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Search: id:A073608
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| A073608 |
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a(1) = 1, a(n) = smallest number such that a(n)-a(n-k) is a prime power > 1 for all k. |
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+0
1
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OFFSET
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1,2
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COMMENT
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Differences |a(i)-a(j)| are prime powers for all i,j. Conjecture: sequence is bounded.
Proof that sequence is complete: Assume there is some k after the term 12. Then {k-1, k-3, k-5} must contain a multiple of 3. Also {k-8,k-10,k-12} also contains a multiple of 3. No prime > 12 is a multiple of 3, so the multiples of 3 are both prime powers. This implies there must be two powers of 3 that have a difference at most 11, but no such pair exists > 12 (only 1,3 and 3,9 qualify.) - Jim Nastos (nastos(AT)gmail.com), Aug 09 2002
There is an elementary proof that no set of seven integers of this kind exists. - Don Reble (djr(AT)nk.ca), Aug 10, 2002.
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EXAMPLE
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a(5) = 10 as 10-8, 10-5, 10-3, 10-1 or 2, 5, 7, 9 are prime powers.
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CROSSREFS
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Cf. A073607.
Sequence in context: A120943 A087792 A141436 this_sequence A155945 A096985 A138829
Adjacent sequences: A073605 A073606 A073607 this_sequence A073609 A073610 A073611
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KEYWORD
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nonn,fini,full
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AUTHOR
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Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 04 2002
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EXTENSIONS
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Sixth term from Jim Nastos, Aug 09, 2002
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