A188540 -id:A188540 - cp's OEIS frontend

A188631 Numbers k such that d(k-1) < d(k) < d(k+1) where d(k) is the number of divisors of k.

62, 63, 74, 164, 188, 194, 195, 207, 255, 275, 278, 279, 284, 314, 356, 362, 363, 398, 399, 404, 422, 423, 428, 455, 458, 459, 483, 494, 524, 539, 584, 614, 615, 662, 674, 692, 734, 747, 758, 759, 764, 824, 854, 867, 890, 927, 935, 944, 956, 998, 999

Offset: 1

Juri-Stepan Gerasimov, Apr 05 2011

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A188540, A075028.

d:=func; [ n: n in [2..1000] | d(n-1) lt d(n) and d(n) lt d(n+1)]; // Bruno Berselli, Apr 05 2011

Select[Range[2,1000], DivisorSigma[0, # - 1] < DivisorSigma[0, #] < DivisorSigma[0, # + 1] &] (* T. D. Noe, Apr 05 2011 *)

A000005(a(n)-1) < A000005(a(n)) < A000005(a(n)+1).

Corrected and extended by T. D. Noe, Apr 05 2011

A188926 Decimal expansion of sqrt((7+sqrt(13))/6).

Original entry on oeis.org

1, 3, 2, 9, 5, 0, 8, 1, 3, 4, 3, 2, 7, 8, 7, 9, 2, 4, 9, 8, 9, 5, 7, 2, 3, 2, 4, 3, 7, 4, 0, 9, 4, 4, 4, 7, 1, 3, 3, 5, 9, 6, 0, 8, 7, 1, 9, 6, 7, 0, 0, 6, 1, 5, 6, 0, 8, 4, 7, 9, 6, 4, 8, 5, 0, 1, 0, 2, 5, 7, 3, 6, 9, 5, 8, 2, 0, 5, 2, 4, 2, 2, 9, 5, 2, 4, 1, 3, 7, 1, 6, 4, 9, 6, 4, 3, 1, 5, 2, 7, 1, 3, 0, 5, 7, 6, 8, 4, 4, 5, 4, 5, 4, 7, 8, 2, 6, 7, 9, 0, 9, 2, 1, 0, 8, 3, 3, 6, 5, 9

Offset: 1

Clark Kimberling, Apr 13 2011

Decimal expansion of the length/width ratio of a sqrt(1/3)-extension rectangle. See A188640 for definitions of shape and r-extension rectangle.

A sqrt(1/3)-extension rectangle matches the continued fraction [1,3,28,1,2,2,42,1,1,1,4,...] for the shape L/W=sqrt((7+sqrt(13))/6). This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,1,...]. Specifically, for the sqrt(1/3)-extension rectangle, 1 square is removed first, then 3 squares, then 28 squares, then 1 square,..., so that the original rectangle of shape sqrt((7+sqrt(13))/6) is partitioned into an infinite collection of squares.

			1.32950813432787924989572324374094447133596...

Cf. A188540, A188927.

r = 3^(-1/2); t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]
RealDigits[Sqrt[(7+Sqrt[13])/6],10,140][[1]] (* Harvey P. Dale, Feb 08 2013 *)

cp's OEIS Frontend

A188631 Numbers k such that d(k-1) < d(k) < d(k+1) where d(k) is the number of divisors of k.

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