A116154 -id:A116154 - 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

A116163 Numbers k such that k concatenated with k+1 gives the product of two numbers which differ by 1.

Original entry on oeis.org

1, 5, 61, 65479, 84289, 106609, 225649, 275599, 453589, 1869505, 2272555, 2738291, 3221951, 1667833021, 2475062749, 2525062249, 3500010739, 9032526511, 9225507211, 1753016898055, 1860598847399, 3233666953849, 3379207972471, 5632076031055, 5823639407489

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 3.

			1 is a member since 12 = 3*4; also 10 = 2*5.
5 is a member since 56 = 7*8; also 54 = 6*9.

Giovanni Resta, Table of n, a(n) for n = 1..2000

Cf. A116154, A115426, A116170, A116294.

Cf. A116143, A102567, A116112, A116130, A116280.

Union @@ ((y /. List@ ToRules@ Reduce[x (x+1) == 10^# y +y+1 && x>0 && 10^(#-1) <= y+1 < 10^#, {x,y}, Integers]) & /@ Range[13] /. y->{}) (* Giovanni Resta, Jul 08 2018 *)

Edited by N. J. A. Sloane, Apr 15 2007, Jun 27 2009

More terms from Giovanni Resta, Jul 08 2018

A116285 Numbers k such that k * (k+1) is the concatenation of a number m with itself.

Original entry on oeis.org

363, 637, 714, 923, 8905, 81818, 336633, 663367, 7272727, 76470589, 333666333, 405436668, 428571429, 447710185, 454545454, 473684211, 526315789, 545454546, 552289815, 571428571, 594563332, 666333667, 692307693, 711446449, 762237762, 834008097, 859982123, 879120879, 902255640, 974025975, 980861244

Offset: 1

Giovanni Resta, Feb 06 2006

From Robert Israel, Apr 08 2025: (Start)
Numbers k such that k * (k + 1) = (10^d + 1) * m for some d and m where m has d digits.
Includes (10^(3*d)-1)/3 + (10^d-1)*10^d/3 and 2*(10^(3*d)-1)/3 - (10^d-1)*10^d/3 + 1 for all d >= 1. (End)

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A116154, A116272, A116286, A116294, A161356.

q:= proc(d,m) local R,t,a,b,x,q;
   t:= 10^d+1;
   R:= NULL;
   for a in numtheory:-divisors(t) do
     b:= t/a;
     if igcd(a,b) > 1 then next fi;
     for x from chrem([0,-m],[a,b]) by t do
       q:= x*(x+m)/t;
       if q >= 10^d then break fi;
       if q >= 10^(d-1) then R:= R, x fi;
   od od;
   sort(convert({R},list));
end proc:
seq(op(q(d,1)),d=1..10); # Robert Israel, Apr 08 2025

A161356(n) = a(n)*(a(n)+1). - Michael S. Branicky, Jul 11 2025

More terms from Robert Israel, Apr 08 2025

A116141 Numbers k such that k concatenated with k-2 gives the product of two numbers which differ by 1.

Original entry on oeis.org

4, 5303944, 6677714, 2070936216988528558, 2969428172738875624, 6685545813563350444, 8013829604553736451395958429212, 9724110515510343256451213152382

Offset: 1

Giovanni Resta, Feb 06 2006

			6677714//6677712 = 8171728 * 8171729, where // denotes
concatenation.

Cf. A116134, A054214, A116154, A116272.

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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A116163 Numbers k such that k concatenated with k+1 gives the product of two numbers which differ by 1.

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A116285 Numbers k such that k * (k+1) is the concatenation of a number m with itself.

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A116141 Numbers k such that k concatenated with k-2 gives the product of two numbers which differ by 1.

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