A226742
Triangular numbers obtained as the concatenation of 2*k and k.
Original entry on oeis.org
21, 105, 2211, 9045, 222111, 306153, 742371, 890445, 1050525, 22221111, 88904445, 107905395, 173808690, 2222211111, 8889044445, 12141260706, 15754278771, 222222111111, 888890444445, 22222221111111, 36734701836735, 65306123265306
Offset: 1
If k=111, 2k=222, 2k//k = 222111 = 666*667/2, a triangular number.
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g:= proc(d) local a, b, n, Res, x, y;
Res:= NULL:
for a in numtheory:-divisors(2*(2*10^d+1)) do
b:= 2*(2*10^d+1)/a;
if igcd(a, b)>1 then next fi;
n:= chrem([0, -1], [a, b]);
x:= n*(n+1)/2;
y:= x/(2*10^d+1);
if y < 10^(d-1) or y >= 10^d then next fi;
Res:= Res, (2*10^d+1)*y
od;
op(sort([Res]))
end proc:
map(g, [$1..10]); # Robert Israel, Feb 06 2025
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TriangularQ[n_] := IntegerQ[Sqrt[1 + 8*n]]; t = {}; Do[s = FromDigits[Join[IntegerDigits[2*n], IntegerDigits[n]]]; If[TriangularQ[s], AppendTo[t, s]], {n, 100000}]; t (* T. D. Noe, Jun 18 2013 *)
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concatint(a,b)=eval(concat(Str(a),Str(b)))
istriang(x)=issquare(8*x+1)
{for(n=1,10^5,a=concatint(2*n,n);if(istriang(a),print(a)))}
A226772
Triangular numbers obtained as the concatenation of n and 2n.
Original entry on oeis.org
36, 1326, 2346, 3570, 125250, 223446, 12502500, 22234446, 1250025000, 2066441328, 2222344446, 2383847676, 3673573470, 125000250000, 222223444446, 5794481158896, 12500002500000, 12857132571426, 22222234444446, 49293309858660, 804878916097578, 933618918672378, 971908519438170
Offset: 1
If n=23, 2n=46, n//2n = 2346 = 68*69/2, a triangular number.
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F:= proc(d) local D,R,M,m,w,x,x1,x2;
R:= NULL;
M:= 10^d/2+1;
D:= numtheory:-divisors(M);
for m in D do if igcd(m,M/m)=1 then
for w in [chrem([-1,1],[8*m,M/m]), chrem([1,-1],[8*m,M/m])] do
x:= (w^2-1)/8;
x1:= x mod 10^d;
x2:= floor(x/10^d);
if x1 = 2*x2 and x1 >= 10^(d-1) then R:= R, x fi
od fi od;
op(sort([R]))
end proc:
36, seq(F(d),d=2..10); # Robert Israel, Nov 09 2020
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TriangularQ[n_] := IntegerQ[Sqrt[1 + 8*n]]; t = {}; Do[s = FromDigits[Join[IntegerDigits[n], IntegerDigits[2*n]]]; If[TriangularQ[s], AppendTo[t, s]], {n, 100000}]; t (* T. D. Noe, Jun 18 2013 *)
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concatint(a,b)=eval(concat(Str(a),Str(b)))
istriang(x)=issquare(8*x+1)
{for(n=1,10^5,a=concatint(n,2*n);if(istriang(a),print(a)))}
A226788
Triangular numbers obtained as the concatenation of n and n+1.
Original entry on oeis.org
45, 78, 4950, 5253, 295296, 369370, 415416, 499500, 502503, 594595, 652653, 760761, 22542255, 49995000, 50025003, 88278828, 1033010331, 1487714878, 4999950000, 5000250003, 490150490151, 499999500000, 500002500003, 509949509950, 33471093347110, 49999995000000, 50000025000003
Offset: 1
If n=295, n//n+1 = 295296 = 768*769/2, a triangular number.
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TriangularQ[n_] := IntegerQ[Sqrt[1 + 8*n]]; t = {}; Do[s = FromDigits[Join[IntegerDigits[n], IntegerDigits[n+1]]]; If[TriangularQ[s], AppendTo[t, s]], {n, 100000}]; t (* T. D. Noe, Jun 18 2013 *)
Select[FromDigits[Join[Flatten[IntegerDigits[#]]]]&/@Partition[ Range[ 5000010],2,1], OddQ[Sqrt[8#+1]]&] (* Harvey P. Dale, Jun 11 2015 *)
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concatint(a,b)=eval(concat(Str(a),Str(b)))
istriang(x)=issquare(8*x+1)
{for(n=1,10^7,a=concatint(n,n+1);if(istriang(a),print(a)))}
Showing 1-3 of 3 results.
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