A086537 Beginning with 1, the smallest number not yet seen such that every partial sum has a distinct prime signature.

1, 2, 3, 6, 4, 8, 12, 13, 11, 10, 26, 16, 32, 24, 42, 30, 48, 55, 17, 36, 52, 64, 118, 18, 27, 45, 9, 39, 72, 56, 104, 80, 40, 140, 84, 96, 160, 128, 192, 240, 144, 216, 120, 60, 180, 245, 75, 256, 114, 14, 304, 112, 320, 288, 292, 220, 280, 360, 384, 156, 261, 159, 210

Offset

1,2

Comments

Conjecture: this is a rearrangement of natural numbers (i.e., every natural number is a term).

Links

Robert Israel, Table of n, a(n) for n = 1..731

Example

The partial sums are 1, 3, 6, 12, 16, 24, 36, 49, 54, ... (A086538), each with a distinct prime signature.

Maple

ps:= proc(n) local F;
 F:= ifactors(n)[2];
 sort(F[.., 2])
end proc:
N:= 1000: # for terms before the first term > N
Cands:= [$1..N]:
R:= NULL: s:= 0: Sigs:= {}: found:= true:
for count from 1 while found do
  found:= false;
  for i from 1 to N+1-count do
    sp:= s+Cands[i];
  x:= ps(sp);
  if member(x, Sigs) then next fi;
    R:= R, Cands[i];
    Sigs:= Sigs union {x};
  Cands:= subsop(i=NULL, Cands);
  found:= true;
  s:= sp;
  break
  od
od:
R; # Robert Israel, Jun 17 2024

Prog

(PARI)
ps(n) = local(f); f = factor(n); vecsort(f[, 2]);
psUsed(v, n) = for (i = 1, n - 1, if (v == P[i], return(1))); 0;
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

Crossrefs

Cf. A086538.

Sequence in context: A370046 A373326 A382357 * A212486 A127562 A096113

Adjacent sequences: A086534 A086535 A086536 * A086538 A086539 A086540

Keyword

nonn

Author

Amarnath Murthy, Aug 19 2003

Extensions

More terms from David Wasserman, Mar 15 2005

Duplicate example deleted by Harvey P. Dale, Jun 17 2023

Definition corrected by Robert Israel, Jun 17 2024

Status

approved