A242403 Decimal expansion of the binary self-numbers density constant.

2, 5, 2, 6, 6, 0, 2, 5, 9, 0, 0, 8, 8, 8, 2, 9, 2, 2, 1, 5, 5, 0, 6, 2, 7, 1, 4, 3, 2, 7, 8, 9, 4, 1, 4, 1, 8, 2, 5, 2, 1, 9, 3, 3, 9, 6, 2, 9, 7, 8, 4, 6, 1, 3, 0, 1, 6, 8, 6, 2, 1, 7, 2, 2, 9, 2, 2, 8, 0, 5, 4, 8, 4, 4, 7, 6, 6, 3, 2, 5, 6, 6, 9, 5, 9, 1, 4, 2, 4, 4, 7, 9, 3, 8, 6, 8, 8, 9, 4, 9

Offset: 0

Author

Jean-François Alcover, May 13 2014

Keywords

• nonn
• cons

Comments

This constant is transcendental (Troi and Zannier, 1999). - Amiram Eldar, Nov 28 2020

Examples

			0.2526602590088829221550627143278941418252...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 179.
• József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.
• G. Troi and U. Zannier, Note on the density constant in the distribution of self-numbers, Bollettino dell'Unione Matematica Italiana, Serie 7, Vol. 9-A, No. 1 (1995), pp. 143-148.

Links

• G. Troi and U. Zannier, Note on the density constant in the distribution of self-numbers - II, Bollettino dell'Unione Matematica Italiana, Serie 8, Vol. 2-B, No. 2 (1999), pp. 397-399; alternative link.
• Umberto Zannier, On the distribution of self-numbers, Proc. Amer. Math. Soc., Vol. 85, No. 1 (1982), pp. 10-14.
• Index entries for transcendental numbers

Crossrefs

Cf. A010061 (binary self numbers), A003052 (decimal self numbers), A010064, A010067, A010070, A092391, A228082.

Programs

Mathematica

m0 = 100; dm = 100; digits = 100; Clear[lambda]; lambda[m_] := lambda[m] = Total[1/2^Union[Table[n + Total[IntegerDigits[n, 2]], {n, 0, m}]]]^2/8 // N[#, 2*digits]& // RealDigits[#, 10, 2*digits]& // First; lambda[m0]; lambda[m = m0 + dm]; While[lambda[m] != lambda[m - dm], Print["m = ", m]; m = m + dm]; lambda[m][[1 ;; digits]]

Formula

Equals (1/8)*(Sum_{n not a binary self-number} 1/2^n)^2.