A115426 Numbers k such that the concatenation of k with k+2 gives a square.

7874, 8119, 69476962, 98010199, 108746354942, 449212110367, 544978035127, 870501316279, 998001001999, 1428394731903223, 1499870932756487, 1806498025502498, 1830668275445687, 1911470478658759, 2255786189655202

Offset

1,1

Comments

Numbers k such that k concatenated with k+1 gives the product of two numbers which differ by 2.

Numbers k such that k concatenated with k-2 gives the product of two numbers which differ by 4.

Numbers k such that k concatenated with k-7 gives the product of two numbers which differ by 6.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Example

8119//8121 = 9011^2, where // denotes concatenation.
98010199//98010200 = 99000100 * 99000102.
98010199//98010197 = 99000099 * 99000103.

Prog

(Python)
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

Crossrefs

Cf. A030465, A102567, A115427, A115428, A115429, A115430, A115431, A115432, A115433, A115434, A115435, A115436, A115437.

Cf. A116106, A116110, A116112, A116125, A116242.

Cf. A116136, A116143, A116107, A116275, A116163, A116099, A116177, A116295.

Sequence in context: A286208 A268583 A045292 * A206069 A237969 A256163

Adjacent sequences: A115423 A115424 A115425 * A115427 A115428 A115429

Keyword

base,nonn

Author

Giovanni Resta, Jan 24 2006

Extensions

Edited by N. J. A. Sloane, Apr 13 2007

Status

approved