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Emirps in A157711

After watching Matt Parker's February 16th Numberphile video on Stephan Schöler's largest currently known emirp, I wondered of course if any of the primes that are the sum of four distinct powers of ten (A157711, on which I have been working now for nine months) were emirps. It turns out there were a healthy number and I decided to create two new OEIS sequences for them: A393530 and A393531.

While more than half (533) of the first 1000 digit-lengths had no solutions, the others had anywhere from 1 to 7 (digit-length 41) solutions. Moreover, the accumulation of emirp-pair counts showed a steady rise, giving me hope that this would continue for larger digit-lengths:

emirp-pair count accumulations for the first 1036 digit-lengths (click to enlarge)

I will see if I can increase my counts to 2000+ digit-lengths. This will take some weeks. At the same time I'll try a limited sampling of solution searches for digit-lengths greater than 10011 to see if I can luck into a record large emirp.

The five-millionth term of A157711

The sum of the first 4006 prime counts of A383675 is 4998755. The sum of the first 4007 prime counts is 5001521. Hence, the five-millionth term of A157711 is a 4007-digit integer:

4998756 10^4006+10^39+10^18+1

4998757 10^4006+10^104+10^53+1

4998758 10^4006+10^108+10^16+1

4998759 10^4006+10^177+10^120+1

4998760 10^4006+10^192+10^59+1

...

4999995 10^4006+10^2671+10^2184+1

4999996 10^4006+10^2672+10^696+1

4999997 10^4006+10^2672+10^1708+1

4999998 10^4006+10^2673+10^392+1

4999999 10^4006+10^2673+10^530+1

5000000 10^4006+10^2673+10^876+1

5000001 10^4006+10^2673+10^1124+1

5000002 10^4006+10^2675+10^1928+1

5000003 10^4006+10^2676+10^1232+1

5000004 10^4006+10^2677+10^1160+1

5000005 10^4006+10^2677+10^2092+1

...

5001517 10^4006+10^4003+10^3570+1

5001518 10^4006+10^4003+10^3818+1

5001519 10^4006+10^4003+10^3854+1

5001520 10^4006+10^4004+10^609+1

5001521 10^4006+10^4005+10^1892+1

A157711(1*10^6) = 10^1793+10^673+10^615+1 [2025 June 19]

A157711(2*10^6) = 10^2535+10^1160+10^398+1 [2025 July 21]

A157711(3*10^6) = 10^3103+10^2747+10^859+1 [2025 September 10]

A157711(4*10^6) = 10^3583+10^3040+10^2776+1 [2025 December 7]

A157711(5*10^6) = 10^4006+10^2673+10^876+1 (above)

A157711(6*10^6) ~ 10^4387

I'll continue to update (here, until the next millionth is reached) my current plot of A383675:

A383675 to n=4007; max=(3993,3278) [updated February 14]  click to enlarge

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