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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A251239 Indices of prime numbers in A098550.

Original entry on oeis.org

2, 3, 9, 15, 22, 23, 30, 43, 51, 61, 62, 79, 87, 88, 101, 114, 127, 132, 142, 153, 158, 167, 175, 194, 204, 215, 222, 233, 238, 247, 274, 283, 296, 301, 324, 329, 338, 355, 364, 375, 386, 393, 414, 423, 430, 435, 452, 479, 490, 497, 506, 523, 528, 541, 550
Offset: 1

Views

Author

Reinhard Zumkeller, Dec 02 2014

Keywords

Comments

It is conjectured that every prime appears in A098550, and if so then A098550(a(n)) = A000040(n). [Comment edited by N. J. A. Sloane, Dec 15 2014] [It is now known that every prime appears in A098550, although it is not known that they appear in their right order. - N. J. A. Sloane, Dec 25 2014]
A010051(A098550(a(n))) = 1; A049084(A098550(a(n))) > 0.
Conjecture: a(n) = A251541(n) + 2 for n > 4. - Reinhard Zumkeller, Dec 16 2014
A253049(n) = A098550(a(n)+1). - Reinhard Zumkeller, Dec 29 2014

Links

Crossrefs

Cf. A098550, A010051, A049084, A251392, A247253 (first differences), A251595, A251541, A253049, A253048.
This is a subsequence of A251391 and A251241,

Programs

  • Haskell
    a251239 n = a251239_list !! (n-1)
a251239_list = filter ((== 1) . a010051' . a098550) [1..]
  • Mathematica
    a098550[lst_List] :=
Block[{k = 4},
  While[GCD[lst[[-2]], k] == 1 || GCD[lst[[-1]], k] > 1 ||
    MemberQ[lst, k], k++]; Append[lst, k]];
a251239[n_] :=
Flatten@Position[Nest[a098550, {1, 2, 3}, n], Integer?PrimeQ]; a251239[550] (* _Michael De Vlieger, Dec 23 2014, based on Robert G. Wilson v at A098550 *)

A251416 a(n) = smallest number not in {A098550(1), A098550(2), ..., A098550(n)}.

Original entry on oeis.org

2, 3, 4, 5, 5, 5, 5, 5, 6, 7, 7, 7, 7, 7, 10, 11, 11, 11, 11, 11, 11, 13, 17, 17, 17, 17, 17, 17, 17, 18, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 31
Offset: 1

Views

Author

N. J. A. Sloane, Dec 03 2014

Keywords

Comments

a(n) = smallest missing number in A098550 once we have found A098550(n).
Bradley Klee conjectures that after a(30)=18, all further terms are primes, that every prime appears, and the primes appear in increasing order.

References

  • Bradley Klee, Posting to Sequence Fans Mailing List, Dec 03 2014

Links

Crossrefs

Cf. A098550, A251415. See A251417 for lengths of runs.
Cf. A251595 (distinct terms).
Cf. A251546, A251549.

Programs

  • Haskell
    import Data.List (delete)
a251416 n = a251416_list !! (n-1)
a251416_list = 2 : 3 : f 2 3 [4..] where
   f u v ws = h ws where
     h (x:xs) = if gcd x u > 1 && gcd x v == 1
                   then (head ws) : f v x (delete x ws) else h xs
-- Reinhard Zumkeller, Dec 05 2014
  • Maple
    # This produces the first 100 terms. Uses b1 = list of terms in A098550, from b-file
b2:={$3..5000}:
b3:=[2]:
for i from 2 to 100 do
b2:=remove('x->x=b1[i]',b2):
b3:=[op(b3),b2[1]];
od:
b3;
  • Mathematica
    terms = 100;
f[lst_List] := Block[{k = 4}, While[GCD[lst[[-2]], k] == 1 || GCD[lst[[-1]], k] > 1 || MemberQ[lst, k], k++]; Append[lst, k]];
A098550 = Nest[f, {1, 2, 3}, terms-3];
a[1] = 2;
a[n_] := a[n] = For[k = a[n - 1], True, k++, If[FreeQ[A098550[[1 ;; n]], k], Return[k]]];
Array[a, terms] (* Jean-François Alcover, Aug 01 2018, after Robert G. Wilson v *)

Formula

a(n) = Min{A251546(n), A251549(n)}. - Reinhard Zumkeller, Dec 19 2014

A251417 Lengths of runs of identical terms in A251416.

Original entry on oeis.org

1, 1, 1, 5, 1, 5, 1, 6, 1, 7, 1, 12, 8, 10, 1, 17, 8, 1, 13, 13, 13, 5, 10, 11, 5, 9, 8, 19, 10, 11, 7, 11, 5, 9, 27, 9, 13, 5, 23, 5, 9, 17, 9, 11, 11, 7, 21, 9, 7, 5, 17, 27, 11, 7, 9, 17, 5, 13, 9, 21, 11, 7, 13, 9, 9
Offset: 1

Views

Author

N. J. A. Sloane, Dec 03 2014

Keywords

Comments

It would be nice to have an alternative description of this sequence, one that is not based on A098550.
It appears (conjecture) that a(n)>1 for n>18. - Alexander R. Povolotsky, Dec 07 2014
Conjecture: a(n) = A247253(n-5) for n>12. - Reinhard Zumkeller, Dec 07 2014
The previous conjecture is equivalent to the statement that A251416(n) lists all primes and only primes after a(30)=18. - M. F. Hasler, Dec 08 2014

Examples

			See A251595.

Links

Crossrefs

Cf. A098550, A251416, A251595, A247253.

Programs

  • Haskell
    import Data.List (group)
a251417 n = a251417_list !! (n-1)
a251417_list = map length $ group a251416_list
-- Reinhard Zumkeller, Dec 05 2014
  • Mathematica
    termsOfA251416 = 700;
f[lst_List] := Block[{k = 4}, While[GCD[lst[[-2]], k] == 1 || GCD[lst[[-1]], k] > 1 || MemberQ[lst, k], k++]; Append[lst, k]];
A098550 = Nest[f, {1, 2, 3}, termsOfA251416 - 3];
b[1] = 2;
b[n_] := b[n] = For[k = b[n-1], True, k++, If[FreeQ[A098550[[1 ;; n]], k], Return[k]]];
A251416 = Array[b, termsOfA251416];
Length /@ Split[A251416] (* Jean-François Alcover, Aug 01 2018, after Robert G. Wilson v *)

Formula

Let f(n)=A098551(A251595(n)). Then one can prove that A251417(n) = f(n) - f(n-1), n>=2. - Vladimir Shevelev, Dec 09 2014
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