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A264962
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Pulsating Checkpoint Sequence: use powers of 2 as checkpoints; place powers of 2, starting with 4, with spacing equal to the previous power of 2. Whenever we encounter a checkpoint, we jump over it; otherwise, we insert four numbers into the sequence: 2p, p, 3p, and 3p+3, where p is the smallest odd prime not yet in the sequence.
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1
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4, 6, 3, 9, 12, 8, 10, 5, 15, 18, 14, 7, 21, 24, 16, 22, 11, 33, 36, 26, 13, 39, 42, 34, 17, 51, 54, 38, 19, 57, 60, 32, 46, 23, 69, 72, 58, 29, 87, 90, 62, 31, 93, 96, 74, 37, 111, 114, 82, 41, 123, 126, 86, 43, 129, 132, 94, 47, 141, 144, 106, 53, 159
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OFFSET
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1,1
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COMMENTS
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The checkpoints, which are the numbers 4, 8, 16, 32, etc., are placed so that the number 2^k is located at the (2^k + k - 5)th position in the sequence, for k>=2; thus:
4 = 2^2 = a(2^2 + 2 - 5) = a( 4 + 2 - 5) = a(1);
8 = 2^3 = a(2^3 + 3 - 5) = a( 8 + 3 - 5) = a(6);
16 = 2^4 = a(2^4 + 4 - 5) = a(16 + 4 - 5) = a(15);
32 = 2^5 = a(2^5 + 5 - 5) = a(32 + 5 - 5) = a(32); etc.
The number of terms that will be placed between successive checkpoints 2^k and 2^(k+1) is (2^(k+1) + (k+1) - 5) - (2^k + k - 5) - 1 = 2^k for each k>=2; i.e., there will be 4 terms placed between 4 and 8, 8 terms placed between 8 and 16, 16 terms placed between 16 and 32, etc
Not every positive integer greater than four will appear in this sequence. If p and q are two consecutive primes with |p-q|>2, then the numbers from (p+2)*3 to (q-1)*3 will not occur in this sequence. No number of the form 2^k*m, where k>1 and m is an odd number not divisible by 3, will occur in this sequence (for example, 20, 28). Also, the numbers of form t^k, where t>3 is an odd prime and k>1 will not occur in this sequence (for example, 25, 49).
No two adjacent terms will share more than one prime factor.
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LINKS
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EXAMPLE
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We begin by placing successive powers of 2, starting with 2^2 = 4, with spacing equal to the value of the previous power of 2, in the sequence as checkpoints:
.
.|4 terms|
.|<----->| |<-- 8 terms -->| |<---------- 16 terms --------->|
4,_,_,_,_,8,_,_,_,_,_,_,_,_,16,_,_,_,_,_,_,_,_,_,_,_,_,_,_,_,_,32,...
Then we fill in the remaining locations sequentially, jumping over each checkpoint as we encounter it. Those remaining locations are filled in sequentially, in sets of four terms at a time (i.e., in quadruples). We begin inserting the quadruples of the form {2p, p, 3p, 3p+3}, where p in the j-th quadruple inserted is the j-th odd prime; thus, the first quadruple is {2*3, 3, 3*3, 3*3+3} = {6,3,9,12}, and inserting it gives
4,6,3,9,12,8,_,_,_,_,_,_,_,_,16,_,_,_,_,_,_,_,_,_,_,_,_,_,_,_,_,32,...
Now we jump over the checkpoint 8 and insert the next two quadruples (which have p=5 and p=7, respectively):
4,6,3,9,12,8,10,5,15,18,14,7,21,24,16,_,_,_,_,_,_,_,_,_,_,_,_,_,_,_,_,32,...
Continuing as above, we insert the next 4 quadruples (16 terms) after the checkpoint term 16, the next 8 quadruples (32 terms) after the checkpoint term 32, etc.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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