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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

%I #15 Oct 19 2017 10:43:02

%S 2,5,13,17,19,31,37,41,43,61,67,71,73,79,89,97,101,109,113,127,131,

%T 137,139,151,157,163,181,193,197,199,211,223,229,233,239,241,251,257,

%U 271,277,281,307,313,331,337,349,353,373,379,397,401,409,419,421,431,433

%N 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)).

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

%C 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

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

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

%H Wouter Meeussen, <a href="/A061303/a061303.txt">Don Reble's equivalence proof</a>

%e 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,...}, ...

%t (* 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 *)

%t Select[Range[2,500], GCD[(#-1)!, EulerPhi[(#-1)!] ]===GCD[ #! ,EulerPhi[#!] ]& ] (* _Wouter Meeussen_, Mar 18 2014 *)

%Y Cf. A061092.

%K nonn

%O 0,1

%A _Amarnath Murthy_, Apr 26 2001

%E Edited by _Dean Hickerson_, Jun 09 2002

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