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A161460
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Positive integers n such that there is no m different from n where both d(n) = d(m) and d(n+1) = d(m+1), where d(n) is the number of positive divisors of n.
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2
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1, 2, 3, 4, 8, 15, 16, 24, 35, 48, 63, 64, 80, 99, 288, 528, 575, 624, 728, 960, 1023, 1024, 1088, 1295, 2303, 2400, 4095, 4096, 5328, 6399, 6723, 9408, 9999, 14640, 15624, 28223, 36863, 38415, 46655, 50175, 50624, 57121, 59048, 59049, 65535, 65536
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Are these values known to be correct, or are they just conjectures? - Leroy Quet, Jun 20 2009
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LINKS
| R. J. Mathar, Entries proven or conjectured (PDF)
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EXAMPLE
| d(15) = 4, and d(15+1) = 5. Any positive integers m+1 with exactly 5 divisors must by of the form p^4, where p is prime. So m = p^4 -1 = (p^2+1)*(p+1)*(p-1). Now, in order for d(m) to have exactly 4 divisors, m must either be of the form q^3 or q*r, where q and r are distinct primes. But no p is such that (p^2+1)*(p+1)*(p-1) = q^3. And the only p where (p^2+1)*(p+1)*(p-1) = q*r is when p = 2 ( and so q=5, r =3). So there is only one m where both d(m) = 4 and d(m+1) = 5, which is when m=15. Therefore 15 is in this sequence.
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CROSSREFS
| Sequence in context: A117395 A006755 A005853 * A097029 A122774 A189740
Adjacent sequences: A161457 A161458 A161459 * A161461 A161462 A161463
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KEYWORD
| nonn
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AUTHOR
| Leroy Quet, Jun 10 2009
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EXTENSIONS
| Extended with J. Brennen's values of Jun 11 2009. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 16 2009
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