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 1-4 of 4 results.

A179019 a(n) = (A179017(n)+1)/2.

Original entry on oeis.org

2, 3, 6, 7, 11, 15, 22, 30, 31, 34, 35, 39, 42, 43, 47, 58, 62, 66, 67, 70, 71, 78, 79, 83, 87, 94, 102, 103, 106, 107, 110, 111, 114, 115, 119, 130, 134, 139, 142, 143, 146, 155, 159, 166, 174, 178, 179, 183, 186, 187, 191, 195, 202, 206, 210, 211, 214, 215, 218, 219
Offset: 1

Views

Author

Artur Jasinski, Jun 24 2010

Keywords

Comments

For numbers a and c, see A172186 and A179017. Numbers b are this sequence.
These numbers c, with distribution a+b=c such that a=(c-1)/2 and b=(c+1)/2, have minimal possible values with function L(a,b,c) = log(c)/log(N[a,b,c]) = log(c)/log((c^2-1)c/4).
There exist no numbers or distributions for which L < log(c)/log((c^2-1)c/4). - Artur Jasinski

Links

Crossrefs

Cf. A172120, A172121, A147298, A147299, A147300, A147301, A147302, A147306, A147307, A147638, A147639, A147640, A147641, A147642, A147643, A143700, A179017, A172186.

Programs

  • Mathematica
    aa = {}; Do[If[(GCD[x, (x - 1)/2] == 1) && (GCD[x, (x + 1)/2] == 1) && (GCD[(x - 1)/2, (x + 1)/2] == 1), If[SquareFreeQ[(x^2 - 1) x/4], AppendTo[aa, (x + 1)/2]]], {x, 2, 1000}]; aa

Formula

a(n) = A179017(n) - A172186(n). - Hugo Pfoertner, Mar 22 2020

A172186 Numbers k such that k, k+1 and 2*k+1 are squarefree.

Original entry on oeis.org

1, 2, 5, 6, 10, 14, 21, 29, 30, 33, 34, 38, 41, 42, 46, 57, 61, 65, 66, 69, 70, 77, 78, 82, 86, 93, 101, 102, 105, 106, 109, 110, 113, 114, 118, 129, 133, 138, 141, 142, 145, 154, 158, 165, 173, 177, 178, 182, 185, 186, 190, 194, 201, 205, 209, 210, 213, 214, 217, 218
Offset: 1

Views

Author

Artur Jasinski, Jan 28 2010

Keywords

Comments

This sequence is similar to A007674. For terms in A007674 which lack in this sequence see A172187.
The asymptotic density of this sequence is 2 * Product_{p prime} (1 - 3/p^2) = 2 * A206256 = 0.250973961811... (Tsang, 1985). - Amiram Eldar, Feb 26 2024

Links

Crossrefs

Cf. A007674, A172187, A179017, A206256.

Programs

  • Mathematica
    Select[Range[250], And @@ SquareFreeQ /@ {#, # + 1, 2 # + 1} &]  (* Harvey P. Dale, Mar 11 2011 *)

Formula

a(n) = (A179017(n)-1)/2.

A269844 Primes equal to the sum of a pair of consecutive integers which are both squarefree.

Original entry on oeis.org

5, 11, 13, 29, 43, 59, 61, 67, 83, 131, 139, 157, 173, 211, 227, 229, 277, 283, 317, 331, 347, 373, 389, 419, 421, 443, 461, 509, 547, 563, 571, 619, 643, 653, 659, 661, 691, 709, 733, 787, 797, 821, 853, 859, 877, 907, 941, 947, 997, 1019, 1021, 1069, 1091, 1093, 1109, 1123, 1163, 1181, 1213
Offset: 1

Views

Author

Bill McEachen, Mar 06 2016

Keywords

Comments

The associated prime factors will always include 2 and 3.
Will every prime number be encountered as a prime factor from the sequence entries?
The sequence appears to share many of it terms with A001122.
What is the asymptotic behavior?
Conjecture: sequence has density A271780/2 = A005597*4/Pi^2 = 0.2675535... in the primes. - Charles R Greathouse IV, Jan 24 2018
The prime terms of A179017 (except 3). - Bill McEachen, Oct 21 2021

Examples

			277 = 138 + 139 = 2*3*23 + 139 is in the sequence since both terms are squarefree.
281 = 140 + 141 = 2^2*5*7 + 3*47 is not in the sequence since the former term is divisible by 2^2.

Links

Crossrefs

Cf. A001122 (primes with primitive root 2), A179017.
Cf. A005597, A271780.

Programs

  • Mathematica
    Select[Prime@ Range[3, 200], PrimeOmega@ # == PrimeNu@ # &[# (# + 1)] &@ Floor[#/2] &] (* Michael De Vlieger, Mar 07 2016 *)
  • PARI
    genit(maxx)={for(i5=3,maxx,n=prime(i5);a=factor(floor(n/2.));b=factor(ceil(n/2.));clear=1;for(j5=1,omega(floor(n/2.)),if(a[j5,2]<>1,clear=0));
for(j7=1,omega(ceil(n/2.)),if(b[j7,2]<>1,clear=0));if(clear>0,print1(n,",")));}
  • PARI
    is(n)=isprime(n) && issquarefree(n\2) && issquarefree(n\2+1) \\ Charles R Greathouse IV, Jan 24 2018
  • PARI
    list(lim)=my(v=List(),t=1); forfactored(k=3,(lim+1)\2, if(vecmax(k[2][,2])>1, t=0, ; if(t && isprime(t=2*k[1]-1), listput(v,t)); t=1)); Vec(v) \\ Charles R Greathouse IV, Jan 24 2018

A375514 Numbers k such that floor((k^3 - k)/4) is squarefree.

Original entry on oeis.org

2, 3, 4, 5, 10, 11, 12, 13, 14, 18, 20, 21, 26, 29, 30, 34, 40, 42, 43, 46, 50, 52, 56, 59, 60, 61, 62, 66, 67, 68, 69, 74, 77, 78, 82, 83, 84, 85, 88, 90, 92, 93, 94, 98, 104, 106, 110, 114, 115, 122, 123, 126, 130, 131, 132, 133, 138, 139, 140, 141, 142, 146
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Aug 18 2024

Keywords

Crossrefs

Supersequence of A179017 and A269844.
Cf. A005117.

Programs

  • Magma
    [k: k in [2..150] | IsSquarefree(Floor((k^3-k) div 4))];
  • Mathematica
    Select[Range[150],SquareFreeQ[Floor[(#^3-#)/4]] &] (* Stefano Spezia, Aug 19 2024 *)
Showing 1-4 of 4 results.

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