A115550
Numbers k such that k^2 is the concatenation of two numbers m and 8*m.
Original entry on oeis.org
18, 36, 168, 252, 336, 1668, 3336, 16668, 33336, 166668, 333336, 1666668, 3333336, 16666668, 22222224, 27777780, 33333336, 113149848, 114678900, 116207952, 117737004, 119266056, 120795108, 122324160, 123853212
Offset: 1
A115551
Numbers k such that the concatenation of 8*k with k gives a square.
Original entry on oeis.org
1, 4, 9, 89, 889, 8889, 88889, 888889, 8888889, 88888889, 888888889, 1026765584, 1066636289, 1107266436, 1148656025, 1190805056, 1233713529, 1277381444, 1321808801, 1366995600, 1412941841, 1459647524, 1507112649, 1555337216
Offset: 1
A115552
Numbers k such that k^2 is the concatenation of two numbers 8*m and m.
Original entry on oeis.org
9, 18, 27, 267, 2667, 26667, 266667, 2666667, 26666667, 266666667, 2666666667, 9063180828, 9237472767, 9411764706, 9586056645, 9760348584, 9934640523, 10108932462, 10283224401, 10457516340, 10631808279, 10806100218
Offset: 1
A115553
Numbers k such that the concatenation of k with 9*k gives a square.
Original entry on oeis.org
2041, 8164, 2366863905325444, 5325443786982249, 9467455621301776, 2040816326530612244897959183673469387755102041, 8163265306122448979591836734693877551020408164
Offset: 1
A115554
Numbers k such that k^2 is the concatenation of two numbers m and 9*m.
Original entry on oeis.org
14287, 28574, 15384615384615386, 23076923076923079, 30769230769230772, 14285714285714285714285714285714285714285714287, 28571428571428571428571428571428571428571428574
Offset: 1
A271637
Squared-squares in base 2: numbers n such that n^2 in base 2 is of the form xx for a string x.
Original entry on oeis.org
6, 820, 104391567, 119304648, 858993460, 900719925474100, 26202761468337432, 29478106651879611, 32753451835421790, 225701339254799219773, 243062980735937621294, 260424622217076022815, 277786263698214424336, 944473296573929042740, 232485734541274841289650
Offset: 1
The number 6 is in the sequence because 36 = 6^2 and 36 in base 2 is 100100, which is xx for x = 100.
- Andrew Bridy, Robert J. Lemke Oliver, Arlo Shallit, and Jeffrey Shallit, The Generalized Nagell-Ljunggren Problem: Powers with Repetitive Representations, Experimental Math, 28 (2019), 428-439.
A369689
a(n) is the least positive number k such that k^2 is the concatenation of m and m + n for some positive number m, or -1 if there is no such k.
Original entry on oeis.org
36363636364, 428, 8874, 5, 310, 4, 39, -1, 7747, 465
Offset: 0
a(3) = 5 because 5^2 = 25 is the concatenation of 2 and 2 + 3 = 5, and 5 is the least m that works.
a(7) = -1 because it can be proven that 7 is not a square mod (10^d + 1) for any d, and therefore there are no k and m such that k^2 is the concatenation of m and m + 7.
-
from itertools import count
from sympy import sqrt_mod
def A369689(n):
for j in count(0):
b = 10**j
a = b*10+1
for k in sorted(sqrt_mod(n,a,all_roots=True)):
m = (k**2-n)//a
if m>0 and b <= m+n < a-1:
return k # Chai Wah Wu, Feb 18 2024
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