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)).

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

Offset: 0

Author

Amarnath Murthy, Apr 26 2001

Keywords

• nonn

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

Primes p such that phi(p) divides phi(Product_{primes q <= p} phi(q)), where phi is A000010. - Richard R. Forberg, Sep 11 2024

Examples

			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,...}, ...
		

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

• Wouter Meeussen, Don Reble's equivalence proof

Crossrefs

Cf. A061092. A000010

Programs

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, jWouter Meeussen, Mar 18 2014 *)
result = {}; prodEPP = 1; Do[prodEPP *= EulerPhi[Prime[i]];
 If[Divisible[EulerPhi[prodEPP], EulerPhi[Prime[i]]],
AppendTo[result, Prime[i]]], {i, 1, 1000}]; result  (* Richard R. Forberg, Sep 16 2024 *)

Extensions

Edited by Dean Hickerson, Jun 09 2002