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
A116105
Numbers k such that k concatenated with k-8 gives the product of two numbers which differ by 5.
Original entry on oeis.org
5303944, 6677714, 2070936216988528558, 2969428172738875624, 6685545813563350444, 8013829604553736451395958429212, 9724110515510343256451213152382
Offset: 1
A116235
n times n+4 gives the concatenation of two numbers m and m-8.
Original entry on oeis.org
79, 8155, 9799, 467345, 532652, 997999, 76450587, 99979999, 7801738413, 8593817621, 9208120791, 9999799999, 346667333465, 401461854013, 598538145984, 653332666532, 945207479451, 999997999999, 48349470735058
Offset: 1
A116096
Numbers k such that k concatenated with k-9 gives the product of two numbers which differ by 4.
Original entry on oeis.org
21, 30, 902406, 959721, 6040059046, 6242406405, 9842410005, 9900249006, 15033519988494, 17250863148969, 22499666270469, 27632040031654, 34182546327286, 37487353123861, 52213551379230, 74230108225630
Offset: 1
9900249006//9900248997 = 9949999499 * 9949999503, where // denotes concatenation.
A116103
Numbers k such that k concatenated with k-8 gives the product of two numbers which differ by 3.
Original entry on oeis.org
58, 1552909498221642960203840517299022287877411846, 1723404686015903064905313664707396376746205762, 2153694542826970975147295558223788704098018912
Offset: 1
58//50 = 75 * 78, 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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