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

61, 83, 89, 593, 1399, 2063, 2287, 2351, 2441, 3491, 5081, 5171, 5479, 6599, 9497, 12073, 16561, 17569, 21377, 23099, 23189, 28573, 29063, 32143, 36293, 36497, 36587, 39569, 49279, 61291, 62383, 65449, 66373, 71167, 72379, 75347, 81457, 88591, 92377, 94261, 104369

Offset: 1

Author

K. D. Bajpai, Dec 29 2014

Keywords

• nonn
• base

Comments

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).

Examples

			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.
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.
		

Links

• K. D. Bajpai, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A033447, A062088, A253140.

Programs

Mathematica

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
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 *)