The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A161460 Positive integers k such that there is no m different from k where both d(k) = d(m) and d(k+1) = d(m+1), where d(k) is the number of positive divisors of k. 3
 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; text; internal format)
 OFFSET 1,2 COMMENTS Are these values known to be correct, or are they just conjectures? - Leroy Quet, Jun 20 2009 Numbers k that are uniquely identified by the values of the ordered pair (d(k), d(k+1)). - Jon E. Schoenfield, Aug 11 2019 Conjecture: 2 is the only term that is neither a square nor 1 less than a square. - Jon E. Schoenfield, Aug 12 2019 LINKS R. J. Mathar, Recurring pairs of consecutive entries in the number-of-divisors function, vixra:1911.0287 (2019) EXAMPLE d(15) = 4, and d(15+1) = 5. Any positive integer m+1 with exactly 5 divisors must be 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 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 m=15. Therefore 15 is in this sequence. CROSSREFS Cf. A000005, A164119. Sequence in context: A117395 A006755 A005853 * A097029 A122774 A274166 Adjacent sequences:  A161457 A161458 A161459 * A161461 A161462 A161463 KEYWORD nonn AUTHOR Leroy Quet, Jun 10 2009 EXTENSIONS Extended with J. Brennen's values of Jun 11 2009 by R. J. Mathar, Jun 16 2009 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified March 31 13:07 EDT 2020. Contains 333151 sequences. (Running on oeis4.)