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A073673 Rearrangement of natural numbers such that every partial product + 1 is a prime.

%I A073673 #36 Jun 21 2025 12:01:04
%S A073673 1,2,3,5,6,9,4,7,10,17,12,11,13,8,19,27,21,26,20,16,14,28,22,18,47,30,
%T A073673 31,23,34,37,41,45,49,33,36,58,24,62,39,56,42,93,54,25,51,53,15,70,72,
%U A073673 73,46,50,64,97,55,57,171,96,79,81,66,71,132,89,121,29,61,60,177,32
%N A073673 Rearrangement of natural numbers such that every partial product + 1 is a prime.
%C A073673 From _Robert G. Wilson v_, Dec 24 2012: (Start)
%C A073673 Records: 1, 2, 3, 5, 6, 9, 10, 17, 19, 27, 28, 47, 49, 58, 62, 93, 97, 171, 177, 184, 221, 243, 470, 512, 573, 768, 856, 999, 1028, 1226, 1659, 2522, ...
%C A073673 Late Records: 1, 2, 3, 4, 7, 8, 14, 15, 29, 32, 35, 59, 75, ... (End)
%H A073673 Michael S. Branicky, <a href="/A073673/b073673.txt">Table of n, a(n) for n = 1..1600</a> (terms 751..954 from Bill McEachen, 1..750 from Robert G. Wilson v)
%F A073673 Conjecture: n/log(n) << a(n) << n*log(n). - _Thomas Ordowski_, Aug 09 2017
%t A073673 f[s_List] := Block[{k = 1, p = Times @@ s}, While[ MemberQ[s, k] || !PrimeQ[k*p + 1], k++]; Append[s, k]]; Nest[f, {1}, 69] (* _Robert G. Wilson v_, Dec 24 2012 *)
%o A073673 (PARI) v=[1];n=1;while(n<100,s=1+n*prod(i=1,#v,v[i]);if(isprime(s)&&!vecsearch(vecsort(v),n),v=concat(v,n);n=0);n++);v \\ _Derek Orr_, Jun 16 2015
%o A073673 (Python)
%o A073673 from gmpy2 import is_prime
%o A073673 from itertools import islice
%o A073673 def agen(startp=1, startset=set()): # generator of terms
%o A073673     aset, p, mink = startset, startp, 1
%o A073673     while True:
%o A073673         an = mink
%o A073673         while an in aset or not is_prime(p*an + 1): an += 1
%o A073673         yield an; aset.add(an); p *= an
%o A073673         while mink in aset: aset.discard(mink); mink += 1
%o A073673 print(list(islice(agen(), 70))) # _Michael S. Branicky_, May 19 2023
%Y A073673 Cf. A073674, A083769.
%K A073673 nonn
%O A073673 1,2
%A A073673 _Amarnath Murthy_, Aug 11 2002
%E A073673 More terms from _Sascha Kurz_, Feb 01 2003