A308055 -id:A308055 - cp's OEIS frontend

A309123 a(1) = 50 and for any n > 0, a(n+1)^2 is the smallest square that begins with a(n).

50, 71, 267, 517, 2274, 1508, 3884, 1971, 444, 667, 817, 286, 535, 732, 856, 2926, 541, 736, 858, 293, 542, 233, 483, 695, 834, 2888, 16995, 41225, 20304, 4506, 6713, 2591, 5091, 22564, 47502, 21795, 46686, 21607, 46484, 215602, 46433, 68142, 261041, 510922

Offset: 1

Rémy Sigrist, Jul 13 2019

This sequence is similar to A308055.

The initial value (50) seems to be the first one for which the iteration of A018796 diverges; there are neither duplicates nor squares among the first 79000 terms.

			The first terms, alongside the square of a(n+1), are:
  n   a(n)  a(n+1)^2
  --  ----  --------
   1    50      5041
   2    71     71289
   3   267    267289
   4   517   5171076
   5  2274   2274064
   6  1508  15085456
   7  3884   3884841
   8  1971    197136
   9   444    444889
  10   667    667489
  11   817     81796
  12   286    286225
  13   535    535824
  14   732    732736
  15   856   8561476
  16  2926    292681

Rémy Sigrist, Table of n, a(n) for n = 1..1000
Rémy Sigrist, PARI program for A309123

Cf. A018796, A308055.

PARI
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a(1) = 50 and then a(n+1) = A018796(a(n)) for n > 0.

A336251 a(1) = 15; for n > 1, a(n)^2 is the smallest square that begins with a(n-1) in base 6.

Original entry on oeis.org

15, 57, 111, 155, 183, 199, 508, 812, 171, 471, 319, 643, 913, 1088, 2909, 11650, 9518, 21074, 31357, 93691, 396693, 55540, 50905, 119374, 182804, 226216, 251647, 265415, 111280, 72055, 142024, 13567, 25160, 34262, 39982, 105795, 172093, 537634

Offset: 1

Sean Lipton, Jul 14 2020

The sequence becomes periodic after 491 terms with a period of 10.
The only square in this sequence is 25 and the sequence becomes periodic two terms later.
The maximum value is a(226) = 325880259349618.

			The first 10 a(n) alongside the base 6 representations of a(n) and squares of a(n+1):
  n   a(n)  a(n) b6  a(n+1)^2 b6
  --  ----  -------  -----------
   1    15       23        23013
   2    57      133       133013
   3   111      303       303121
   4   155      415       415013
   5   183      503       503201
   6   199      531      5310424
   7   508     2204     22044304
   8   812     3432       343213
   9   171      443      4431013
  10   471     2103      2103041

Sean Lipton, Table of n, a(n) for n = 1..600

Cf. A308055, A309123.

lista(nn) = {my(a = 15); for (n=2, nn, print1(a, ", "); my(i=1); while((sqrtint((a+1)*6^i-1) <= sqrtint(a*6^i-1)), i++); a = ceil(sqrt(a*6^i)););} \\ Michel Marcus, Jul 15 2020

a(n+1) = ceiling(sqrt(a(n)*6^i)) such that floor(sqrt((a(n)+1)*6^i-1)) > floor(sqrt(a(n)*6^i-1)) where i is a whole number and minimized.

For n > 481, a(n) = a(n+10).

cp's OEIS Frontend

A309123 a(1) = 50 and for any n > 0, a(n+1)^2 is the smallest square that begins with a(n).

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