A115426
Numbers k such that the concatenation of k with k+2 gives a square.
Original entry on oeis.org
7874, 8119, 69476962, 98010199, 108746354942, 449212110367, 544978035127, 870501316279, 998001001999, 1428394731903223, 1499870932756487, 1806498025502498, 1830668275445687, 1911470478658759, 2255786189655202
Offset: 1
8119//8121 = 9011^2, where // denotes concatenation.
98010199//98010200 = 99000100 * 99000102.
98010199//98010197 = 99000099 * 99000103.
Cf.
A030465,
A102567,
A115427,
A115428,
A115429,
A115430,
A115431,
A115432,
A115433,
A115434,
A115435,
A115436,
A115437.
-
from itertools import count, islice
from sympy import sqrt_mod
def A115426_gen(): # generator of terms
for j in count(0):
b = 10**j
a = b*10+1
for k in sorted(sqrt_mod(2,a,all_roots=True)):
if a*(b-2) <= k**2-2 < a*(a-3):
yield (k**2-2)//a
A115426_list = list(islice(A115426_gen(),40)) # Chai Wah Wu, Feb 20 2024
A116163
Numbers k such that k concatenated with k+1 gives the product of two numbers which differ by 1.
Original entry on oeis.org
1, 5, 61, 65479, 84289, 106609, 225649, 275599, 453589, 1869505, 2272555, 2738291, 3221951, 1667833021, 2475062749, 2525062249, 3500010739, 9032526511, 9225507211, 1753016898055, 1860598847399, 3233666953849, 3379207972471, 5632076031055, 5823639407489
Offset: 1
1 is a member since 12 = 3*4; also 10 = 2*5.
5 is a member since 56 = 7*8; also 54 = 6*9.
-
Union @@ ((y /. List@ ToRules@ Reduce[x (x+1) == 10^# y +y+1 && x>0 && 10^(#-1) <= y+1 < 10^#, {x,y}, Integers]) & /@ Range[13] /. y->{}) (* Giovanni Resta, Jul 08 2018 *)
A116128
Numbers k such that k concatenated with k-4 gives the product of two numbers which differ by 3.
Original entry on oeis.org
5303944, 6677714, 2070936216988528558, 2969428172738875624, 6685545813563350444, 8013829604553736451395958429212, 9724110515510343256451213152382
Offset: 1
A116274
Numbers k such that k*(k+3) gives the concatenation of two numbers m and m-2.
Original entry on oeis.org
9, 99, 362, 636, 713, 922, 999, 8904, 9999, 81817, 99999, 336632, 663366, 999999, 7272726, 9999999, 76470588, 99999999, 333666332, 405436667, 428571428, 447710184, 454545453, 473684210, 526315788, 545454545, 552289814
Offset: 1
Showing 1-4 of 4 results.
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