A075114 -id:A075114 - cp's OEIS frontend

A084703 Squares k such that 2*k+1 is also a square.

0, 4, 144, 4900, 166464, 5654884, 192099600, 6525731524, 221682772224, 7530688524100, 255821727047184, 8690408031080164, 295218051329678400, 10028723337177985444, 340681375412721826704, 11573138040695364122500, 393146012008229658338304, 13355391270239113019379844

Offset: 0

Amarnath Murthy, Jun 08 2003

With the exception of 0, a subsequence of A075114. - R. J. Mathar, Dec 15 2008
Consequently, A014105(k) is a square if and only if k = a(n). - Bruno Berselli, Oct 14 2011
From M. F. Hasler, Jan 17 2012: (Start)
Bisection of A079291. The squares 2*k+1 are given in A055792.
A204576 is this sequence written in binary. (End)
a(n+1), n >= 0, is the perimeter squared (x(n) + y(n) + z(n))^2 of the ordered primitive Pythagorean triple (x(n), y(n) = x(n) + 1, z(n)). The first two terms are (x(0)=0, y(0)=1, z(0)=1), a(1) = 2^2, and (x(1)=3, y(1)=4, z(1)=5), a(2) = 12^2. - George F. Johnson, Nov 02 2012

D. W. Wilson, Table of n, a(n-1) for n = 1..100 (offset=1)
Emrah Kılıç, Yücel Türker Ulutaş, and Neşe Ömür, A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters, J. Int. Seq. 14 (2011), Article 11.5.6, table 3, k=2.
Thomas Koshy, Products Involving Reciprocals of Gibonacci Polynomials, The Fibonacci Quarterly, Vol. 60, No. 1 (2022), pp. 15-24.
Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Cf. A000129, A001109, A001110, A001333, A001541, A001542, A001652, A001653.
Cf. A014105, A029549, A046090, A046176, A055792, A075114, A079291, A084702.
Cf. A089928, A204576.
Cf. similar sequences with closed form ((1 + sqrt(2))^(4*r) + (1 - sqrt(2))^(4*r))/8 + k/4: this sequence (k=-1), A076218 (k=3), A278310 (k=-5).

[4*Evaluate(ChebyshevU(n), 3)^2: n in [0..30]]; // G. C. Greubel, Aug 18 2022

b[n_]:= b[n]= If[n<2, n, 34*b[n-1] -b[n-2] +2]; (* b=A001110 *)
a[n_]:= 4*b[n]; Table[a[n], {n, 0, 30}]
4*ChebyshevU[Range[-1,30], 3]^2 (* G. C. Greubel, Aug 18 2022 *)

[4*chebyshev_U(n-1, 3)^2 for n in (0..30)] # G. C. Greubel, Aug 18 2022

a(n) = 4*A001110(n) = A001542(n)^2.
a(n+1) = A001652(n)*A001652(n+1) + A046090(n)*A046090(n+1) = A001542(n+1)^2. - Charlie Marion, Jul 01 2003
a(n) = A001653(k+n)*A001653(k-n) - A001653(k)^2, for k >= n >= 0; e.g. 144 = 5741*5 - 169^2. - Charlie Marion, Jul 16 2003
G.f.: 4*x*(1+x)/((1-x)*(1-34*x+x^2)). - R. J. Mathar, Dec 15 2008
a(n) = A079291(2n). - M. F. Hasler, Jan 16 2012
From George F. Johnson, Nov 02 2012: (Start)
a(n) = ((17+12*sqrt(2))^n + (17-12*sqrt(2))^n - 2)/8.
a(n+1) = 17*a(n) + 4 + 12*sqrt(a(n)*(2*a(n) + 1)).
a(n-1) = 17*a(n) + 4 - 12*sqrt(a(n)*(2*a(n) + 1)).
a(n-1)*a(n+1) = (a(n) - 4)^2.
2*a(n) + 1 = (A001541(n))^2.
a(n+1) = 34*a(n) - a(n-1) + 8 for n>1, a(0)=0, a(1)=4.
a(n+1) = 35*a(n) - 35*a(n-1) + a(n-2) for n>0, a(0)=0, a(1)=4, a(2)=144.
a(n)*a(n+1) = (4*A029549(n))^2.
a(n+1) - a(n) = 4*A046176(n).
a(n) + a(n+1) = 4*(6*A029549(n) + 1).
a(n) = (2*A001333(n)*A000129(n))^2.
Limit_{n -> infinity} a(n)/a(n-r) = (17+12*sqrt(2))^r. (End)
Empirical: a(n) = A089928(4*n-2), for n > 0. - Alex Ratushnyak, Apr 12 2013
a(n) = 4*A001109(n)^2. - G. C. Greubel, Aug 18 2022
Product_{n>=2} (1 - 4/a(n)) = sqrt(2)/3 + 1/2 (Koshy, 2022, section 3, p. 19). - Amiram Eldar, Jan 23 2025

Edited and extended by Robert G. Wilson v, Jun 15 2003

A075014 Smallest k such that the concatenation k, k-1 is divisible by n; or 0 if no such number exists.

Original entry on oeis.org

1, 1, 2, 3, 1, 5, 2, 3, 5, 1, 17, 17, 6, 9, 11, 3, 16, 5, 7, 21, 2, 17, 18, 29, 26, 17, 5, 33, 8, 11, 35, 3, 17, 33, 26, 41, 11, 7, 17, 21, 13, 47, 4, 17, 41, 41, 27, 29, 9, 51, 50, 17, 21, 5, 61, 61, 35, 27, 52, 41, 29, 35, 68, 45, 6

Offset: 1

Amarnath Murthy, Sep 01 2002

			a(14) = 9 as 14 divides 98.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A075114, A075115, A075116, A075117.

skc[n_]:=Module[{k=1},While[Mod[FromDigits[Flatten[IntegerDigits/@{k,k-1}]],n] != 0,k++];k]; Array[skc,70] (* Harvey P. Dale, Mar 04 2023 *)

Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 19 2003

A075015 Smallest k such that the concatenation k, k+1, k+2 is divisible by n; or 0 if no such number exists.

Original entry on oeis.org

1, 2, 1, 4, 3, 2, 5, 4, 2, 8, 8, 4, 2, 104, 3, 18, 17, 2, 4, 18, 5, 8, 3, 4, 23, 2, 5, 118, 37, 8, 39, 18, 8, 34, 118, 14, 110, 4, 2, 18, 1, 104, 47, 10, 8, 32, 49, 18, 104, 48, 17, 142, 48, 8, 8, 118, 4, 66, 21, 18, 48, 70, 5, 50

Offset: 1

Amarnath Murthy, Sep 01 2002

			a(13) = 2 as 13 divides 234.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A075113, A075114, A075116, A075117.

Table[Module[{k=1},While[!Divisible[FromDigits[Flatten[ IntegerDigits/@ Range[k,k+2]]],n],k++];k],{n,70}] (* Harvey P. Dale, May 10 2012 *)

Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 19 2003

A075016 Smallest k such that the concatenation k, k-1,k-2 is divisible by n; or 0 if no such number exists.

Original entry on oeis.org

2, 2, 2, 4, 2, 2, 2, 4, 4, 2, 12, 4, 105, 2, 2, 4, 7, 4, 18, 22, 2, 12, 11, 4, 27, 118, 4, 106, 21, 2, 23, 14, 12, 34, 2, 4, 112, 18, 105, 22, 15, 2, 39, 34, 7, 14, 9, 4, 141, 52, 7, 118, 58, 4, 12, 106, 18, 50, 38, 22, 10, 54, 106, 14, 157

Offset: 1

Amarnath Murthy, Sep 01 2002

			a(11) = 12 as 11 divides 121110.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A075113, A075114, A075115, A075117.

skc[n_]:=Module[{k=2},While[Mod[FromDigits[Flatten[IntegerDigits/@ Range[ k,k-2,-1]]],n]!=0,k++];k]; Array[skc,70] (* Harvey P. Dale, Nov 01 2019 *)

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 19 2003

A075127 Safe perfect powers: perfect powers n such that (n-1)/2 is also a perfect power.

Original entry on oeis.org

9, 243, 289, 9801, 332929, 11309769, 384199201, 13051463049, 443365544449, 15061377048201, 511643454094369, 17380816062160329, 590436102659356801, 20057446674355970889, 681362750825443653409

Offset: 1

Zak Seidov, Oct 11 2002

nonn

If both powers are squares, the smaller square is a triangular number, and all square triangular numbers (A001110) correspond to a member in this sequence. This proves that this sequence is infinite. Are there only finitely many other members, i.e., is A075127 \ A055792 finite? - Charles R Greathouse IV, Dec 12 2010

Cf. A001110, A001597, A055792, A070428, A075114.

pp = Select[ Range[10^8], Apply[ GCD, Last[ Transpose[ FactorInteger[ # ]]]] > 1 & ]; Select[pp, Apply[GCD, Last[ Transpose[ FactorInteger[( # - 1)/2]]]] > 1 & ]

for(n=1, 1e10, if(ispower(n) && ispower((n-1)/2), print1(n, ", "))) \\ Altug Alkan, Oct 28 2015

Conjectures from Colin Barker, Oct 28 2015: (Start)
a(n) = 35*a(n-1)-35*a(n-2)+a(n-3) for n>5.
G.f.: x*(234*x^4-8182*x^3+7901*x^2+72*x-9) / ((x-1)*(x^2-34*x+1)).
(End)

One more term from Robert G. Wilson v, Oct 16 2002

a(7)-a(15) from Donovan Johnson, Mar 10 2010

A117547 Numbers n such that 2*n^2+1 is a perfect power.

Original entry on oeis.org

0, 2, 11, 12, 70, 408, 2378, 13860, 80782, 470832, 2744210, 15994428, 93222358, 543339720, 3166815962, 18457556052, 107578520350, 627013566048, 3654502875938, 21300003689580, 124145519261542, 723573111879672, 4217293152016490, 24580185800219268

Offset: 1

T. D. Noe, Mar 29 2006

nonn

The value of y in the solution of the Diophantine equation x^a - 2*y^b = 1. All solutions have b=2. Sequence A075114 gives n^2. The only known solution for a>2 is y=11. See A075114 for more details.

Vec(x^2*(2-x-52*x^2+9*x^3)/ (1-6*x+x^2) + O(x^66))
/* Joerg Arndt, Apr 28 2012, using Colin Barker's g.f. */

Conjecture: a(n) = 6*a(n-1) - a(n-2) for n>5; g.f.: x^2*(2-x-52*x^2+9*x^3)/ (1-6*x+x^2). - Colin Barker, Apr 28 2012

More terms from T. D. Noe, Nov 19 2006

A075013 Smallest k such that the decimal concatenation of k and k+1 is divisible by n.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 5, 5, 4, 9, 16, 1, 7, 5, 4, 19, 3, 13, 22, 19, 16, 27, 2, 19, 24, 7, 31, 5, 31, 19, 27, 19, 16, 3, 9, 31, 26, 41, 7, 19, 28, 37, 20, 27, 4, 51, 20, 19, 16, 49, 52, 35, 32, 31, 49, 5, 22, 31, 66, 19, 32, 27, 58, 19, 9, 49, 6, 35, 28, 9, 26, 67, 13, 63, 49, 79, 16

Offset: 1

Amarnath Murthy, Sep 01 2002

			a(16) = 19 as 16 divides 1920.

Cf. A075114, A075115, A075116, A075117.

With[{cnct=Table[FromDigits[Flatten[IntegerDigits/@{n,n+1}]],{n,100}]},Table[ Position[cnct,?(Divisible[#,k]&),1,1],{k,100}]]//Flatten (* _Harvey P. Dale, Apr 11 2020 *)

for(n=1,100,k=1:while((k*(10^floor(log(k+1)/log(10)+1))+k+1)%n>0,k=k+1):print1(k","))

Corrected and extended by Ralf Stephan, Mar 23 2003

A075017 Smallest k such that the concatenation k, k+1,k+2,k+3 is divisible by n; or 0 if no such number exists.

Original entry on oeis.org

1, 1, 3, 3, 2, 3, 2, 3, 3, 7, 8, 3, 7, 17, 12, 3, 5, 3, 19, 17, 24, 15, 17, 3, 22, 7, 3, 17, 17, 27, 6, 3, 15, 5, 2, 3, 9, 19, 15, 17, 15, 45, 44, 37, 12, 17, 39, 3, 45, 47, 21, 41, 52, 3, 37, 17, 57, 17, 62, 57, 53, 53, 66, 3, 7, 15, 2, 21, 63, 17, 12, 3, 6, 9, 72

Offset: 1

Amarnath Murthy, Sep 01 2002

			a(7) = 2 as 7 divides 2345.

Cf. A075113, A075114, A075115, A075116.

isok(k, n) = my(v = Str(k)); for (j=1, 3, v = concat(v, Str(k+j))); (eval(v) % n) == 0;
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, Jan 21 2017

Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 19 2003

A075018 Smallest k such that the concatenation k, k-1,k-2,k-3 is divisible by n; or 0 if no such number exists.

Original entry on oeis.org

3, 3, 3, 5, 3, 3, 5, 5, 6, 3, 18, 9, 11, 5, 3, 15, 13, 15, 19, 23, 18, 29, 29, 15, 28, 11, 33, 5, 4, 3, 40, 15, 18, 13, 18, 15, 28, 19, 24, 23, 26, 39, 7, 51, 33, 29, 55, 15, 53, 53, 30, 63, 54, 33, 18, 5, 57, 41, 56, 63, 69, 71, 60, 15, 63

Offset: 1

Amarnath Murthy, Sep 01 2002

			a(7) = 5 as 7 divides 5432.

Cf. A075113, A075114, A075115, A075116, A075117.

Transpose[Flatten[Table[Select[Reverse[Partition[Range[100,0,-1], 4,1]], Divisible[FromDigits[ Flatten[IntegerDigits/@#]],n]&,1],{n,70}],1]] [[1]] (* Harvey P. Dale, Nov 22 2011 *)

Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 19 2003

cp's OEIS Frontend

A084703 Squares k such that 2*k+1 is also a square.

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A075014 Smallest k such that the concatenation k, k-1 is divisible by n; or 0 if no such number exists.

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A075015 Smallest k such that the concatenation k, k+1, k+2 is divisible by n; or 0 if no such number exists.

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A075016 Smallest k such that the concatenation k, k-1,k-2 is divisible by n; or 0 if no such number exists.

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A075127 Safe perfect powers: perfect powers n such that (n-1)/2 is also a perfect power.

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PARI

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A117547 Numbers n such that 2*n^2+1 is a perfect power.

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PARI

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A075013 Smallest k such that the decimal concatenation of k and k+1 is divisible by n.

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PARI

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A075017 Smallest k such that the concatenation k, k+1,k+2,k+3 is divisible by n; or 0 if no such number exists.

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