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.

Showing 1-5 of 5 results.

A085901 Duplicate of A077224.

Original entry on oeis.org

1, 2, 4, 9, 13, 29, 33, 101, 105, 109, 157, 177, 253, 289, 301, 353, 409, 429, 465, 501
Offset: 0

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Author

Keywords

A077225 Starting with a(0) = 1, smallest squarefree number k such that, for all a(m), m < n, k + a(m) is not squarefree.

Original entry on oeis.org

1, 3, 15, 17, 233, 291, 577, 723, 1455, 3615, 8117, 8835, 9505, 30833, 128773, 130827, 239595, 273435, 426891, 654135, 676297, 926117, 1455533, 1662533, 2389517, 2762427, 2820927, 7994449, 8098527, 14319073, 16766835, 20506733, 27606617, 31627817, 43558023, 55566015
Offset: 0

Views

Author

Amarnath Murthy, Nov 03 2002

Keywords

Examples

			17 belongs to this sequence as 17 + 1, 17 + 3, 17 + 15 all are divisible by some square.

Crossrefs

Cf. A013929, A077224, A080793, A085902.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Module[{t = Array[a, n, 0], k = a[n - 1] + 1}, While[! SquareFreeQ[k] || AnyTrue[t, SquareFreeQ[k + #] &], k++]; k]; Array[a, 20, 0] (* Amiram Eldar, Aug 21 2023 *)
  • PARI
    v=vector(60); v[1]=1; print1("1,");for(n=2,60, for(k=1,10^15,if(issquarefree(k),s=0;for(l=1,n-1,if(issquarefree(k+v[l]),break);s=s+1)); if(s==n-1,print1(k",");v[n]=k;break)))

Extensions

Edited by Ralf Stephan, Mar 25 2003
More terms from Sam Alexander, Dec 12 2003
a(22) corrected and a(33)-a(35) added by Amiram Eldar, Aug 21 2023

A080793 Starting with a(0) = 1, smallest number k > a(n-1) such that, for all a(m) with m < n, k + a(m) is not squarefree.

Original entry on oeis.org

1, 3, 15, 17, 147, 233, 577, 735, 3615, 4335, 8117, 9505, 10947, 14403, 25215, 30833, 128773, 220647, 251173, 522003, 617547, 770977, 926117, 967335, 1455533, 1662533, 1960215, 2389517, 2469747, 3528147, 4674747, 5556747, 5730135, 5859903, 10892847, 12127647, 15170547
Offset: 0

Views

Author

Ralf Stephan, Mar 25 2003

Keywords

Links

  • Amiram Eldar, Table of n, a(n) for n = 0..10000
  • C. B. Lacampagne, C. A. Nicol, and J. L. Selfridge, Sets With Non-Squarefree Sums, in: Richard Mollin (ed.), Number Theory: Proceedings of the First Conference of the Canadian Number Theory Association, Held at the Banff Center, Banff, Alberta, April 17-27, 1988, Berlin; New York: W. de Gruyter, 1990, pp. 299-310.

Crossrefs

Cf. A013929, A077224, A077225, A085902.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Module[{t = Array[a, n, 0], k = a[n - 1] + 1}, While[AnyTrue[t, SquareFreeQ[k + #] &], k++]; k]; Array[a, 20, 0] (* Amiram Eldar, Aug 21 2023 *)
  • PARI
    v=vector(60); v[1]=1; print1("1, "); for(n=2, 60, for(k=v[n-1]+1, 10^15, s=0; for(l=1, n-1, if(issquarefree(k+v[l]), break); s=s+1); if(s==n-1, print1(k", "); v[n]=k; break)))

Extensions

More terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net) and lambert.herrgesell(AT)aschendorff.de, Jul 26 2005
a(33)-a(36) from Amiram Eldar, Aug 21 2023

A085902 a(0) = 2, a(n) is the smallest squarefree number > a(n-1) such that the sum a(n) + a(i) for all i = 1 to (n-1) is squarefree. Or, sum of any two terms is a squarefree number.

Original entry on oeis.org

2, 3, 11, 19, 55, 59, 83, 111, 127, 155, 163, 199, 203, 219, 263, 299, 307, 311, 371, 383, 399, 455, 515, 803, 883, 919, 983, 1063, 1499, 1559, 1927, 2019, 2063, 2183, 2215, 2271, 2359, 2503, 2703, 2755, 2999, 3459, 3899, 3927, 4271, 4303, 4411, 4519, 4559
Offset: 0

Views

Author

Amarnath Murthy, Jul 09 2003

Keywords

Comments

It can easily be proved that a(n) == 3 (mod 4) for all n > 2.

Links

Crossrefs

Cf. A005117, A077224, A077225, A080793.

Programs

  • Mathematica
    a[0] = 2; a[n_] := a[n] = Module[{t = Array[a, n, 0], k = a[n - 1] + 1}, While[! SquareFreeQ[k] || AnyTrue[t, ! SquareFreeQ[k + #] &], k++]; k]; Array[a, 100, 0] (* Amiram Eldar, Aug 21 2023 *)

Extensions

More terms from Ray Chandler, Sep 13 2003

A214773 Primes such that all pairwise sums are squarefree.

Original entry on oeis.org

2, 3, 11, 19, 59, 83, 127, 163, 199, 227, 271, 311, 383, 419, 443, 811, 911, 919, 1063, 1163, 1171, 1319, 1427, 1559, 2099, 2143, 2543, 2683, 2999, 3259, 4519, 5099, 5171, 5711, 5783, 6211, 6719, 8111, 8219, 9203, 11003, 12227, 12511, 12659, 13259, 13883
Offset: 1

Views

Author

Zak Seidov, Jul 28 2012

Keywords

Comments

a(n+1) is the smallest prime p > a(n) such that all sums a(i)+p, i-1..n are squarefree. All odd terms = 3 mod 4.
The sequence is apparently infinite.

Links

Crossrefs

Cf. A005117, A077224, A214735.

Programs

  • Mathematica
    sumsSqFree[t_, p_] := And @@ SquareFreeQ /@ (t + p); t = {2}; Do[p = NextPrime[t[[-1]]]; While[! sumsSqFree[t, p], p = NextPrime[p]]; AppendTo[t, p], {50}]; t (* T. D. Noe, Jul 30 2012 *)
Showing 1-5 of 5 results.

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