The Journal of the Indian Mathematical Society

C. G. Khatri ¹, A. M. Vaidya ¹

Affiliations
1 Department of Mathematics and Statistics, University School of Sciences, Gujarat University, Ahmedabad, IN

Abstract

For a positive integer n, the nth. triangular number T_n is defined by T_n ==1/2n(n + 1). For a given integer t, we consider the equation

T_a/T_b = t  (1)

If there is a solution T_a, T_b of (1),

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A. M. Vaidya ¹, V. S. Joshi ²

Affiliations
1 Department of Mathematics, University School of Sciences, Gujarat University, Ahmedabad-380009, IN
2 Department of Mathematics, South Gujarat University, Surat-395007, IN

Abstract

Let n be any natural number. For n>1 , let its prime power factorization be n=πp₁^α1. If the digit 2 does not appear in the ternary expansion (i.e. in the representation in the base 3) of α₁ for every i, then n is called an Exponentially Ternary 2-Free (ETF₂) number. By convention we regard 1 as an ETF₂ number. Let E be the class of all ETF₂ numbers. If we denote by E(x), the number of (positive) integers ≤x contained in E, then it is implicit in some results of Murty [3] that
E(x)=Ax+O(x^1/2).

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