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.

A181492 Primes of the form p=3*2^k+1 such that p-2 is also a prime.

Original entry on oeis.org

7, 13, 193, 786433
Offset: 1

Views

Author

M. F. Hasler, Oct 30 2010

Keywords

Comments

Sequence A181490 lists the exponents k, sequences A181491 and A181493 the corresponding lesser twin prime and their average.
a(5) > 3 * 2^3000 + 1. - Max Z. Scialabba, Dec 24 2023

Crossrefs

Cf. A001097, A001359, A006512, A014574.

Programs

  • Mathematica
    Select[3 2^Range[100]+1,And@@PrimeQ[{#,#-2}]&] (* Harvey P. Dale, Jun 19 2013 *)
  • PARI
    for( k=1,999, ispseudoprime(3<

Formula

A181492 = A181491 + 2 = A181493 + 1 = 3*2^A181490 + 1 = intersection of A004119 or A103204 or A181495 with A006512 or A001097.

A181491 Primes of the form p = 3*2^k - 1 such that p+2 is also prime.

Original entry on oeis.org

5, 11, 191, 786431
Offset: 1

Views

Author

M. F. Hasler, Oct 30 2010

Keywords

Comments

Sequence A181490 lists the exponents k, sequences A181492 and A181493 the corresponding upper twin prime and their average.
a(5) > 3 * 2 ^ 3000 + 1. - Max Z. Scialabba, Dec 24 2023

Crossrefs

Cf. A001097, A001359, A006512, A014574.

Programs

  • PARI
    for( k=1,999, ispseudoprime(3<

Formula

A181491 = A007283 intersect A014574 = A181492 - 2 = A181493 - 1 = 3*2^A153890 - 1.

A275418 Numbers n such that n - 1 has exactly as many odd divisors as n + 1.

Original entry on oeis.org

3, 4, 6, 11, 12, 13, 18, 21, 23, 25, 27, 30, 34, 39, 42, 45, 47, 56, 57, 60, 72, 75, 81, 86, 87, 92, 93, 94, 95, 99, 102, 105, 108, 109, 117, 123, 124, 131, 135, 138, 139, 142, 144, 147, 150, 155, 159, 160, 165, 169, 177, 180, 184, 186, 192, 193, 198, 202, 204, 207, 213, 214, 216
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jul 27 2016

Keywords

Comments

Numbers n > 1 such that d(2n - 2) + d(n + 1) = d(2n + 2) + d(n - 1) where d = A000005.
Conjectures:
(1) There are only finitely many terms n such that A001227(n - 1) = A001227(n + 1) is odd: 3, 99, 577, 3363, ... (see A276188).
(2) There are only finitely many terms n such that A001227(n - 1) = A001227(n) = A001227(n + 1) = 2: 6, 11, 12, 13, 23, 47, 192, 193, 383, 786432, ... (see also A181490-A181493, A276136).
(3) There are only finitely many prime terms p such that A001227(p - 1) = A001227(p + 1) is prime: 11, 13, 23, 47, 193, 383, 577, ... (see also A275598).
I don't find any more for conjecture #3 up to 10^10. - Charles R Greathouse IV, Aug 22 2016

Examples

			3 is in this sequence because 2 and 4 both have only one odd divisor, 1.
4 is in this sequence because 3 and 5 both have exactly two odd divisors each (1 and 3 for the former, 1 and 5 for the latter).

Links

Crossrefs

Supersequence of A014574.
Cf. A000005, A001227, A067888, A181490-A181493, A273401, A275598, A276136, A276188.

Programs

  • Magma
    [n: n in [2..216] | NumberOfDivisors(2*(n-1))+ NumberOfDivisors(n+1) eq NumberOfDivisors(2*(n+1))+ NumberOfDivisors(n-1)];
  • Maple
    N:= 1000: # to get all terms < N
nod:= proc(n) numtheory:-tau(n/2^padic:-ordp(n,2)) end proc:
X:= map(nod,[$1..N]):
select(t -> X[t+1]=X[t-1], [$2..N-1]); # Robert Israel, Aug 04 2016
  • Mathematica
    f[n_] := Count[Divisors@ n, k_ /; OddQ@ k]; Select[Range[2, 240], f[# - 1] == f[# + 1] &] (* Michael De Vlieger, Jul 28 2016 *)
Flatten[Position[Partition[Table[Count[Divisors[n],?OddQ],{n,300}],3,1],?(#[[1]]==#[[3]]&),{1},Heads->False]]+1 (* Harvey P. Dale, Nov 02 2016 *)
  • PARI
    a001227(n) = sumdiv(n, d, d%2);
is(n) = a001227(n-1)==a001227(n+1) \\ Felix Fröhlich, Jul 27 2016
  • PARI
    is(n)=numdiv((n-1)>>valuation(n-1,2)) == numdiv((n+1)>>valuation(n+1,2)) \\ Charles R Greathouse IV, Jul 29 2016

Extensions

Name edited by Alonso del Arte, Aug 23 2016

A276136 Numbers m > 1 such that the largest odd divisors of m-1, m, and m+1 are prime.

Original entry on oeis.org

6, 11, 12, 13, 23, 47, 192, 193, 383, 786432
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Aug 22 2016

Keywords

Comments

Conjecture: this sequence is finite.
Any further terms are greater than 10^11. - Charles R Greathouse IV, Aug 22 2016
From Robert Israel, Apr 27 2020: (Start)
Each term is either of the form 3*2^k with 3*2^k-1 and 3*2^k+1 prime, or 3*2^k-1 with 3*2^k-1 prime and 3*2^(k-1)-1 prime, or 3*2^k+1 with 3*2^k+1 prime and 3*2^(k-1)+1 prime.
Any further terms > 10^2000.
(End)

Examples

			6 is in this sequence because the largest odd divisor of 5 is 5, the largest odd divisor of 6 is 3 and the largest odd divisor of 7 is 7, and all three are prime.

Crossrefs

Supersequence of A181493. Subsequence of A038550.
Cf. A001227, A181490, A181491, A181492, A275418.

Programs

  • Magma
    [n: n in [2..3000000] | NumberOfDivisors(2*(n-1))- NumberOfDivisors(n-1)eq 2 and NumberOfDivisors(2(n))-NumberOfDivisors(n) eq 2 and NumberOfDivisors(2*(n+1))- NumberOfDivisors(n+1) eq 2];
  • Maple
    Res:= 6:
for k from 2  while length(3*2^k-1)<1000 do
  if (isprime(3*2^k-1) and isprime(3*2^(k-1)-1)) then Res:= Res, 3*2^k-1
    fi;
  if (isprime(3*2^k-1) and isprime(3*2^k+1)) then Res:= Res, 3*2^k;
    fi;
  if (isprime(3*2^k+1) and isprime(3*2^(k-1)+1)) then Res:= Res, 3*2^k+1;
    fi;
od:
Res; # Robert Israel, Apr 27 2020
  • Mathematica
    Select[Range[2, 10^6], Function[n, Times @@ Boole@ PrimeQ@ Map[First@ Reverse@ DeleteCases[Divisors@ #, d_ /; EvenQ@ d] &, n + Range[-1, 1]] == 1]] (* Michael De Vlieger, Aug 22 2016 *)
SequencePosition[Table[If[PrimeQ[Max[Select[Divisors[n],OddQ]]],1,0],{n,800000}],{1,1,1}][[;;,1]]+1 (* Harvey P. Dale, Jun 27 2023 *)
  • PARI
    isA038550(n)=isprime(n>>valuation(n,2))
is(n)=isA038550(n-1) && isA038550(n) && isA038550(n+1) \\ Charles R Greathouse IV, Aug 22 2016
  • PARI
    forprime(p=2,1e11, my(a=isA038550(p-1),b=isA038550(p+1)); if(a && isA038550(p-2), print1(p-1", ")); if(a && b, print1(p", ")); if(b && isA038550(p+2), print1(p+1", "))) \\ may print numbers several times, but won't skip numbers; Charles R Greathouse IV, Aug 22 2016

Formula

A038550(a(n-1)) + 1 = A038550(a(n)) = A038550(a(n+1)) - 1.
a(n) >> n log n. - Charles R Greathouse IV, Aug 22 2016

A181494 Twin primes (A001097) of the form 3*2^k +- 1.

Original entry on oeis.org

5, 7, 11, 13, 191, 193, 786431, 786433
Offset: 1

Views

Author

M. F. Hasler, Oct 30 2010

Keywords

Crossrefs

Cf. A001097, A001359, A006512, A014574.

Programs

  • PARI
    for( k=1,999, ispseudoprime(3<

Formula

A181494 = A181491 union A181492.
A181494(n) = A181493(ceiling(n/2)) + (-1)^n = A007283(A181490(ceiling(n/2))) + (-1)^n.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.