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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A219547 Numbers k such that 2 times the least prime factor of 2^k + 1 is not the least m > 1 that divides sigma_k(m).

Original entry on oeis.org

8, 16, 32, 40, 48, 56, 64, 80, 88, 96, 104, 112, 128, 136, 152, 160, 176, 184, 192, 200, 208, 224, 232, 240, 248, 256, 272, 280, 296, 304, 320, 328, 336, 344, 352, 368, 376, 384, 392, 400, 416, 424, 440, 448, 464, 472, 480, 488, 496
Offset: 1

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Author

Jonathan Sondow, Nov 24 2012

Keywords

Comments

Numbers k with 2*A002586(k) unequal to A066135(k).
A066135(n) <= 2*A002586(n) for all n (see Comments in A066135). Sequence gives those k for which A066135(k) < 2*A002586(k).
The corresponding least prime factors of 2^k + 1 are A219548.
See A007691 for references, links, and additional comments.

Examples

			A066135(n) = 6,10,6,34,6,10,6 = 2*A002586(n) for n = 1,2,3,4,5,6,7, and A066135(8) = 84 < 2*257 = 2*A002586(8), so a(1) = 8.

Crossrefs

Cf. A002586, A007691, A066135, A219548.

Formula

A066135(a(n)) < 2*A002586(a(n)).
A002586(a(n)) = A219548(n).

A219548 Smallest prime factor of 2^A219547(n) + 1.

Original entry on oeis.org

257, 65537, 641, 257, 193, 257, 274177, 65537, 257, 641, 257, 449, 59649589127497217, 257, 257, 641, 65537, 257, 769, 257, 65537, 641, 257, 193, 257, 1238926361552897, 5441, 257, 257, 65537, 274177, 257, 193, 257, 641, 65537, 257, 59649589127497217, 257, 65537, 641, 257, 257, 274177, 65537, 257, 641, 257, 5953
Offset: 1

Views

Author

Jonathan Sondow, Nov 24 2012

Keywords

Comments

2p divides sigma_k(2p), where p = a(n) and k = A219547(n). But 2p is not the least m > 1 that divides sigma_k(m).

Crossrefs

Cf. A002586, A219547.

Formula

a(n) = A002586(A219547(n)).

A219549 Smallest prime factor of 2^(8n) + 1.

Original entry on oeis.org

257, 65537, 97, 641, 257, 193, 257, 274177, 97, 65537, 257, 641, 257, 449, 97, 59649589127497217, 257, 193, 257, 641, 97, 65537, 257, 769, 257, 65537, 97, 641, 257, 193, 257, 1238926361552897, 97, 5441, 257, 641, 257, 65537, 97, 274177, 257, 193, 257, 641, 97, 65537, 257, 59649589127497217, 257, 65537, 97, 641, 257, 193, 257, 274177, 97, 65537, 257, 641, 257, 5953
Offset: 1

Views

Author

Jonathan Sondow, Nov 28 2012

Keywords

Comments

The smallest prime factor of 2^(8n+k) + 1 does not depend on n if 0 < k < 8 (see Formula in A002586).
For references and links, see A002586.

Examples

			a(1) = 2^8 + 1 = 257 is the Fermat prime A019434(3).
a(2) = 2^16 + 1 = 65537 is the Fermat prime A019434(4).

Links

Crossrefs

Cf. A000215, A002586, A019434, A020639.

Programs

  • Mathematica
    Table[FactorInteger[2^(8*n) + 1][[1, 1]], {n, 20}] (* T. D. Noe, Nov 29 2012 *)

Formula

a(n) = A002586(8n) = A020639(2^(8n) + 1).
a(2^(k-3)) = A020639(A000215(k)) is the smallest prime factor of the k-th Fermat number 2^(2^k) + 1.

A260218 a(1) = 2; for n > 1 if n is even a(n) = spf(1 + Product_{odd m,m

Original entry on oeis.org

2, 3, 2, 5, 2, 3, 2, 17, 2, 3, 2, 5, 2, 3, 2, 257, 2, 3, 2, 5, 2, 3, 2, 17, 2, 3, 2, 5, 2, 3, 2, 65537, 2, 3, 2, 5, 2, 3, 2, 17, 2, 3, 2, 5, 2, 3, 2, 97, 2, 3, 2, 5, 2, 3, 2, 17, 2, 3, 2, 5, 2, 3, 2, 641, 2, 3, 2, 5, 2, 3, 2, 17, 2, 3, 2, 5, 2, 3
Offset: 1

Views

Author

Anders Hellström, Jul 19 2015

Keywords

Links

Crossrefs

Cf. A000945, A005265, A007978, A053669, A075019, A258581.

Programs

  • Mathematica
    f[n_] := Block[{a = {2}, k, m}, Do[If[EvenQ@ k, AppendTo[a, FactorInteger[Product[a[[m]], {m, 1, k - 1, 2}] + 1][[1, 1]]], AppendTo[a, FactorInteger[Product[a[[m]], {m, 2, k - 1, 2}] + 1][[1, 1]]]], {k, 2, n}]; a]; f@ 80 (* Michael De Vlieger, Jul 20 2015 *)
  • PARI
    spf(n)=factor(n)[1, 1]
first(m)=my(v=vector(m), i, odd=2, even=1); v[1]=2; for(i=2, m, if(i%2==0, v[i]=spf(odd+1); even*=v[i], v[i]=spf(even+1); odd*=v[i])); v; /* Anders Hellström, Jul 19 2015 */
  • PARI
    A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
memoA260218 = Map();
A260218(n) = if(1==n,2,if(mapisdefined(memoA260218,n),mapget(memoA260218,n), my(k, m, v = if(!(n%2), k=1; m=1; while(kA260218(k); k += 2); A020639(m+1), k=2; m=1; while(kA260218(k); k += 2); A020639(m+1))); mapput(memoA260218,n,v); (v))); \\ (An incrementally memoized version). Antti Karttunen, Sep 30 2018

Formula

It appears that for odd k, a(k) = 2 and for even k, a(k) = A002586(k/2). - Michel Marcus, Jul 20 2015
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