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A123055
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Lexicographically earliest sequence of nonprime positive integers whose first differences represent all prime numbers with exactly one appearance of each prime.
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1
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1, 4, 6, 25, 30, 77, 84, 95, 108, 125, 148, 177, 208, 245, 286, 329, 382, 441, 502, 573, 640, 713, 792, 875, 964, 1065, 1162, 1265, 1372, 1485, 1594, 1725, 1852, 1989, 2128, 2277, 2428, 2585, 2748, 2915, 3088, 3267, 3448, 3639, 3832, 4029, 4228, 4439, 4662
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OFFSET
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1,2
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COMMENTS
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The n-th prime appears in the sequence of first differences (A075568) not later than at the 2n-th position. To see this it is enough to notice that in the original sequence (excluding the first element) odd and even numbers alternate. Therefore from each odd element m the sequence simply jumps to an even element m+p where p is the smallest previously unused prime. - Max Alekseyev, Sep 26 2006
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LINKS
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EXAMPLE
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The sequence S is constructed like this:
Start with "1": S = 1, ...
Add the smallest prime not added so far, in order to get a composite:
- Can we add 2? No, because 1+2=3 and 3 is prime.
- Can we add 3? Yes, because 1+3=4 and 4 is composite.
So we now have S = 1,4, ...
Add the smallest prime not added so far in order to get a composite:
- Can we add 2 (smallest available prime)? Yes, because 4+2=6 and 6 is composite.
So we now have S = 1,4,6, ...
- Can we add 5? No, because 6+5=11 and 11 is prime.
- Can we add 7? No, because 6+7=13 and 13 is prime.
- Can we add 11? No, because 6+11=17 and 17 is prime.
- Can we add 13? No, because 6+13=19 and 19 is prime.
- Can we add 17? No, because 6+17=23 and 23 is prime.
- Can we add 19? Yes, because 6+19=25 and 25 is composite.
So we now have S = 1,4,6,25, ...
- Can we add 5 (smallest available prime)? Yes, because 25+5=30 and 30 is composite.
So we now have S = 1,4,6,25,30, ...
etc.
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CROSSREFS
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KEYWORD
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base,easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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