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Numbers k such that the numerator and denominator of the reduced fraction Sum_{i=1..k} 1/i have no digits in common.
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%I #43 Oct 30 2023 15:55:31

%S 2,3,5,6,7,8,10,15

%N Numbers k such that the numerator and denominator of the reduced fraction Sum_{i=1..k} 1/i have no digits in common.

%e 2 is in the sequence since Sum_{i=1..2} 1/i = 1 + 1/2 = 3/2 and the numerator and denominator have no digits in common.

%e 3 is in the sequence since Sum_{i=1..3} 1/i = 1 + 1/2 + 1/3 = 11/6 and the numerator and denominator have no digits in common.

%t Select[Range@100, Intersection @@ IntegerDigits[ Through[ {Numerator, Denominator}[ HarmonicNumber@ #]]] == {} &] (* _Giovanni Resta_, Sep 13 2019 *)

%o (PARI) isok(n) = my(h=sum(k=1, n, 1/k)); #setintersect(Set(digits(numerator(h))), Set(digits(denominator(h)))) == 0; \\ _Michel Marcus_, Sep 13 2019

%Y Cf. A001008, A002805.

%K nonn,more,base

%O 1,1

%A _Wesley Ivan Hurt_, Sep 11 2019