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A253232 Smallest of five consecutive primes in arithmetic progression with common difference 90 and equal digit sums.

%I A253232 #13 May 13 2022 14:51:49
%S A253232 61,83,89,593,1399,2063,2287,2351,2441,3491,5081,5171,5479,6599,9497,
%T A253232 12073,16561,17569,21377,23099,23189,28573,29063,32143,36293,36497,
%U A253232 36587,39569,49279,61291,62383,65449,66373,71167,72379,75347,81457,88591,92377,94261,104369
%N A253232 Smallest of five consecutive primes in arithmetic progression with common difference 90 and equal digit sums.
%C A253232 90 is the smallest common difference (d) to get a set of five consecutive primes in arithmetic progression {p, p+d, p+2d, p+3d, p+4d} having digit sums equal; for p < prime(10^5).
%H A253232 K. D. Bajpai, <a href="/A253232/b253232.txt">Table of n, a(n) for n = 1..10000</a>
%e A253232 a(1) = 61: 61+90 = 151; 151+90 = 241; 241+90 = 331; 331+90 = 421; all five are prime. Their digit sums 6+1 = 1+5+1 = 2+4+1 = 3+3+1 = 4+2+1 = 7 are all equal.
%e A253232 a(2) = 83: 83+90 = 173; 173+90 = 263; 263+90 = 353; 353+90 = 443; all five are prime. Their digit sums 8+3 = 1+7+3 = 2+6+3 = 3+5+3 = 4+4+3 = 11 are all equal.
%t A253232 A253232 = {}; Do[d = 90; k = Prime[n]; k1 = k + d; k2 = k + 2 d; k3 = k + 3 d; k4 = k + 4 d; s = Plus @@ IntegerDigits[k]; s1 = Plus @@ IntegerDigits[k1]; s2 = Plus @@ IntegerDigits[k2]; s3 = Plus @@ IntegerDigits[k3]; s4 = Plus @@ IntegerDigits[k4]; If[PrimeQ[k1] && PrimeQ[k2] && PrimeQ[k3] && PrimeQ[k4] && s == s1 && s1 == s2 && s2 == s3 && s3 == s4, AppendTo[A253232, k]], {n, 50000}]; A253232
%t A253232 cd90Q[p_]:=Module[{q=p+90,r=p+180,s=p+270,t=p+360},AllTrue[{p,q,r,s,t},PrimeQ] && Length[Union[Total/@(IntegerDigits/@{p,q,r,s,t})]]==1]; Select[ Prime[ Range[ 10000]],cd90Q] (* _Harvey P. Dale_, May 13 2022 *)
%Y A253232 Cf. A000040, A033447, A062088, A253140.
%K A253232 nonn,base
%O A253232 1,1
%A A253232 _K. D. Bajpai_, Dec 29 2014