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
A115428
Numbers k such that the concatenation of k with k+5 gives a square.
Original entry on oeis.org
1, 4, 20, 31, 14564, 38239, 69919, 120395, 426436, 902596, 7478020, 9090220, 6671332084, 8114264059, 8482227259, 9900250996, 2244338786836, 2490577152964, 2509440638591, 2769448208395, 7012067592220
Offset: 1
Cf.
A030465,
A102567,
A115426,
A115437,
A115429,
A115430,
A115431,
A115432,
A115433,
A115434,
A115435,
A115436,
A115439.
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.
A116099
Numbers k such that k concatenated with k-9 gives the product of two numbers which differ by 7.
Original entry on oeis.org
69, 59898667, 79493157, 13412927190959690154913903, 14163000698458955079906403, 38895475965785687555173929, 40165600438484442828161229, 74294440818366638194239027
Offset: 1
79493157//79493154 = 89158933 * 89158938, where // denotes concatenation.
79493157//79493158 = 89158934 * 89158937.
79493157//79493160 = 89158935 * 89158936.
79493157//79493148 = 89158932 * 89158939.
A116308
Numbers k such that k*(k+2) is the concatenation of two numbers m and m+3.
Original entry on oeis.org
452, 547, 690, 855, 4381, 5618, 72729, 346532, 653467, 9090907, 94117645, 334665332, 336032386, 378253327, 390977441, 439928490, 483516485, 516483514, 560071509, 609022558, 621746672, 663967613, 665334667
Offset: 1
Showing 1-5 of 5 results.
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