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

Author

Giovanni Resta, Jan 24 2006

Keywords

• base
• nonn

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.

Examples

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

Links

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

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.

Programs

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

Extensions

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