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
Primes p such that phi(p) divides phi(Product_{primes q <= p} phi(q)), where phi is A000010. - Richard R. Forberg, Sep 11 2024
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
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 *)
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Amarnath Murthy, Apr 26 2001
EXTENSIONS
Edited by Dean Hickerson, Jun 09 2002
STATUS
approved