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n-th partial sum of A086537.
2

%I #19 Jul 14 2024 17:03:37

%S 1,3,6,12,16,24,36,49,60,70,96,112,144,168,210,240,288,343,360,396,

%T 448,512,630,648,675,720,729,768,840,896,1000,1080,1120,1260,1344,

%U 1440,1600,1728,1920,2160,2304,2520,2640,2700,2880,3125,3200,3456,3570,3584,3888

%N n-th partial sum of A086537.

%C Every term has a unique prime signature. Conjecture: Every prime signature occurs in the sequence.

%H Robert Israel, <a href="/A086538/b086538.txt">Table of n, a(n) for n = 1..731</a>

%e a(4) = 1+2+3+6 = 12, where 1, 2, 3 and 6 are the first four terms of A086537.

%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 of A086537

%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, sp;

%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) 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(s, ", "); while(used[x], x++)); \\ _David Wasserman_, Mar 15 2005

%Y Cf. A086537.

%K nonn

%O 1,2

%A _Amarnath Murthy_, Aug 19 2003

%E Corrected and extended by _David Wasserman_, Mar 15 2005