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Smallest of four consecutive primes in arithmetic progression with common difference 42 and all digit sums prime.
1

%I #17 Oct 31 2016 13:34:45

%S 5,47,157,227,317,337,557,2027,3037,3217,5147,6047,7457,12527,13757,

%T 14657,20357,21017,23747,32057,35027,47417,57047,84137,115727,125627,

%U 127247,136337,147137,149027,212057,219937,225257,230017,240047,242357,264137,284117,304127

%N Smallest of four consecutive primes in arithmetic progression with common difference 42 and all digit sums prime.

%e a(1) = 5: 5 + 42 = 47; 47 + 42 = 89; 89 + 42 = 131; all four are prime. Their digit sums 5, 4 + 7 = 11, 8 + 9 = 17 and 1 + 3 + 1 = 5 are also prime.

%e a(2) = 47: 47 + 42 = 89; 89 + 42 = 131; 131 + 42 = 173; all four are prime. Their digit sums 4 + 7 = 11, 8 + 9 = 17, 1 + 3 + 1 = 5 and 1 + 7 + 3 = 11 are also prime.

%t A277607 = {}; Do[d = 42; k = Prime[n]; k1 = k + d; k2 = k + 2 d; k3 = k + 3 d; If[PrimeQ[k1] && PrimeQ[k2] && PrimeQ[k3] && PrimeQ[Plus @@ IntegerDigits[k]] && PrimeQ[Plus @@ IntegerDigits[k1]] && PrimeQ[Plus @@ IntegerDigits[k2]] && PrimeQ[Plus @@ IntegerDigits[k3]], AppendTo[A25, k]], {n, 30000}]; A277607

%t FCPQ[n_] := Module[{a = n + 42, b = n + 84, c = n + 126}, AllTrue[{a, b, c}, PrimeQ] && AllTrue[Total /@ (IntegerDigits /@ {n, a, b, c}), PrimeQ]]; Select[Prime[Range[30000]], FCPQ]

%Y Cf. A000040, A033447, A062088, A253140.

%K nonn,base

%O 1,1

%A _K. D. Bajpai_, Oct 31 2016