A116112
Numbers k such that k concatenated with k-7 gives the product of two numbers which differ by 7.
Original entry on oeis.org
17, 35, 10408517, 45884051, 62918301, 1116290522645838319925, 1491109615209578451401, 2254276950187476704727, 2758431647767103545151, 3768131911733856383477, 4434103687048263321737, 5230580700713956424051
Offset: 1
62918301//62918300 = 79321055 * 79321060, where // denotes concatenation.
62918301//62918304 = 79321056 * 79321059.
62918301//62918306 = 79321057 * 79321058.
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
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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.
A116242
Numbers k such that k*(k+6) gives the concatenation of two numbers m and m-7.
Original entry on oeis.org
8871, 9008, 83352839, 99000098, 329767122285, 670232877710, 738226276370, 933006600338, 999000000998, 3779410975143112, 3872816717528064, 4250291784692547, 4278630943941864, 4372036686326816, 4749511753491299
Offset: 1
99000098 * 99000104 = 98010199//98010192, where // denotes concatenation.
A116250
n times n+7 gives the concatenation of two numbers m and m-6.
Original entry on oeis.org
95, 384, 428, 566, 610, 813, 995, 4520, 5474, 9995, 90909, 99995, 316831, 683163, 999995, 3636363, 6363631, 9999995, 82352941, 99999995, 331668331, 368421052, 395604390, 442767753, 461538461, 488721799, 511278195, 538461533
Offset: 1
Showing 1-5 of 5 results.
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