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%I #16 Jun 22 2024 08:01:07
%S 1,2,3,6,4,8,12,13,11,10,26,16,32,24,42,30,48,55,17,36,52,64,118,18,
%T 27,45,9,39,72,56,104,80,40,140,84,96,160,128,192,240,144,216,120,60,
%U 180,245,75,256,114,14,304,112,320,288,292,220,280,360,384,156,261,159,210
%N Beginning with 1, the smallest number not yet seen such that every partial sum has a distinct prime signature.
%C Conjecture: this is a rearrangement of natural numbers (i.e., every natural number is a term).
%H Robert Israel, <a href="/A086537/b086537.txt">Table of n, a(n) for n = 1..731</a>
%e The partial sums are 1, 3, 6, 12, 16, 24, 36, 49, 54, ... (A086538), each with a distinct prime signature.
%p ps:= proc(n) local F;
%p F:= ifactors(n)[2];
%p sort(F[..,2])
%p end proc:
%p N:= 1000: # for terms before the first term > N
%p Cands:= [$1..N]:
%p R:= NULL: s:= 0: Sigs:= {}: found:= true:
%p for count from 1 while found do
%p found:= false;
%p for i from 1 to N+1-count do
%p sp:= s+Cands[i];
%p x:= ps(sp);
%p if member(x,Sigs) then next fi;
%p R:= R, Cands[i];
%p Sigs:= Sigs union {x};
%p Cands:= subsop(i=NULL, Cands);
%p found:= true;
%p s:= sp;
%p break
%p od
%p od:
%p R; # _Robert Israel_, Jun 17 2024
%o (PARI)
%o ps(n) = local(f); f = factor(n); vecsort(f[,2]);
%o psUsed(v, n) = for (i = 1, n - 1, if (v == P[i], return(1))); 0;
%o print1(1, ", "); P = vector(70); used = vector(10000); x = 2; s = 1; for (n = 1, 70, i = x; v = ps(s + i); while (psUsed(v, n), i++; while (used[i], i++); v = ps(s + i)); used[i] = 1; P[n] = v; s += i; print1(i, ", "); while(used[x], x++)); \\ _David Wasserman_, Mar 15 2005
%Y Cf. A086538.
%K nonn
%O 1,2
%A _Amarnath Murthy_, Aug 19 2003
%E More terms from _David Wasserman_, Mar 15 2005
%E Duplicate example deleted by _Harvey P. Dale_, Jun 17 2023
%E Definition corrected by _Robert Israel_, Jun 17 2024
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