A339076 Numbers which are coprime to their digital sum (A007953).

1, 10, 11, 13, 14, 16, 17, 19, 23, 25, 29, 31, 32, 34, 35, 37, 38, 41, 43, 47, 49, 52, 53, 56, 58, 59, 61, 65, 67, 71, 73, 74, 76, 79, 83, 85, 89, 91, 92, 94, 95, 97, 98, 100, 101, 103, 104, 106, 107, 109, 113, 115, 119, 121, 122, 124, 125, 127, 128, 131, 137

Offset: 1

Author

Amiram Eldar, Nov 22 2020

Keywords

• nonn
• base

Comments

Numbers k such that gcd(k, A007953(k)) = 1.

Olivier (1975, 1976) proved that the asymptotic density of this sequence is 9/(2*Pi^2) = 0.455945... (A088245).

None of the terms are divisible by 3.

The powers of 10 (A011557) are terms. These are also the only Niven numbers (A005349) in this sequence.

Includes all the prime numbers above 7.

Examples

			10 is a term since A007953(10) = 1 + 0 = 1, and gcd(10, 1) = 1.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Curtis Cooper and Robert E. Kennedy, On the set of positive integers which are relatively prime to their digital sum and its complement, J. Inst. Math. & Comp. Sci. (Math. Ser.), Vol. 10 (1997), pp. 173-180.
• 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. A005349, A007953, A088245.

Subsequence of A001651.

Subsequence: A011557.

Binary version: A094387.

Programs

Mathematica

Select[Range[200], CoprimeQ[#, Plus @@ IntegerDigits[#]] &]