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.
%I A061303 #34 Sep 16 2024 23:58:21
%S A061303 2,5,13,17,19,31,37,41,43,61,67,71,73,79,89,97,101,109,113,127,131,
%T A061303 137,139,151,157,163,181,193,197,199,211,223,229,233,239,241,251,257,
%U A061303 271,277,281,307,313,331,337,349,353,373,379,397,401,409,419,421,431,433
%N 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)).
%C A061303 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 A061303 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
%C A061303 Primes p such that phi(p) divides phi(Product_{primes q <= p} phi(q)), where phi is A000010. - _Richard R. Forberg_, Sep 11 2024
%D A061303 Amarnath Murthy, On the divisors of Smarandache Unary Sequence. Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000.
%D A061303 Amarnath Murthy, Smarandache Prime Generator Sequence (to be published in Smarandache Notions Journal).
%H A061303 Wouter Meeussen, <a href="/A061303/a061303.txt">Don Reble's equivalence proof</a>
%e A061303 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 A061303 (* 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 A061303 Select[Range[2,500], GCD[(#-1)!, EulerPhi[(#-1)!] ]===GCD[ #! ,EulerPhi[#!] ]& ] (* _Wouter Meeussen_, Mar 18 2014 *)
%t A061303 result = {}; prodEPP = 1; Do[prodEPP *= EulerPhi[Prime[i]];
%t A061303 If[Divisible[EulerPhi[prodEPP], EulerPhi[Prime[i]]],
%t A061303 AppendTo[result, Prime[i]]], {i, 1, 1000}]; result (* _Richard R. Forberg_, Sep 16 2024 *)
%Y A061303 Cf. A061092. A000010
%K A061303 nonn
%O A061303 0,1
%A A061303 _Amarnath Murthy_, Apr 26 2001
%E A061303 Edited by _Dean Hickerson_, Jun 09 2002