cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A068856 First of three consecutive Ulam numbers (A002858) in arithmetic progression with difference 22.

Original entry on oeis.org

905, 1208, 2134, 3109, 8294, 12482, 12917, 15744, 16047, 16543, 19353, 19871, 19893, 19915, 19937, 20499, 29073, 37239, 37261, 37283, 38534, 39333, 39355, 40931, 40953, 43328, 49031, 59230, 64823, 67005, 67027, 71628, 71650, 72234, 81326, 82213, 82406, 82428
Offset: 1

Views

Author

Naohiro Nomoto, Mar 29 2002

Keywords

Comments

Numbers k such that A002858(k+2)-A002858(k+1) = A002858(k+1)-A002858(k) = 22. Sequence gives A002858(k).
A002858(1587+k) = 19871 + 22*k (0<=k<=5). !!

Links

Crossrefs

Cf. A002858.

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Oct 10 2003
More terms from Amiram Eldar, Aug 23 2020

A307330 Indices of Ulam prime triples, where u(k), u(k+1) and u(k+2) are all primes, and u(k) = A002858(k) are the Ulam numbers.

Original entry on oeis.org

858, 1547, 2515, 3059, 9503, 9781, 11899, 11900, 13622, 15413, 18928, 22829, 25462, 32020, 33021, 33963, 36598, 37811, 37854, 40046, 44447, 49104, 55249, 56526, 60847, 61741, 80658, 84414, 86235, 87971, 92832, 96674, 97185, 100004, 102684, 108025, 108273
Offset: 1

Views

Author

Amiram Eldar, Apr 02 2019

Keywords

Links

Crossrefs

Cf. A002858, A068820, A307328, A307329.

Programs

  • Mathematica
    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]]&&p[[n+2]], AppendTo[s,n]], {n,1,len-2}]; s (* after Jean-François Alcover at A002858 *)

A330909 Floor of area of triangle whose sides are consecutive Ulam numbers (A002858).

Original entry on oeis.org

0, 2, 5, 11, 23, 43, 70, 100, 141, 227, 361, 478, 670, 826, 1044, 1183, 1405, 1668, 1960, 2272, 2545, 2889, 3351, 3819, 4267, 4523, 4955, 5669, 6558, 7474, 8203, 8914, 9633, 10813, 12245, 13611, 13972, 14587, 15473, 16798, 17987, 19298, 20229, 21909, 23166
Offset: 1

Views

Author

Frank M Jackson, May 01 2020

Keywords

Comments

It has been proved that three consecutive Ulam numbers U(n) for n > 1 satisfy the triangle inequality. See Wikipedia link below.

Examples

			a(2) = 2 because the triangle with sides (2, 3, 4) has area 3*sqrt(15)/4 = 2.9047...

Links

Crossrefs

Cf. A002858, A331729.

Programs

  • Mathematica
    lst1 = ReadList["https://oeis.org/A002858/b002858.txt", {Number,Number}]; lst={}; Do[{a, b, c}={lst1[[n]][[2]], lst1[[n+1]][[2]], lst1[[n+2]][[2]]}; s = (a+b+c)/2; A=Sqrt[s(s-a)(s-b)(s-c)]; AppendTo[lst, Floor@A], {n, 1, 50}]; lst

Formula

Given a triangle with sides a, b and c, the area A = sqrt(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2.

A333676 Least positive integer k such that k*prime(n) is an Ulam number (A002858).

Original entry on oeis.org

1, 1, 29, 4, 1, 1, 6, 2, 3, 3, 2, 4, 2, 6, 1, 1, 3, 12, 6, 7, 3, 4, 13, 4, 1, 13, 2, 21, 12, 3, 16, 1, 5, 5, 19, 8, 22, 15, 4, 13, 2, 5, 2, 26, 1, 15, 38, 6, 16, 12, 10, 13, 1, 2, 8, 4, 38, 5, 15, 205, 2, 5, 10, 31, 4, 6, 16, 8, 31, 20, 5, 7, 2, 4, 4, 18, 3, 3, 4, 1, 3, 8, 1, 2, 10, 8, 15, 12
Offset: 1

Views

Author

Frank M Jackson, Apr 01 2020

Keywords

Comments

Conjecture: For any prime number p there exists a positive integer k such that k*p is an Ulam number.

Examples

			a(3) = 29 because 5*29 gives the Ulam number 145 and this is the smallest Ulam number divisible by 5.

Links

Crossrefs

Cf. A002858, A322372.

Programs

  • Mathematica
    lst1 = ReadList["https://oeis.org/A002858/b002858.txt", {Number, Number}]; lst = {}; Do[n=1; While[!IntegerQ[k=lst1[[n]][[2]]/Prime[m]], n++]; AppendTo[lst, k], {m, 1, 100}]; lst

A346216 a(n) = (10^n)-th Ulam number A002858.

Original entry on oeis.org

1, 18, 690, 12294, 132788, 1351223, 13509072, 135160791, 1351856726, 13517664323
Offset: 0

Views

Author

Frank M Jackson, Oct 16 2021

Keywords

Comments

The terms a(0)-a(4) are from the b-file for A002858. The remaining terms are from the paper by Philip Gibbs. As the sequence settles, its terms lie very close to the straight line 13.51*n.

Links

Crossrefs

Cf. A002858.

Formula

a(n) = A002858(10^n).

Extensions

a(5)-a(9) from Philip Gibbs and Jud McCranie, Feb 29 2012 - Sep 09 2015

A033628 Numbers that are in both the 1-additive and 0-additive sequences (A002858 and A033627).

Original entry on oeis.org

1, 2, 4, 13, 16, 28, 82, 97, 106, 145, 148, 175, 238, 241, 253, 316, 319, 358, 370, 382, 400, 409, 412, 451, 502, 544, 568, 607, 646, 673, 685, 688, 739, 751, 781, 820, 847, 949, 991, 1018, 1030, 1081, 1186, 1252, 1360, 1387, 1462, 1465, 1489, 1492
Offset: 1

Views

Author

Jud McCranie

Keywords

References

  • R. K. Guy, Unsolved Problems in Number Theory, C4

A080598 For numbers k such that A080570(k) and 2*A080571(k) are distinct Ulam numbers, sequence gives A002858(k).

Original entry on oeis.org

3, 6, 8, 11, 36, 47
Offset: 1

Views

Author

Naohiro Nomoto, Feb 23 2003

Keywords

Comments

No others terms <= 41000.
A002858(k) = A080570(k) + 2*A080571(k).

Crossrefs

Cf. A002858, A080570, A080571.

Extensions

Incorrect terms removed by Jinyuan Wang, Sep 05 2020

A322605 Numbers k such that all k - u are Ulam numbers (A002858) where u is an Ulam number in the range k/2 <= u < k.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 17, 19, 24, 29, 34, 39, 44
Offset: 1

Views

Author

Frank M Jackson, Dec 20 2018

Keywords

Comments

The following is a quotation from Hage-Hassan in his paper (see Link below). "The (concept of) right and left symmetry is fundamental in physics. This incites us to ask whether this symmetry is in (the) primes. Find the numbers n with a + a' = n. a, a' are primes and {a} are all the primes with: n/2 <= a < n and n = 2,3, ..."
This sequence is analogous to A320447. Instead of the sequence of primes it uses the sequence of Ulam numbers (A002858). It is conjectured that the sequence is finite and full.

Examples

			a(10)=12, because the Ulam numbers u in the range 6 <= u < 12 are {6, 8, 11}. Also the complementary set {6, 4, 1} has all its members Ulam numbers. This is the 10th occurrence of such a number.

Links

Crossrefs

Cf. A002858, A320447.

Programs

  • Mathematica
    Ulam[n_] := Module[{ulams={1, 2}, p}, Do[AppendTo[ulams, p=Last[ulams]; While[p++; Length[DeleteCases[Intersection[ulams, p-ulams], p/2, 1, 1]]!=2]; p], {n-2}]; ulams]; ulst=Ulam[1000]; plst[n_] := Select[ulst, Ceiling[n/2]<=#

A334763 Ceiling of circumradius of triangle whose sides are consecutive Ulam numbers (A002858).

Original entry on oeis.org

3, 4, 5, 6, 7, 9, 10, 14, 15, 19, 21, 24, 26, 29, 31, 34, 37, 40, 43, 45, 48, 52, 55, 58, 60, 63, 68, 72, 77, 80, 84, 87, 93, 99, 103, 104, 107, 110, 115, 118, 123, 126, 131, 134, 138, 139, 142, 146, 149, 153, 158, 168, 176, 182, 185, 190, 194, 200, 204, 208
Offset: 2

Views

Author

Frank M Jackson, May 10 2020

Keywords

Comments

It has been proved that three consecutive Ulam numbers U(n) for n > 1 satisfy the triangle inequality. See Wikipedia link below. Consequently it is possible to create n-gons using n consecutive Ulam numbers. The sequence starts at offset 2 because using the first Ulam number generates a triangle with sides (1,2,3) that is degenerate with infinite circumradius.
Conjecture: Triangles whose sides are consecutive Ulam numbers are acute apart from (1,2,3), (2,3,4), (3,4,6), (4,6,8), (6,8,11) and (16,18,26).

Examples

			a(2)=3 because a triangle with sides 2,3,4 has area = (1/4)*sqrt((2+3+4)(2+3-4)(2-3+4)(-2+3+4)) = 2.904... and circumradius = 2*3*4/(4A) = 2.065...

Links

Crossrefs

Cf. A002858, A331676.

Programs

  • Mathematica
    lst1=ReadList["https://oeis.org/A002858/b002858.txt", {Number, Number}]; lst={}; Do[{a, b, c}={lst1[[n]][[2]], lst1[[n+1]][[2]], lst1[[n+2]][[2]]}; s=(a+b+c)/2; A=Sqrt[s(s-a)(s-b)(s-c)]; R=a*b*c/(4 A); AppendTo[lst, Ceiling@R], {n, 2, 100}]; lst

Formula

Circumradius of a triangle with sides a, b, c is given by R = a*b*c/(4A) where the Area A is given by Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) and where s = (a+b+c)/2.

A347212 a(n) is the next Ulam number (A002858) after 2^(n-1).

Original entry on oeis.org

1, 2, 3, 6, 11, 18, 36, 69, 131, 258, 522, 1025, 2054, 4101, 8194, 16394, 32817, 65542, 131085, 262183, 524313, 1048588, 2097164, 4194310, 8388624, 16777218, 33554433, 67108881, 134217730, 268435473, 536870927, 1073741861
Offset: 0

Views

Author

Frank M Jackson, Sep 02 2021

Keywords

Comments

This sequence appears to be a "complete" (sic) sequence as defined in the Wikipedia link.

Examples

			a(0) = 1 because 1 is the next Ulam number after 2^(-1) = 1/2.
a(6) = 36 because 36 is the next Ulam number after 2^5 = 32.

Links

  • Wikipedia, "Complete" sequence. [Wikipedia calls a sequence "complete" (sic) if every positive integer is a sum of distinct terms. This name is extremely misleading and should be avoided. - _N. J. A. Sloane_, May 20 2023]

Crossrefs

Cf. A002858, A331729.

Programs

  • Mathematica
    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}]; lst=ulams; nextu[n_] := Module[{m = 1}, While[lst[[m]]<=n, m++]; lst[[m]]]; Join[{1}, Table[nextu[2^(n - 1)], {n, 1, 18}]]

Extensions

a(22)-a(31) from Amiram Eldar, Sep 02 2021
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