A116300
n times n+9 gives the concatenation of two numbers m and m+1.
Original entry on oeis.org
26, 66, 3416102887775247376839416334668635, 3756559953325598880263233435801764, 4313503800489302411917772257282208
Offset: 1
66 * 75 = 49//50, where // denotes concatenation.
-
F:= proc(d) local t, g, Cands:
t:= 10^d+1;
if NumberTheory:-QuadraticResidue(85,t) <> 1 then return NULL fi;
Cands:= map(s -> rhs(op(s)), [msolve(x^2 + 9*x - 1, t)]);
g:= proc(r) local v; v:= r^2 + 9*r - 1; v >= t*(t-11)/10 and v < t*(t-2) end proc;
op(sort(select(g, Cands)));
end proc:
map
map(F, [$1..82]); # Robert Israel, Aug 25 2023
-
from itertools import count, islice
from sympy import sqrt_mod_iter
def A116300_gen(): # generator of terms
for l in count(1):
m = 10**l+1
k, r, dlist = m*(m-11)/10, m*(m-2), []
for a in sqrt_mod_iter(85,m):
d = ((a if a&1 else a+m)>>1)-4
if kA116300_list = list(islice(A116300_gen(),14)) # Chai Wah Wu, May 07 2024
A116305
Numbers k such that k*(k+8) gives the concatenation of two numbers m and m+2.
Original entry on oeis.org
3377, 6616, 7027, 49941473, 50058520, 97000297, 785321170880, 799019801017, 989301366860, 997000002997, 4044997109223941, 4136014586911892, 4248535260473901, 4366325506691845, 4457342984379796, 5542657015620197
Offset: 1
97000297 * 97000305 = 94090583//94090585, where // denotes concatenation.
A116175
Numbers k such that k concatenated with k+2 gives the product of two numbers which differ by 9.
Original entry on oeis.org
1122070, 1538990, 3692998, 4422608, 1661985831086046580, 3508493742205242178, 5949206094811297768, 11324756194731856386028, 44020205992737767475178, 55147546334394567747178
Offset: 1
4422608//4422610 = 6650265 * 6650274, where // denotes
concatenation.
A116313
n times n+9 gives the concatenation of two numbers m and m+3.
Original entry on oeis.org
3, 38569, 61423, 88696, 3686022, 6313970, 7223061, 84319381, 98033553, 345087246071675410527410, 403910775483440116409763, 596089224516559883590229, 654912753928324589472582
Offset: 1
98033553 * 98033562 = 96105783//96105786, where // denotes
concatenation.
Showing 1-4 of 4 results.