

A127629


Numbers m such that a divisor, together with its quotient and remainder, are consecutive terms (in that order) in a geometric sequence.


5



9, 28, 34, 58, 65, 75, 110, 126, 132, 201, 205, 217, 224, 246, 254, 258, 294, 344, 384, 399, 436, 498, 502, 513, 516, 520, 579, 657, 680, 690, 730, 786, 810, 866, 880, 978, 979, 1001, 1008, 1028, 1038, 1105, 1128, 1164, 1330, 1332, 1365, 1370, 1374, 1388
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OFFSET

1,1


COMMENTS

The sequence misses the primes.
When m is a term, then m = d*q + r and r<q<d are in geometric progression; but also, in this case, m = d'*q' + r with r<d'<q' that are in geometric progression with d'=q and q'=d (see examples).  Bernard Schott, May 15 2020


LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000
C. Hughes, Geometric Division
Wikipedia, Euclidean division


EXAMPLE

58 is in the sequence because 58 = 9*6 + 4, where 9, 6 and 4 are consecutive terms in a geometric sequence.
For a(4) = 58 with noninteger ratio = 3/2:
58  9 58  6
 
4  6 4  9
For a(16) = 258 with integer ratio = 4:
258  32 258  8
 
2  8 2  32


MATHEMATICA

mx = 1388; m = Ceiling @ Sqrt[mx]; s={}; Do[r = Select[Divisors[k^2], #<k &]; q = k^2/r; v = q * k + r; s = Join[s, v], {k, 1, m}]; Select[Union[s], # <= mx &] (* Amiram Eldar, Aug 28 2019 *)


PROG

(PARI) is(n)={for(d=1, n, if((n\d)*(n%d)==d^2, return(1))); return(0)}


CROSSREFS

Disjoint union of A334185 and A334186.
Subsequence: A001093 \ {0, 1, 2} (for remainder = 1).
Sequence in context: A031454 A044999 A155473 * A334185 A267686 A024670
Adjacent sequences: A127626 A127627 A127628 * A127630 A127631 A127632


KEYWORD

easy,nonn


AUTHOR

Nick Hobson (nickh(AT)qbyte.org), Jan 20 2007


STATUS

approved



