A358975 Numbers that are coprime to their digital sum in base 3 (A053735).

1, 3, 5, 7, 9, 11, 13, 17, 19, 23, 27, 29, 31, 37, 41, 43, 47, 49, 51, 53, 55, 59, 61, 67, 69, 71, 73, 79, 81, 83, 85, 89, 91, 97, 101, 103, 107, 109, 113, 119, 121, 123, 125, 127, 129, 131, 137, 139, 141, 143, 147, 149, 151, 153, 155, 157, 159, 161, 163, 167, 169

Offset: 1

Author

Amiram Eldar, Dec 07 2022

Keywords

• nonn
• base

Comments

Numbers k such that gcd(k, A053735(k)) = 1.

All the terms are odd since if k is even then A053735(k) is even and so gcd(k, A053735(k)) >= 2.

Olivier (1975, 1976) proved that the asymptotic density of this sequence is 4/Pi^2 = 0.40528... (A185199).

The powers of 3 (A000244) are terms. These are also the only ternary Niven numbers (A064150) in this sequence.

Includes all the odd prime numbers (A065091).

Examples

			3 is a term since A053735(3) = 1, and gcd(3, 1) = 1.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Christian Mauduit, Carl Pomerance, and András Sárközy, On the distribution in residue classes of integers with a fixed sum of digits, The Ramanujan Journal, Vol. 9, No. 1-2 (2005), pp. 45-62; alternative link.
• Michel Olivier, Sur la probabilité que n soit premier à la somme de ses chiffres, C. R. Math. Acad. Sci. Paris, Série A, Vol. 280 (1975), pp. 543-545.
• Michel Olivier, Fonctions g-additives et formule asymptotique pour la propriété (n, f(n)) = q, Acta Arithmetica, Vol. 31, No. 4 (1976), pp. 361-384; alternative link.

Crossrefs

Cf. A064150, A053735, A185199, A332880.

Subsequences: A000244, A065091.

Similar sequences: A094387, A339076, A358976, A358977, A358978.

Programs

Mathematica

q[n_] := CoprimeQ[n, Plus @@ IntegerDigits[n, 3]]; Select[Range[200], q]

PARI

is(n) = gcd(n, sumdigits(n, 3)) == 1;