A060289 Number of distinct (non-overlapping) twin Harshad numbers whose sum is prime and where the 2nd Harshad is <= 10^n.
4, 5, 17, 53, 250, 1404, 9013, 58608, 401614, 2908740, 21832530
Offset: 1
Examples
a(1)=4 because there are four pairs of Harshads whose sum is prime and the 2nd Harshad in the pair is <=10; these are 1+2=3, 3+4=7, 5+6=11, 9+10=19. 8+9=17 is not included because this pair overlaps 7+8=15, which also happens to be not prime. (Another sequence might include such overlapping pairs.)
Programs
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Mathematica
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, If[PrimeQ[2*k - 1],c++;If[k > p, n++; AppendTo[s, c-1]; p *= 10]]; q1 = False, q1 = q2]; k++]; s (* Amiram Eldar, Jan 19 2021 *) -
PARI
Niven(n)=n%sumdigits(n)==0 a(n)=my(t,s); for(k=1,10^n,if(Niven(k), if(isprime(t+k), t=-10^n; s++); t=k)); s \\ Charles R Greathouse IV, Jan 23 2014
Formula
Generate the twin Harshads whose sum is prime. Count how many there are where the 2nd Harshad in the pair is <= a consecutive power of 10.
Extensions
a(8)-a(11) from Amiram Eldar, Jan 19 2021