 is the product of 3 distinct primes and 114 is an Ulam number.
273 is a term because 273=3*7*13 is the product of 3 distinct primes and 273 is an Ulam number.

Robert Israel, Table of n, a(n) for n = 1..10000

Intersection of A002858 and A007304.

Cf. A068820, A378795.

N:= 10^4: # for terms <= N
U:= [1, 2]: V:= Vector(N): V[3]:= 1: R:= NULL:
for i from 3 do
   for k from U[-1]+1 to N do
     if V[k] = 1 then
       J:= select(`<=`, U +~ k, N);
       V[J]:= V[J] +~ 1;
       U:= [op(U), k];
       F:= ifactors(k)[2]:
       if F[.., 2] = [1, 1, 1] then R:= R, k; break fi
   od;
   if k > N then break fi;
od:
R; # Robert Israel, Jan 03 2025

seq[numUlams_] := Module[{ulams = {1, 2}}, Do[AppendTo[ulams, n = Last[ulams]; While[n++; Length[DeleteCases[Intersection[ulams, n - ulams], n/2, 1, 1]] != 2]; n], {numUlams}]; Select[ulams, FactorInteger[#][[;; , 2]] == {1, 1, 1} &]]; seq[300] (* Amiram Eldar, Dec 17 2024, after Jean-François Alcover at A002858 *)

A068799 Ulam numbers such that n/2 is also an Ulam number.

Original entry on oeis.org

2, 4, 6, 8, 16, 26, 36, 72, 106, 114, 138, 412, 546, 800, 1966, 2616, 5162, 6410, 8236, 27062, 29784, 177104, 278256, 342486, 556126, 664364, 1388708, 6833546, 8622762, 9734048

Offset: 1

Naohiro Nomoto, Mar 29 2002

Cf. A002858, A068791.

a(n) = 2 * A068791(n). - Sean A. Irvine, Mar 15 2024

More terms from Jud McCranie, Feb 12 2003

cp's OEIS Frontend

A183528 An Ulam-type sequence: a(n) = n if n<=5; for n>5, a(n) = least number > a(n-1) which is a unique sum of 5 distinct earlier terms.

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Maple

A274522 Index of the smaller Ulam number of record gaps (A080287).

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A287612 Ulam numbers k such that 4*k is also an Ulam number.

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Mathematica

A336528 a(1) = 1; a(2) = 2; for n > 2, a(n) is the least number > a(n-1) whose decimal representation is uniquely the concatenation of the decimal representations of two distinct earlier terms.

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PARI

A349462 The a_i coefficients of the standard form decomposition of U(1,X), the Ulam sequence starting with 1, X in the ordered abelian group of linear integer polynomials in X, where the ordering is lexigraphical.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A349463 The b_i coefficients of the standard form decomposition of U(1,X), the Ulam sequence starting with 1, X in the ordered abelian group of linear integer polynomials in X, where the ordering is lexigraphical.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A349464 The c_i coefficients of the standard form decomposition of U(1,X), the Ulam sequence starting with 1, X in the ordered abelian group of linear integer polynomials in X, where the ordering is lexigraphical.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A349465 The d_i coefficients of the standard form decomposition of U(1,X), the Ulam sequence starting with 1, X in the ordered abelian group of linear integer polynomials in X, where the ordering is lexigraphical.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A379162 Ulam numbers that are sphenics.

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Maple