A115432
Numbers k such that the concatenation of k with k-4 gives a square.
Original entry on oeis.org
65, 6653, 9605, 218413, 283720, 996005, 58446925, 99960005, 6086712229, 7385370133, 8478948853, 9999600005, 120178240093, 161171620229, 358247912200, 426843573160, 893417179213, 999996000005, 23376713203604
Offset: 1
Cf.
A030465,
A102567,
A115426,
A115437,
A115428,
A115429,
A115430,
A115431,
A115433,
A115434,
A115435,
A115436,
A115443.
-
f:= proc(d) uses NumberTheory; local m,r;
m:= 10^d + 1;
if QuadraticResidue(-4,m) = -1 then return NULL fi;
r:= ModularSquareRoot(-4, m);
op(sort(select(t -> t >= 10^(d-1)+4 and t < 10^d+4, map(t -> ((r*t mod m)^2+4)/m, convert(RootsOfUnity(2,m),list)))))
end proc:
map(f, [$1..20]); # Robert Israel, Sep 12 2023
A116116
Numbers k such that k concatenated with k-6 gives the product of two numbers which differ by 2.
Original entry on oeis.org
21, 30, 902406, 959721, 6040059046, 6242406405, 9842410005, 9900249006, 15033519988494, 17250863148969, 22499666270469, 27632040031654, 34182546327286, 37487353123861, 52213551379230, 74230108225630
Offset: 1
9900249006//9900249000 = 9949999500 * 9949999502, where // denotes concatenation.
A116252
n times n+2 gives the concatenation of two numbers m and m-5.
Original entry on oeis.org
80, 8156, 9800, 467346, 532653, 998000, 76450588, 99980000, 7801738414, 8593817622, 9208120792, 9999800000, 346667333466, 401461854014, 598538145985, 653332666533, 945207479452, 999998000000, 48349470735059
Offset: 1
A116120
Numbers k such that k concatenated with k-5 gives the product of two numbers which differ by 1.
Original entry on oeis.org
7, 55, 5255, 6295, 75511, 92137, 5551405, 7943911, 1257669432489239675, 2148743977942130765, 2877834685014859837, 4164942361780146967, 10235898967817894552965, 15389481849681116007757
Offset: 1
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.
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
Showing 1-6 of 6 results.
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