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A061303 Given a prime p, let s(p,0)=p and let s(p,n+1) be the smallest prime == 1 (mod s(p,n)). Let S(p) be the sequence {s(p,n): n=0,1,...}. Let a(0)=2 and let a(n+1) be the smallest prime not in any of the sequences S(a(0)), ..., S(a(n)). 3
2, 5, 13, 17, 19, 31, 37, 41, 43, 61, 67, 71, 73, 79, 89, 97, 101, 109, 113, 127, 131, 137, 139, 151, 157, 163, 181, 193, 197, 199, 211, 223, 229, 233, 239, 241, 251, 257, 271, 277, 281, 307, 313, 331, 337, 349, 353, 373, 379, 397, 401, 409, 419, 421, 431, 433 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

It is conjectured for primes p and q the sequences S(p) and S(q) are disjoint, unless one is contained in the other.

Also values of n such that gcd(n! , phi(n!)) equals gcd((n-1)! , phi((n-1)!)), see proof by Don Reble. - Wouter Meeussen, Mar 18 2014

REFERENCES

Amarnath Murthy, On the divisors of Smarandache Unary Sequence. Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000.

Amarnath Murthy, Smarandache Prime Generator Sequence (to be published in Smarandache Notions Journal).

LINKS

Table of n, a(n) for n=0..55.

Wouter Meeussen, Don Reble's equivalence proof

EXAMPLE

a(0)=2 so S(a(0))={2,3,7,29,...}, which is A061092. Hence a(1)=5 so S(a(1))={5,11,23,47,...}. Hence a(2)=13 so S(a(2))={13,53,107,643,...}, ...

MATHEMATICA

(* start *) s[p_, 0] := s[p, 0]=p; s[p_, n_] := s[p, n]=Module[{q}, For[q=s[p, n-1]+1, !PrimeQ[q], q+=s[p, n-1], Null]; q]; ins[q_, p_] := Module[{k}, For[k=0, s[p, k]<=q, k++, If[s[p, k]==q, Return[True]]]; False]; a[0]=2; a[n_] := a[n]=Module[{i, j, q}, For[i=1, True, i++, q=Prime[i]; For[j=0, j<n, j++, If[ins[q, a[j]], Break[]]]; If[j==n, Return[q]]]]; (* end *)

Select[Range[2, 500], GCD[(#-1)!, EulerPhi[(#-1)!] ]===GCD[ #! , EulerPhi[#!] ]&  ] (* Wouter Meeussen, Mar 18 2014 *)

CROSSREFS

Cf. A061092.

Sequence in context: A175256 A139254 A266987 * A173626 A215424 A215428

Adjacent sequences:  A061300 A061301 A061302 * A061304 A061305 A061306

KEYWORD

nonn

AUTHOR

Amarnath Murthy, Apr 26 2001

EXTENSIONS

Edited by Dean Hickerson, Jun 09 2002

STATUS

approved

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Last modified April 6 05:38 EDT 2020. Contains 333267 sequences. (Running on oeis4.)