A116267 -id:A116267 - cp's OEIS frontend

A116136 Numbers k such that k concatenated with k-3 gives the product of two numbers which differ by 4.

9, 99, 183, 328, 528, 715, 999, 6099, 9999, 13224, 40495, 99999, 106755, 453288, 999999, 2066115, 2975208, 9999999, 22145328, 28027683, 99999999, 110213248, 110667555, 147928995, 178838403, 226123528, 275074575, 333052608, 378698224, 445332888, 446245635, 518348515

Offset: 1

Giovanni Resta, Feb 06 2006

Also numbers k such that k concatenated with itself gives the product of two numbers which differ by 2.

			8315420899//8315420896 = 9118892968 * 9118892972, where // denotes concatenation.

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

Cf. A116129, A115431, A116099, A115426, A116267.

Cf. also A102567, A116154, A116130, A116286.

from itertools import count, islice
from sympy import sqrt_mod
def A116136_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**2-1)//a
A116136_list = list(islice(A116136_gen(),40)) # Chai Wah Wu, Feb 19 2024

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

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

A116260 n times n+4 gives the concatenation of two numbers m and m-4.

Original entry on oeis.org

9, 99, 999, 9999, 99999, 999999, 9999999, 99999999, 999999999, 9999999999, 36363636362, 45454545453, 54545454544, 63636363635, 72727272726, 81818181817, 90909090908, 99999999999, 999999999999, 9999999999999

Offset: 1

Giovanni Resta, Feb 06 2006

			99999999 * 100000003 = 100000001//99999997, where // denotes concatenation.

Chai Wah Wu, Table of n, a(n) for n = 1..121

Cf. A116129, A116254, A116259, A116261, A116267.

from itertools import count, islice
from sympy import sqrt_mod
def A116260_gen(): # generator of terms
    for j in count(0):
        b = 10**j
        a = b*10+1
        for k in sorted(sqrt_mod(0,a,all_roots=True))+[a]:
            if a*(b+4) <= k**2 < a*(a+3):
                yield k-2
A116260_list = list(islice(A116260_gen(),20)) # Chai Wah Wu, Feb 19 2024

10^d-1 for d>0 are terms. - Chai Wah Wu, Feb 19 2024

A116268 Numbers k such that k*(k+5) gives the concatenation of two numbers m and m-3.

Original entry on oeis.org

81, 77394227, 89158933, 36623663376237623663376335, 37633762366336633762366235, 62366237633663366237633761, 63376336623762376336623661, 86194223018927804587702128, 88063202723646452838040443, 35574229497606875609044578088011, 35693849662968953146129859753682, 42317841210726174031503123524229

Offset: 1

Giovanni Resta, Feb 06 2006

			89158933 * 89158938 = 79493157//79493154, where // denotes concatenation.

Robert Israel, Table of n, a(n) for n = 1..220

Cf. A028557, A116099, A116261, A116267, A116269, A116276.

f:= proc(d) local k, K;
      K:= map(t -> rhs(op(t)), [msolve(k^2+5*k+3=0,10^d+1)]);
      op(sort(select(k -> k^2 + 5*k + 3 >= (10^d+1)*10^(d-1), K)));
end proc:
map(f, [$1..62]); # Robert Israel, Jul 10 2025

More terms from Robert Israel, Jul 10 2025

A116275 Numbers k such that k*(k+4) gives the concatenation of two numbers m and m-2.

Original entry on oeis.org

8872, 9009, 83352840, 99000099, 329767122286, 670232877711, 738226276371, 933006600339, 999000000999, 3779410975143113, 3872816717528065, 4250291784692548, 4278630943941865, 4372036686326817, 4749511753491300, 5250488246508697, 5627963313673180, 5721369056058132

Offset: 1

Giovanni Resta, Feb 06 2006

			99000099 * 99000103 = 98010199//98010197, where // denotes concatenation.

Chai Wah Wu, Table of n, a(n) for n = 1..1187

Cf. A115426, A116267, A116274, A116276, A116288.

from itertools import count, islice
from sympy import sqrt_mod
def A116275_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+3) <= k**2-2 < a*(a+2):
                yield k-2
A116275_list = list(islice(A116275_gen(),30)) # Chai Wah Wu, Feb 19 2024

a(16)-a(18) from Chai Wah Wu, Feb 19 2024

A116266 n times n+2 gives the concatenation of two numbers m and m-3.

Original entry on oeis.org

7, 7311, 8990, 32523, 67476, 76567, 4891729, 5108270, 8528093, 71588335, 98999900, 399659933006, 600340066993, 723627738226, 877712329767, 998999999000, 3485626998113, 3787100274613, 6212899725386, 6514373001886

Offset: 1

Giovanni Resta, Feb 06 2006

			98999900 * 98999902 = 98009803//98009800, where // denotes
concatenation.

Cf. A115431, A116252, A116265, A116267, A116273.

cp's OEIS Frontend

A116136 Numbers k such that k concatenated with k-3 gives the product of two numbers which differ by 4.

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A116260 n times n+4 gives the concatenation of two numbers m and m-4.

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A116268 Numbers k such that k*(k+5) gives the concatenation of two numbers m and m-3.

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A116275 Numbers k such that k*(k+4) gives the concatenation of two numbers m and m-2.

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A116266 n times n+2 gives the concatenation of two numbers m and m-3.

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