A116107 -id:A116107 - 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

A115429 Numbers k such that the concatenation of k with k+8 gives a square.

Original entry on oeis.org

6001, 6433, 11085116, 44496481, 96040393, 115916930617, 227007035017, 274101929528, 434985419768, 749978863753, 996004003993, 1365379857457948, 1410590590957816, 1762388551055953, 2307340946901148, 2700383162251217

Offset: 1

Giovanni Resta, Jan 24 2006

Also numbers k such that k concatenated with k+7 gives the product of two numbers which differ by 2.
Also numbers k such that k concatenated with k+4 gives the product of two numbers which differ by 4.
Also numbers k such that k concatenated with k-1 gives the product of two numbers which differ by 6.
Also numbers k such that k concatenated with k-8 gives the product of two numbers which differ by 8.

			6001//6009 = 7747^2, where // denotes concatenation.
96040393//96040400 = 98000200 * 98000202.
96040393//96040397 = 98000199 * 98000203.
96040393//96040392 = 98000198 * 98000204.

Cf. A030465, A102567, A115426, A115437, A115428, A115430, A115431, A115432, A115433, A115434, A115435, A115436, A115440.
Cf. A116107, A116109, A116113, A116239.
Cf. A116205, A115430, A116335, A116185, A116187, A116317.
Cf. A116131, A116112, A116152, A116159, A116282.

Edited by N. J. A. Sloane, Apr 15 2007

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

Giovanni Resta, Feb 06 2006

Also numbers k such that k concatenated with k-1 gives the product of two numbers which differ by 5.
Also numbers k such that k concatenated with k+3 gives the product of two numbers which differ by 3.
Also numbers k such that k concatenated with k+5 gives the product of two numbers which differ by 1.

			62918301//62918300 = 79321055 * 79321060, where // denotes concatenation.
62918301//62918304 = 79321056 * 79321059.
62918301//62918306 = 79321057 * 79321058.

Cf. A116107, A115426, A116113, A116119, A116243.
Cf. A116163, A115429, A116158, A116281.
Cf. A116171, A116177, A116179, A116185, A116309, A115430, A116322.

Edited by N. J. A. Sloane, Apr 15 2007

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

Giovanni Resta, Feb 06 2006

Also numbers k such that k concatenated with k-3 gives the product of two numbers which differ by 5.
Also numbers k such that k concatenated with k+1 gives the product of two numbers which differ by 3.
Also numbers k such that k concatenated with k+3 gives the product of two numbers which differ by 1.

			79493157//79493154 = 89158933 * 89158938, where // denotes concatenation.
79493157//79493158 = 89158934 * 89158937.
79493157//79493160 = 89158935 * 89158936.
79493157//79493148 = 89158932 * 89158939.

Cf. A116098, A116193, A116107, A116230, A116130, A116136, A116205, A116107, A116268.

Cf. A115426, A115428, A116171, A116296, A116170, A116177, A116307.

Edited by N. J. A. Sloane, Apr 12 2007

A116106 Numbers k such that k concatenated with k-8 gives the product of two numbers which differ by 6.

Original entry on oeis.org

9, 13, 99, 103, 183, 328, 528, 715, 999, 1003, 6099, 9999, 10003, 13224, 40495, 99999, 100003, 106755, 453288, 999999, 1000003, 2066115, 2975208, 9999999, 10000003, 22145328, 28027683, 99999999, 100000003, 110213248, 110667555

Offset: 1

Giovanni Resta, Feb 06 2006

			999999999//999999991 = 999999997 * 1000000003, where // denotes concatenation.

Cf. A116098, A116105, A116107, A115426, A116237.

A116146 Numbers k such that k concatenated with k-2 gives the product of two numbers which differ by 8.

Original entry on oeis.org

47, 550, 802, 570035, 623387, 1327222, 4041011, 8252210, 164398539591831062, 173868055738777586, 339918283297349107, 353476744122425611, 846974882088186070, 868300386379144450, 1875603455546506870

Offset: 1

Giovanni Resta, Feb 06 2006

			8252210//8252208 = 9084164 * 9084172, where // denotes concatenation.

Cf. A116139, A116107, A116147, A116161, A116277.

A116238 Numbers k such that k*(k+7) gives the concatenation of two numbers m and m-8.

Original entry on oeis.org

69, 79822842, 69852478553064869297984899963804, 77473062193002372448027740546436, 77747359197583788609974143907616, 84341826458653210947638195982112, 85367942837521291760016984490248

Offset: 1

Giovanni Resta, Feb 06 2006

			79822842 * 79822849 = 63716866//63716858, where // denotes concatenation.

Cf. A116107, A116230, A116237, A116239, A116243.

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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A115429 Numbers k such that the concatenation of k with k+8 gives a square.

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A116112 Numbers k such that k concatenated with k-7 gives the product of two numbers which differ by 7.

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A116099 Numbers k such that k concatenated with k-9 gives the product of two numbers which differ by 7.

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A116106 Numbers k such that k concatenated with k-8 gives the product of two numbers which differ by 6.

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A116146 Numbers k such that k concatenated with k-2 gives the product of two numbers which differ by 8.

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A116238 Numbers k such that k*(k+7) gives the concatenation of two numbers m and m-8.

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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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