A116275 -id:A116275 - cp's OEIS frontend

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

Giovanni Resta, Jan 24 2006

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.

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

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

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.

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

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

A116267 Numbers k such that k*(k+4) gives the concatenation of two numbers m and m-3.

Original entry on oeis.org

8, 98, 426, 571, 725, 844, 998, 7808, 9998, 36363, 63634, 99998, 326732, 673265, 999998, 4545452, 5454545, 9999998, 47058821, 52941176, 99999998, 331983805, 332667332, 384615384, 422892896, 475524475, 524475522, 577107101, 615384613, 667332665, 668016192, 719964244

Offset: 1

Giovanni Resta, Feb 06 2006

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A116136, A116260, A116266, A116268, A116275.

from itertools import count, islice
from sympy import sqrt_mod
def A116267_gen(): # generator of terms
    for j in count(0):
        b = 10**j
        a = b*10+1
        for k in sorted(sqrt_mod(1,a,all_roots=True)):
            if a*(b+3) <= k**2-1 < a*(a+2):
                yield k-2
A116267_list = list(islice(A116267_gen(),20)) # Chai Wah Wu, Feb 19 2024

a(29)-a(32) from Chai Wah Wu, Feb 19 2024

A116288 Numbers k such that k*(k+4) gives the concatenation of a number m with itself.

Original entry on oeis.org

7, 97, 451, 546, 689, 854, 997, 4380, 5617, 9997, 72728, 99997, 346531, 653466, 999997, 9090906, 9999997, 94117644, 99999997, 334665331, 336032385, 378253326, 390977440, 439928489, 483516484, 516483513, 560071508

Offset: 1

Giovanni Resta, Feb 06 2006

Contains 10^m - 3 hence the sequence is infinite. - David A. Corneth, Feb 20 2024

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A116125, A116275, A116287, A116289, A116297.

from itertools import count, islice
from sympy import sqrt_mod
def A116288_gen(): # generator of terms
    for j in count(0):
        b = 10**j
        a = b*10+1
        for k in sorted(sqrt_mod(4,a,all_roots=True)):
            if a*b <= k**2-4 < a*(a-1):
                yield k-2
A116288_list = list(islice(A116288_gen(),40)) # Chai Wah Wu, Feb 20 2024

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

Giovanni Resta, Feb 06 2006

Cf. A116143, A116259, A116273, A116275, A116280.

A116276 Numbers k such that k*(k+5) gives the concatenation of two numbers m and m-2.

Original entry on oeis.org

70, 79822843, 69852478553064869297984899963805, 77473062193002372448027740546437, 77747359197583788609974143907617, 84341826458653210947638195982113, 85367942837521291760016984490249

Offset: 1

Giovanni Resta, Feb 06 2006

			79822843 * 79822848 = 63716866//63716864, where // denotes concatenation.

Cf. A028557, A116107, A116268, A116275, A116277, A116281.

cp's OEIS Frontend

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

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A116267 Numbers k such that k*(k+4) gives the concatenation of two numbers m and m-3.

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A116288 Numbers k such that k*(k+4) gives the concatenation of a number m with itself.

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A116274 Numbers k such that k*(k+3) gives the concatenation of two numbers m and m-2.

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A116276 Numbers k such that k*(k+5) gives the concatenation of two numbers m and m-2.

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