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Beginning with 1, the smallest number not yet seen such that every partial sum has a distinct prime signature.
2

%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