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A270877
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Numbers surviving a decaying sieve.
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6
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1, 2, 4, 5, 6, 8, 13, 16, 17, 19, 22, 23, 24, 27, 28, 29, 32, 34, 38, 39, 40, 41, 42, 44, 49, 50, 51, 52, 56, 59, 60, 61, 64, 65, 68, 71, 72, 73, 74, 80, 89, 92, 94, 95, 96, 104, 107, 109, 113, 116, 118, 128, 131, 134, 137, 139, 142, 149, 151, 155
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OFFSET
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1,2
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COMMENTS
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In the normal sieve of Eratosthenes, for a given number p, we cross out all multiples of p; that is, p, p + p, p + p + p, .... In this decaying sieve, we cross out p, p + (p-1), p + (p-1) + (p-2), ..., p + (p-1) + (p-2) + ... + 1 (a finite list of p numbers). The sequence gives those values which are not crossed out by a sum initiated by a lesser integer. They are the "primes" of this decaying sieve.
Geometrical interpretation: in the sieve of Eratosthenes, each surviving integer p can be seen as eliminating those numbers that enumerate a rectangular area dot pattern one side of which has length p. In this sieve, each surviving integer k eliminates each number that enumerates a trapezoidal area dot pattern (on a triangular grid) with longest side k, plus the limiting case of the triangular area dot pattern with side k (the k-th triangular number). - Peter Munn, Jan 05 2017
If such a pattern has m dots, the possible lengths (number of dots) for the longest side are the nonzero numbers that occur in row m of A286013 after the number m in column 1. Thus m is in this sequence if and only if none of the other numbers in row m of A286013 are in this sequence. - Peter Munn, Jun 18 2017
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LINKS
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FORMULA
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Lexicographically earliest sequence of positive integers such that for n >= 1, 1 <= m < n, k >= 1, A286013(a(n),k) <> a(m). - Peter Munn, Jun 19 2017
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EXAMPLE
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The sieve starts as follows. Initially no numbers are crossed out. Take a(1)=1 and cross it out. The next uncrossed number is 2, so a(2)=2. Now cross out 2 and 2+1. The next uncrossed number is 4, so a(3)=4. Then cross out 4, 4+3, 4+3+2, 4+3+2+1. The next uncrossed number is 5, and so on.
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MATHEMATICA
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nn = 200; a = Range@ nn; Do[If[Length@a >= n, a = Complement[a, Function[k, Rest@ Map[Total, MapIndexed[Take[k, #] &, Range@ Max@ k]]]@ Reverse@ Range@ a[[n]]]], {n, 2, nn}]; a (* Michael De Vlieger, Mar 25 2016 *)
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PROG
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(Java)
int limit = 15707; //highest number in the sieve (inclusive)
boolean[] n = new boolean[limit + 1];
int index = 1;
for ( int i = 1; i < n.length; i++ ) {
if ( !n[i] ) {
System.out.println(index++ + " " + i);
int j = i, k = i;
while ( k + j - 1 < n.length && j > 0 ) {
k += --j;
n[k] = true;
}
}
}
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CROSSREFS
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Cf. A281256 for tabulation of its runs of consecutive integers.
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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Essential qualification added to definition by Peter Munn, Jan 19 2017
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STATUS
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approved
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