A225947
Lexicographically least sequence of primes (including 1) that are sum-free.
Original entry on oeis.org
1, 2, 5, 11, 23, 43, 47, 137, 157, 293, 439, 1163, 1201, 2339, 3529, 5867, 9391, 23623, 24659, 49477, 72953, 147083, 195511, 392059, 538001, 1052479, 1590467, 2520503, 4503007, 5041007, 14047027, 15637483, 28239989, 55404001, 115994933, 210773399
Offset: 1
a(8)=137 as 137 is the next prime after a(7)=47 that cannot be formed from distinct sums of a(1),...,a(7) (1,2,5,11,23,43,47).
- Giovanni Resta, Table of n, a(n) for n = 1..40
- H. L. Abbott, On sum-free sequences, Acta Arithmetica, 1987, Vol 48, Issue 1, pp. 93-96.
- Carlos Rivera, Puzzle 127. Non adding prime sequences, The Prime Puzzles & Problems Connection.
- Eric Weisstein's World of Mathematics, A-Sequence
- Wikipedia, Sum-free sequence
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memberQ[n1_, k1_] := If[Select[IntegerPartitions[Prime[n1], Length[k1], k1], Sort@#==Union@# &]=={}, False, True]; k={1}; n=1; While[Length[k]<15, (If[!memberQ[n, k], k=Append[k, Prime[n]]]; n++)]; k
A379045
a(1) = 1. For n > 1, a(n) is the least odd prime p which cannot be represented as the sum of a subset of the previous terms.
Original entry on oeis.org
1, 3, 5, 7, 17, 19, 53, 67, 173, 211, 439, 997, 1993, 2801, 4969, 6791, 13697, 18661, 50849, 50971, 106669, 152729, 310127, 412333, 826097, 1134841, 2271053, 2991883, 4952809, 7223627, 18574201, 20534933, 40243939, 60778433, 100713031, 222270319, 241670423, 563829493
Offset: 1
11 is not a term since 11 = 7 + 3 + 1.
13 is not a term since 13 = 7 + 5 + 1.
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b:= proc(n, i) option remember; n=0 or i>0 and s(i)>=n
and (b(n, i-1) or a(i)<=n and b(n-a(i), i-1))
end:
s:= proc(n) option remember; `if`(n<1, 0, s(n-1)+a(n)) end:
a:= proc(n) option remember; local p; p:= a(n-1);
while b(p, n-1) do p:= nextprime(p) od; p
end: a(1), a(2):=1, 3:
seq(a(n), n=1..26); # Alois P. Heinz, Dec 15 2024
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b[n_, i_] := b[n, i] = n == 0 || i > 0 && s[i] >= n
&& (b[n, i-1] || a[i] <= n && b[n - a[i], i-1]);
s[n_] := s[n] = If[n < 1, 0, s[n-1] + a[n]];
a[n_] := a[n] = Module[{p = a[n-1]},
While[b[p, n-1], p = NextPrime[p]]; p];
{a[1], a[2]} = {1, 3};
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 02 2025, after Alois P. Heinz *)
Showing 1-2 of 2 results.
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