A116180 -id:A116180 - cp's OEIS frontend

A116133 Numbers k such that k concatenated with k-4 gives the product of two numbers which differ by 9.

64, 1876, 3214, 8776, 17227120144584448354, 19391087990479091186, 21557280540171744040, 23970390649400564404, 26051259424903756364, 28697643366269876506, 31320461807899711990, 34215988012600009664

Offset: 1

Giovanni Resta, Feb 06 2006

Also numbers k such that k concatenated with k+4 gives the product of two numbers which differ by 7.

Since k*(10^j+1) + 4 = m*(m+7) iff k*(10^j+1) - 4 = m*(m+7) - 8 = (m-1)*(m+8) = M*(M+9) this is trivially the same sequence. - N. J. A. Sloane, Apr 11 2007

			8776//8772 = 9364 * 9373, where // denotes concatenation. Also 8776//8780 = 9365 * 9372.

Cf. A116132, A116140, A116264.

Cf. A116180, A116187, A116189, A116195, A116319.

Edited by N. J. A. Sloane, Apr 11 2007

A116174 Numbers k such that k concatenated with k+2 gives the product of two numbers which differ by 8.

Original entry on oeis.org

1143, 4382, 4943, 24941511, 25058558, 94090583, 616729341438, 638432642423, 978717194478, 994009005983, 1636200161363007, 1710661666314798, 1805005185949007, 1906479843038783, 1986790648039982, 3072104679280383

Offset: 1

Giovanni Resta, Feb 06 2006

Also numbers k such that k concatenated with k+9 gives the product of two numbers which differ by 6.

			94090583//94090585 = 97000297 * 97000305, where // denotes concatenation.
94090583//94090592 = 97000298 * 97000304.

Cf. A116168, A116173, A116175, A116181, A116305.

Cf. A116180, A116169, A116352.

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

A116187 Numbers k such that k concatenated with k+4 gives the product of two numbers which differ by 6.

Original entry on oeis.org

12, 43, 20440836, 30017751, 61336887, 52400871197755334426147587, 53651708763838760619655612, 56652002793835820319625612, 57952296063256269823192087, 17684775866714240650923831481623

Offset: 1

Giovanni Resta, Feb 06 2006

Also numbers k such that k concatenated with k+9 gives the product of two numbers which differ by 4. For proof that this is the same sequence compare A116133.

			61336887//61336891 = 78317867 * 78317873, where // denotes concatenation. 61336887//61336896 = 78317868 * 78317872.

Cf. A116173, A115429, A116133, A116194, A116318.

Cf. A116193, A116180, A116350.

Edited by N. J. A. Sloane, Apr 11 2007

A116126 Numbers k such that k concatenated with k-5 gives the product of two numbers which differ by 8.

Original entry on oeis.org

10, 6752089, 6448802889351008245, 18894512461523256139943105859903480218905, 31958875438439894736354375209245786214798

Offset: 1

Giovanni Resta, Feb 06 2006

Cf. A116113, A116125, A116180, A116139, A116257.

A116167 Numbers k such that k concatenated with k+1 gives the product of two numbers which differ by 7.

Original entry on oeis.org

7, 493, 607, 629, 757, 17927, 33247, 93869, 19467217, 31223879, 72757727, 13454739732766891651472740499, 40093333713615672956030023507, 48089152118689474641229584727, 66424317743191484432891678269

Offset: 1

Giovanni Resta, Feb 06 2006

			72757727//72757728 = 85298137 * 85298144, where // denotes concatenation.

Cf. A116160, A115428, A116168, A116180, A116298.

A116179 Numbers k such that k concatenated with k+3 gives the product of two numbers which differ by 5.

Original entry on oeis.org

1, 3, 81, 1353, 3997, 7723, 23761, 26271, 76771, 1415683, 3890571, 8495497, 1066870443, 1239366513, 4198438981, 4534273891, 6502317141, 6918679731, 2199164200036329043, 2820114781174460091, 5500888421709400741

Offset: 1

Giovanni Resta, Feb 06 2006

Cf. A116172, A116112, A116180, A116193, A116310.

A116207 Numbers k such that k concatenated with k+7 gives the product of two numbers which differ by 5.

Original entry on oeis.org

493, 607, 629, 757, 17927, 33247, 93869, 19467217, 31223879, 72757727, 13454739732766891651472740499, 40093333713615672956030023507, 48089152118689474641229584727, 66424317743191484432891678269

Offset: 1

Giovanni Resta, Feb 06 2006

From Robert Israel, Nov 27 2024: (Start)
If 10^d + 1 has a prime factor p such that 53 is not a square mod p, then there are no terms k where k + 7 has d digits.
For example, there are no terms where d == 2 (mod 4), since in that case 10^d + 1 is divisible by 101, and 53 is not a square mod 101. (End)

			72757727//72757734 = 85298138 * 85298143, where // denotes concatenation.

Robert Israel, Table of n, a(n) for n = 1..85 (all terms with up to 113 digits).

Cf. A116167, A116114, A116200, A116173, A116208, A116180, A116338.

f:= proc(d) # terms where k+7 has d digits
    local S,x,R,k;
    S:= map(t -> rhs(op(t)), [msolve(x*(x+5) = 7, 10^d+1)]);
    R:= NULL:
    for x in S do
      k := (x*(x+5)-7)/(10^d+1);
      if ilog10(k+7) = d - 1 then R:= R,k fi
    od:
    op(sort([R]))
end proc:
map(f, [$1..31]); # Robert Israel, Nov 27 2024

A116114 Numbers k such that k concatenated with k-7 gives the product of two numbers which differ by 9.

Original entry on oeis.org

13, 493, 607, 629, 757, 17927, 33247, 93869, 19467217, 31223879, 72757727, 13454739732766891651472740499, 40093333713615672956030023507, 48089152118689474641229584727, 66424317743191484432891678269

Offset: 1

Giovanni Resta, Feb 06 2006

Cf. A116109, A116113, A116180, A116245.

A116181 Numbers k such that k concatenated with k+3 gives the product of two numbers which differ by 8.

Original entry on oeis.org

57, 1062, 4545, 9517, 65040816823705919148274557212824575357, 73854453524198329292125744115286819630, 78444109191135472482770868099397646205

Offset: 1

Giovanni Resta, Feb 06 2006

			9517//9520 = 9752 * 9760, where // denotes concatenation.

Cf. A116174, A116180, A116182, A116189, A116312.

A116311 Numbers k such that k*(k+7) gives the concatenation of two numbers m and m+3.

Original entry on oeis.org

8445, 8810, 69125298546226023971, 69855225553525294044, 74604750601020544519, 75334677608319814592, 92496418993920707746, 93226346001219977819, 97975871048715228294, 98705798056014498367

Offset: 1

Giovanni Resta, Feb 06 2006

			8810 * 8817 = 7767//7770, where // denotes concatenation.

Cf. A116180, A116298, A116310, A116312, A116319.

cp's OEIS Frontend

A116133 Numbers k such that k concatenated with k-4 gives the product of two numbers which differ by 9.

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A116174 Numbers k such that k concatenated with k+2 gives the product of two numbers which differ by 8.

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A116187 Numbers k such that k concatenated with k+4 gives the product of two numbers which differ by 6.

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A116126 Numbers k such that k concatenated with k-5 gives the product of two numbers which differ by 8.

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A116167 Numbers k such that k concatenated with k+1 gives the product of two numbers which differ by 7.

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A116179 Numbers k such that k concatenated with k+3 gives the product of two numbers which differ by 5.

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A116207 Numbers k such that k concatenated with k+7 gives the product of two numbers which differ by 5.

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A116114 Numbers k such that k concatenated with k-7 gives the product of two numbers which differ by 9.

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A116181 Numbers k such that k concatenated with k+3 gives the product of two numbers which differ by 8.

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A116311 Numbers k such that k*(k+7) gives the concatenation of two numbers m and m+3.

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