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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A173927 Smallest integer k such that the number of iterations of Carmichael lambda function (A002322) needed to reach 1 starting at k (k is counted) is n.

Original entry on oeis.org

1, 2, 3, 5, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 531441, 1594323, 4782969, 14348907, 43046721, 86093443, 258280327, 688747547
Offset: 1

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Author

Michel Lagneau, Nov 26 2010

Keywords

Comments

Smallest number k such that the trajectory of k under iteration of Carmichael lambda function contains exactly n distinct numbers (including k and the fixed point).
The first 13 terms are 1 or a prime. The next five terms are powers of 3. Then another prime. What explains this behavior? - T. D. Noe, Mar 23 2012
A185816(a(n)) = n - 1. - Reinhard Zumkeller, Sep 02 2014
If a(n) (n > 1) is either a prime or a power of 3, then a(n) is also the smallest integer k such that the number of iterations of Euler's totient function (A000010) needed to reach 1 starting at k (k is counted) is n. - Jianing Song, Jul 10 2019

Examples

			for n=5, a(5)=11 gives a chain of length 5 because the trajectory is 11 -> 10 -> 4 -> 2 -> 1.

Links

Crossrefs

Cf. A002322, A027763, A056637.
Cf. A185816 (number of iterations of Carmichael lambda function needed to reach 1), A003434 (number of iterations of Euler's totient function needed to reach 1).

Programs

  • Haskell
    import Data.List (elemIndex); import Data.Maybe (fromJust)
a173927 = (+ 1) . fromJust . (`elemIndex` map (+ 1) a185816_list)
-- Reinhard Zumkeller, Sep 02 2014
  • Mathematica
    f[n_] := Length@ NestWhileList[ CarmichaelLambda, n, Unequal, 2] - 1; t = Table[0, {30}]; k = 1; While[k < 2100000001, a = f@ k; If[ t[[a]] == 0, t[[a]] = k; Print[a, " = ", k]]; k++] (* slightly modified by Robert G. Wilson v, Sep 01 2014 *)

Extensions

a(20)-a(21) from Robert G. Wilson v, Sep 01 2014

A101231 a(n) = n-th prime of Erdős-Selfridge classification n-.

Original entry on oeis.org

2, 29, 67, 179, 941, 4079, 20389, 65267, 224563, 978863, 6448979, 47247763, 309550999, 2150787839, 13925010299
Offset: 1

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Author

Jonathan Vos Post, Dec 15 2004

Keywords

Comments

Diagonalization of the Erdős-Selfridge classification of primes.

Examples

			a(1) = 2 because 2 is the first element of A005109.
a(2) = 29 because 29 is the 2nd element of A005110.
a(3) = 67 because 67 is the 3rd element of A005111.
a(4) = 179 because 179 is the 4th element of A005112.

References

  • R. K. Guy, Unsolved Problems in Number Theory, A18.

Crossrefs

Cf. A056637, A005109, A005110, A005111, A005112, A005113, A081424, A081425, A081426, A081427, A081429.

Extensions

More terms from R. J. Mathar, May 02 2007

A129250 Primes of Erdős-Selfridge class 16-.

Original entry on oeis.org

22111003847, 25782283783, 34824831403, 42970472971, 44905511759, 45490491349, 52486961911, 54560052479, 55437374381, 65803884467, 66333011539
Offset: 1

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Author

M. F. Hasler, Apr 21 2007

Keywords

Comments

Knowledge of a(k), k=1..9 allows us to establish A056637(17) = 1 + 2*a(9) = 110874748763.

Links

Crossrefs

Cf. A056637, A081640, A081641, A129248, A129249.

Programs

  • PARI
    nextclass( a, s=-1, p, n=[] )={ if( !p, p=nextprime(a[ #a]+1)); print("Computing all primes of next class up to ",2*p-s ); for( i=1,#a, for( k=1,p/a[i], if( is/*pseudo*/prime(2*k*a[i]-s), n=concat(n,2*k*a[i]-s); ) ) ); vecsort(n) }; A129250=nextclass(A129249)

Formula

{ a(n) } = { p=1+2*k*A129249(n); n=1,2,3..., k=1,2,3... such that p is prime and k has no factor of class > 15- }.

A177854 Smallest prime of rank n.

Original entry on oeis.org

2, 3, 11, 131, 1571, 43717, 5032843, 1047774137, 7128418089643
Offset: 0

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Author

Lei Zhou, May 14 2010

Keywords

Comments

The Brillhart-Lehmer-Selfridge algorithm provides a general method for proving the primality of P as long as one can factor P+1 or P-1. Therefore for any prime number, when P+1 or P-1 is completely factored, the primality of any factors of P+1 or P-1 can also be proved by the same algorithm. The shortest recursive primality proving chain depth is called the rank of P (cf. A169818).

Examples

			The "trivial" prime 2 has rank 0. 3 = 2+1 takes one step to reduce to 2, so 3 has rank 1.
P=131: P+1=132=2^2*3*11. P1[1]=2 has rank 0; P1[2]=3 has rank 1; P1[3]=11: P1[3]+1=12=2^2*3; is one step from 3 and has recursion depth = 2. So P=131 has total maximum recursion depth 2+1 = 3 and therefore has rank 3.

Links

Crossrefs

These are the primes where records occur in A169818.
Cf. A005113, A056637. - Robert G. Wilson v, May 28 2010

Programs

  • Mathematica
    (* The following program runs through all prime numbers until it finds the first rank n prime. (It took about a week for n = 7.) *) Fr[n_]:= Module[{nm, np, fm, fp, szm, szp, maxm, maxp, thism, thisp, res, jm, jp}, If[n == 2, res = 0, nm = n - 1; np = n + 1; fm = FactorInteger[nm]; fp = FactorInteger[np]; szm = Length[fm]; szp = Length[fp]; maxm = 0; Do[thism = Fr[fm[[jm]][[1]]]; If[maxm < thism, maxm = thism], {jm, 1, szm}]; maxp = 0; Do[thisp = Fr[fp[[jp]][[1]]]; If[maxp < thisp, maxp = thisp], {jp, 1, szp}]; res = Min[maxm, maxp] + 1 ]; res]; target=0; Do[p = Prime[i]; s = Fr[p]; If[s == target, Print[p]; target++], {i, Infinity}]
  • PARI
    rank(p)=if(p<8, return(p>2)); vecmin(apply(k->vecmax(apply(rank, factor(k)[,1])), [p-1,p+1]))+1
print1(2); r=0;forprime(p=3,, t=rank(p); if(t>r, r=t; print1(", "p))) \\ Charles R Greathouse IV, Oct 03 2016

Extensions

Partially edited by N. J. A. Sloane, May 15 2010, May 28 2010
Definition corrected by Robert Gerbicz, May 28 2010
a(8) from Florian Baur, Sep 05 2024

A102444 Smallest number m such that A102442(m)=n.

Original entry on oeis.org

1, 5, 11, 23, 47, 149, 359, 719, 1439, 2879, 12097, 24197, 48407, 96821, 193649, 968237, 2614327, 5809201, 11618413, 25055857, 75167579, 225502703, 451005407, 2109888899, 4510054327, 14500925539, 43502776619, 87005553241, 174011106487
Offset: 1

Views

Author

Reinhard Zumkeller, Jan 09 2005

Keywords

Comments

Primes: a(n+1) > 2*a(n).

Crossrefs

Cf. A102440.
A056637 describes another iteration that reduces primes to 2 and 3.

Extensions

More terms from David Wasserman, Apr 04 2008
Previous Showing 21-25 of 25 results.

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

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