A116098
Numbers k such that k concatenated with k-9 gives the product of two numbers which differ by 6.
Original entry on oeis.org
11, 101, 1001, 10001, 100001, 1000001, 10000001, 100000001, 1000000001, 10000000001, 13223140496, 20661157025, 29752066116, 40495867769, 52892561984, 66942148761, 82644628100, 100000000001, 1000000000001
Offset: 1
100000001//99999992 = 99999998 * 100000004, where // denotes
concatenation.
-
g:= proc(d) local r,c,a,b;
r:= mul(t[1],t=select(s -> s[2]::odd, ifactors(10^d+1)[2]))
c:= ceil((10^(d-1)+9)/r);
a:= isqrt(c);
if a^2 < c then a:= a+1 fi;
c:= floor((10^d+8)/r);
b:= isqrt(c);
if b^2 > c then b:= b-1 fi;
seq(r*y^2, y = a..b)
end proc:
seq(g(d),d=1..60); # Robert Israel, Aug 13 2018
A116129
Numbers k such that k concatenated with k-4 gives the product of two numbers which differ by 4.
Original entry on oeis.org
11, 101, 1001, 10001, 100001, 1000001, 10000001, 100000001, 1000000001, 10000000001, 13223140496, 20661157025, 29752066116, 40495867769, 52892561984, 66942148761, 82644628100, 100000000001, 1000000000001
Offset: 1
100000001//99999997 = 99999999 * 100000003, where // denotes concatenation.
-
g:= proc(d) local r,c,a,b;
r:= mul(t[1],t=select(s -> s[2]::odd, ifactors(10^d+1)[2]));
c:= ceil((10^(d-1)+4)/r);
a:= isqrt(c);
if a^2 < c then a:= a+1 fi;
c:= floor((10^d+3)/r);
b:= isqrt(c);
if b^2 > c then b:= b-1 fi;
seq(r*y^2, y = a..b)
end proc:
map(g, [$1..60]); # Robert Israel, Aug 13 2018
A116254
k times k+4 gives the concatenation of two numbers m and m-5.
Original entry on oeis.org
89, 9077, 9899, 733672, 998999, 88225293, 99989999, 8900869206, 9296908810, 9604060395, 9999899999, 326666333265, 673333666732, 700730927006, 972603739725, 999998999999, 34519562953735, 39737862788836, 49917309624954, 50082690375043, 60262137211161, 65480437046262
Offset: 1
-
from itertools import count, islice
from sympy import sqrt_mod
def A116254_gen(): # generator of terms
for j in count(0):
b = 10**j
a = b*10+1
for k in sorted(sqrt_mod(-1,a,all_roots=True)):
m = (k**2+1)//a
if a*(b+4) <= k**2+1 < a*(a+3):
yield k-2
A116254_list = list(islice(A116254_gen(),40)) # Chai Wah Wu, Feb 19 2024
A116122
Numbers k such that k concatenated with k-5 gives the product of two numbers which differ by 3.
Original entry on oeis.org
92185, 1453156572932210152879253333913, 3829098407015032018435618903285, 1017438814759112270449904796121753809
Offset: 1
92185//92180 = 96012 * 96015, where // denotes concatenation.
A116124
Numbers k such that k concatenated with k-5 gives the product of two numbers which differ by 5.
Original entry on oeis.org
61, 65479, 84289, 106609, 225649, 275599, 453589, 1869505, 2272555, 2738291, 3221951, 1667833021, 2475062749, 2525062249, 3500010739, 9032526511, 9225507211, 1753016898055, 1860598847399, 3233666953849, 3379207972471
Offset: 1
9225507211//9225507206 = 9604950394 * 9604950399, where // denotes concatenation.
A116143
Numbers k such that k concatenated with k-2 gives the product of two numbers which differ by 3.
Original entry on oeis.org
10, 100, 132, 406, 510, 852, 1000, 7930, 10000, 66942, 100000, 113322, 440056, 1000000, 5289256, 10000000, 58477510, 100000000, 111333222, 164378892, 183673470, 200444410, 206611570, 224376732, 277008310, 297520662
Offset: 1
A116273
Numbers k such that k*(k+2) gives the concatenation of two numbers m and m-2.
Original entry on oeis.org
90, 9078, 9900, 733673, 999000, 88225294, 99990000, 8900869207, 9296908811, 9604060396, 9999900000, 326666333266, 673333666733, 700730927007, 972603739726, 999999000000, 34519562953736, 39737862788837
Offset: 1
90 is a member since 90*92 = 8280 = 82 80.
9078 is a member since 9078*9080 = 82428240 = 8242 8240.
A116141
Numbers k such that k concatenated with k-2 gives the product of two numbers which differ by 1.
Original entry on oeis.org
4, 5303944, 6677714, 2070936216988528558, 2969428172738875624, 6685545813563350444, 8013829604553736451395958429212, 9724110515510343256451213152382
Offset: 1
6677714//6677712 = 8171728 * 8171729, where // denotes
concatenation.
Original entry on oeis.org
6, 5346, 8083, 10578, 45531, 58626, 2392902, 2609443, 7272838, 51248898, 98009803, 159728062051, 360408196038, 523637103531, 770378933826, 998000998003, 1214959556998, 1434212848998, 3860012299771, 4243705560771
Offset: 1
Original entry on oeis.org
6, 5346, 8083, 10578, 45531, 58626, 2392902, 2609443, 7272838, 51248898, 98009803, 159728062051, 360408196038, 523637103531, 770378933826, 998000998003, 1214959556998, 1434212848998, 3860012299771, 4243705560771
Offset: 1
Comments