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Number of twin Harshads, including overlaps, whose sum is prime and where the 2nd Harshad is <= 10^n.
3

%I #13 Jan 19 2021 03:58:34

%S 7,8,21,60,278,1520,9583,61835,419705,3023007,22597508

%N Number of twin Harshads, including overlaps, whose sum is prime and where the 2nd Harshad is <= 10^n.

%F Generate the twin Harshads whose sum is prime. Count how many there are where the 2nd Harshad in the pair is <= to a consecutive power of 10.

%e a(1)=7 because there are 7 pairs of Harshads whose sum is prime and the 2nd Harshad in the pair is <= 10; these are 1+2=3, 2+3=5, 3+4=7, 5+6=11, 6+7=13, 8+9=17, 9+10=19. (Another sequence does not permit overlapping Harshad numbers.)

%t harshadQ[n_] := Divisible[n, Plus @@ IntegerDigits[n]]; c = 0; p = 10; s = {}; n = 0; k = 2; q1 = True; While[n < 6, q2 = harshadQ[k]; If[q1 && q2 && PrimeQ[2*k-1], c++; If[k > p, n++; AppendTo[s, c-1]; p *= 10]]; q1 = q2; k++]; s (* _Amiram Eldar_, Jan 19 2021 *)

%Y Cf. A060288, A060289, A060290, A005349, A060159.

%K nonn,base,more

%O 1,1

%A _Enoch Haga_, Mar 24 2001

%E a(8)-a(11) from _Amiram Eldar_, Jan 19 2021