A323061 Numbers n that are not multiples of 10 and such that 10*n*R(n) is a square, where R(n) = A004086(n) is the decimal digits of n reversed.

544968, 547658, 560106, 601065, 856745, 869445, 2495295, 4601685, 5606106, 5861064, 5925942, 6016065, 20861005, 21778875, 22972005, 29389855, 42251835, 50016802, 50027922, 51826326, 53815224, 55898392, 56066106, 56570706, 56873466, 57887712, 60166065, 60707565

Offset: 1

Author

Chai Wah Wu, Jan 07 2019

Keywords

• nonn
• base

Comments

10*a(n) are exactly the terms in A322835 that are not multiples of 100.

m is a term if and only if R(m) is a term.

The product of the first and last digits of a term is a multiple of 10, i.e., the first and last digits of a term are the digit 5 and an even nonzero digit.

The sequence has an infinite number of terms. For instance, 601x065 is a term where x is a string of k repeated digits 6 and k >= 0, i.e., 601065, 6016065, 60166065, etc. Similarly numbers of the form 560x106 are also terms.

To see this, let a = 601*10^(3+k) + 65 + 6000*(10^k-1)/9. Then R(a) = 56*10^(4+k) + 106 + 6000*(10^k-1)/9. The number 10*a*R(a) can be written as 30360100*(10^(k + 3) - 1)^2/9 whose square root is 5510*(10^(k + 3) - 1)/3.

From Chai Wah Wu, Feb 18 2019: (Start)

22994x77005 is a term where x is a string of k repeated digits 9 and k >= 0. Let a = 22994*10^(5+k) + 77005 + 100000*(10^k-1). Then R(a) = 50077*10^(5+k) + 49922 + 100000*(10^k-1). The number 10*a*R(a) can be written as 11515436100*(10^(k+5) - 1)^2, whose square root is 107310*(10^(k+5) - 1).

23804x76195 is a term where x is a string of k repeated digits 9 and k >= 0. Let a = 23804*10^(5+k) + 76195 + 100000*(10^k-1). Then R(a) = 59167*10^(5+k) + 40832 + 100000*(10^k-1). The number 10*a*R(a) can be written as 14084942400*(10^(k+5) - 1)^2, whose square root is 118680*(10^(k+5) - 1).

If w is a term with decimal representation a, then the number n corresponding to the string axa is also a term, where x is a string of k repeated digits 0 and k >= 0. The number n = w*10^(k+m)+w = w*(10^(k+m)+1) where m is the number of digits of w. Note that n is also not a multiple of 10. Then R(n) = R(w)*10^(k+m)+R(w) = R(w)(10^(k+m)+1). Then 10*n*R(n) = 10*w*R(w)(10^(k+m)+1)^2 which is a square since w is a term.

The same argument shows that numbers corresponding to axaxa, axaxaxa, ..., are also terms.

For example, since 544968 is a term, so are 544968544968, 5449680544968, 54496800544968, 5449680054496800544968, etc.

(End)

Examples

			238026195 * 591620832 * 10 = 1186681320^2.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..200 (first 54 terms from Chai Wah Wu)

Crossrefs

Cf. A004086, A322835.

Programs

Mathematica

Select[Range[61*10^6],Mod[#,10]!=0&&IntegerQ[Sqrt[10# IntegerReverse[ #]]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 16 2020 *)

PARI

isok(n) = (n % 10) && issquare(10*n*fromdigits(Vecrev(digits(n)))); \\ Michel Marcus, Jan 10 2019