A115429
Numbers k such that the concatenation of k with k+8 gives a square.
Original entry on oeis.org
6001, 6433, 11085116, 44496481, 96040393, 115916930617, 227007035017, 274101929528, 434985419768, 749978863753, 996004003993, 1365379857457948, 1410590590957816, 1762388551055953, 2307340946901148, 2700383162251217
Offset: 1
6001//6009 = 7747^2, where // denotes concatenation.
96040393//96040400 = 98000200 * 98000202.
96040393//96040397 = 98000199 * 98000203.
96040393//96040392 = 98000198 * 98000204.
Cf.
A030465,
A102567,
A115426,
A115437,
A115428,
A115430,
A115431,
A115432,
A115433,
A115434,
A115435,
A115436,
A115440.
A116244
Numbers k such that k * (k + 8) is the concatenation of two numbers m and m-7.
Original entry on oeis.org
94, 461, 532, 714, 818, 994, 3424, 6569, 9994, 90903, 99994, 980198, 999994, 3636357, 6363636, 9999994, 41176464, 58823529, 99999994, 413533834, 426573426, 428571422, 432620005, 567379988, 571428571, 573426567
Offset: 1
-
F:= proc(d) local R, t,alpha, beta, gamma, delta, B,C,n,m,i0,i,gamma0, delta0;
R:= NULL;
t:= 10^d+1;
for alpha in numtheory:-divisors(t) do
beta:= t/alpha;
if igcd(alpha,beta) > 1 then next fi;
delta0:= 6/beta mod alpha;
gamma0:= (beta*delta0-6)/alpha;
B:= 2*alpha*gamma0 + 6;
C:= gamma0*delta0 - 10^(d-1) - 7;
if C < 0 then i0:= 0 else i0:= ceil((-B + sqrt(B^2-4*t*C))/(2*t)) fi;
for i from i0 do
gamma:= gamma0 + i*beta;
delta:= delta0 + i*alpha;
m:= gamma*delta;
if m -7 >= 10^d then break fi;
if m - 7 >= 10^(d-1) then R:= R, alpha*gamma-1 fi;
od
od;
sort(convert({R},list))
end proc:
seq(op(F(d)),d=1..10); # Robert Israel, Aug 22 2023
-
a[n_] := Module[{solutions = {}, kvalues, e = 2}, While[Length[solutions] < n, sol = Solve[{a*b == 10^e + 1, 10^(e - 1) <= c*d < 10^e, a*c + 6 == b*d, a > 0, b > 0, c > 0, d > 0}, {a, b, c, d}, Integers]; kvalues = (a*c - 1) /. sol; solutions = Union[solutions, kvalues]; e++]; Take[solutions, n]]; a[26] (* Robert P. P. McKone, Aug 22 2023 *)
A116238
Numbers k such that k*(k+7) gives the concatenation of two numbers m and m-8.
Original entry on oeis.org
69, 79822842, 69852478553064869297984899963804, 77473062193002372448027740546436, 77747359197583788609974143907616, 84341826458653210947638195982112, 85367942837521291760016984490248
Offset: 1
79822842 * 79822849 = 63716866//63716858, where // denotes concatenation.
A116240
n times n+9 gives the concatenation of two numbers m and m-8.
Original entry on oeis.org
93, 454, 538, 993, 7664, 9993, 72720, 99993, 356428, 643564, 999993, 9090909, 9999993, 35294117, 64705875, 99999993, 335664328, 664335664, 684210519, 818181818, 838056673, 846153846, 866028701, 980125138, 999999993
Offset: 1
Showing 1-4 of 4 results.
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