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A337065 Infinite sum of the prime numbers, compacted (see the Comments line for an explanation).

%I A337065 #29 Aug 09 2023 07:06:31
%S A337065 5,12,24,59,97,84,159,128,144,162,186,420,647,457,503,360,1214,1677,
%T A337065 532,548,564,600,624,648,1033,1079,752,772,798,828,852,1315,906,924,
%U A337065 1924,3096,1667,3496,1208,1230,3834,1993,1360,2101,1446,1472,2251,1530,3977,2471,1668,2569
%N A337065 Infinite sum of the prime numbers, compacted (see the Comments line for an explanation).
%C A337065 If the successive terms of the present sequence are expressed as the sum of k>1 consecutive primes in only one way and added, the end result will be 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + ... (conjectured to extend ad infinitum).
%C A337065 This is the lexicographically earliest sequence of distinct positive terms with this property.
%C A337065 a(n) is the smallest term of A084146 such that the set of prime parts of a(n) has (i) no primes in common with the union of the prime parts of a(1), ..., a(n-1) and (ii) contains the smallest prime excluded in the union of the prime parts of a(1), ..., a(n-1). - _R. J. Mathar_, Aug 19 2020
%H A337065 Jean-Marc Falcoz, <a href="/A337065/b337065.txt">Table of n, a(n) for n = 1..201</a>
%e A337065 The 1st term is 5 and 5 = 2+3.
%e A337065 The 2nd term is 12 and 12 = 5+7.
%e A337065 The 3rd term is 24 and 24 = 11+13.
%e A337065 The 4th term is 59 and 59 = 17+19+23.
%e A337065 The 5th term is 97 and 97 = 29+31+37.
%e A337065 The 6th term is 84 and 84 = 41+43; etc.
%e A337065 (The 4th term is NOT a(4) = 36 as 36 is the sum of consecutive primes in more than one way: 36 = 17+19 and 36 = 5+7+11+13).
%p A337065 # the set of prime partitions of A084146(n)
%p A337065 A084146Pset := proc(n::integer)
%p A337065     option remember;
%p A337065     local pset,k,i,spr ;
%p A337065     pset := {} ;
%p A337065     if isA084146(n) then
%p A337065         for k from 2 do
%p A337065             if add(ithprime(i),i=1..k) > n then
%p A337065                 break;
%p A337065             end if;
%p A337065             for i from 1 do
%p A337065                 spr := add( ithprime(j),j=i..i+k-1) ;
%p A337065                 if spr > n then
%p A337065                     break;
%p A337065                 elif spr = n then
%p A337065                     return {seq(ithprime(j),j=i..i+k-1)} ;
%p A337065                 end if;
%p A337065             end do:
%p A337065         end do:
%p A337065     end if;
%p A337065     pset ;
%p A337065 end proc:
%p A337065 A337065 := proc(n)
%p A337065     option remember;
%p A337065     local pprev,i,pmex,uni, thiss ;
%p A337065     # the set of all primes needed to represent all previous terms
%p A337065     pprev := {} ;
%p A337065     for i from 1 to n-1 do
%p A337065         pprev := pprev union A084146Pset(procname(i)) ;
%p A337065     end do ;
%p A337065     # smallest prime not in the representation of previous terms
%p A337065     for i from 1 do
%p A337065         if not ithprime(i) in pprev then
%p A337065             pmex := ithprime(i) ;
%p A337065             break;
%p A337065         end if;
%p A337065     end do:
%p A337065     for uni from 1 do
%p A337065         thiss := A084146Pset(uni) ;
%p A337065         if pmex in thiss and thiss intersect(pprev) = {} then
%p A337065             return uni ;
%p A337065         end if;
%p A337065     end do:
%p A337065 end proc:
%p A337065 for n from 1 do
%p A337065     print(A337065(n)) ;
%p A337065 end do: # _R. J. Mathar_, Aug 19 2020
%Y A337065 Cf. A336897, A050936, A084146.
%K A337065 nonn
%O A337065 1,1
%A A337065 _Eric Angelini_ and _Jean-Marc Falcoz_, Aug 14 2020