A253216
Smallest of four primes in arithmetic progression with common difference 6 and digit sum prime.
Original entry on oeis.org
1091, 15791, 30091, 369991, 421691, 501191, 661091, 1101091, 1539991, 2042591, 2210291, 2542091, 2811191, 3351191, 3512291, 3864691, 4411391, 4675591, 5960791, 5992291, 5998691, 6884191, 6918391, 7516891, 8608591, 8697791, 9297091, 9622891, 9646291, 12013091
Offset: 1
a (1) = 1091: 1091 + 6 = 1097; 1097 + 6 = 1103; 1103 + 6 = 1109; all four are prime. Their digit sums 1+0+9+1 = 11; 1+0+9+7 = 17; 1+1+0+3 = 5 and 1+1+0+9 = 11 are also prime.
a(2) = 15791: 15791 + 6 = 15797; 15797 + 6 = 15803; 15803 + 6 = 15809; all four are prime. Their digit sums 1+5+7+9+1 = 23, 1+5+7+9+7 = 29, 1+5+8+0+3 = 17 and 1+5+8+0+9 = 23 are also prime.
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A253216 = {}; Do[d = 6; k = Prime[n]; k1 = k + d; k2 = k + 2d; k3 = k + 3d; If[PrimeQ[k1] && PrimeQ[k2] && PrimeQ[k3] && PrimeQ[Plus @@ IntegerDigits[k]] && PrimeQ[Plus @@ IntegerDigits[k1]] && PrimeQ[Plus @@ IntegerDigits[k2]] && PrimeQ[Plus @@ IntegerDigits[k3]], AppendTo[A253216, k]], {n, 1000000}]; A253216
prQ[{a_,b_,c_,d_}]:=AllTrue[{b,c,d},PrimeQ]&&AllTrue[Total/@ (IntegerDigits/@ {a,b,c,d}),PrimeQ]; Select[#+{0,6,12,18}& /@Prime[Range[800000]],prQ][[All,1]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 26 2018 *)
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for( n=1, 10^6, k=prime(n); k1=k+6; k2=k+12; k3=k+18; if(isprime(k1)&isprime(k2)&isprime(k3) &isprime(eval(Str(sumdigits(k)))) &isprime(eval(Str(sumdigits(k1)))) &isprime(eval(Str(sumdigits(k2)))) &isprime(eval(Str(sumdigits(k3)))), print1(k,", ")))
A253232
Smallest of five consecutive primes in arithmetic progression with common difference 90 and equal digit sums.
Original entry on oeis.org
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
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.
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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 *)
A277607
Smallest of four consecutive primes in arithmetic progression with common difference 42 and all digit sums prime.
Original entry on oeis.org
5, 47, 157, 227, 317, 337, 557, 2027, 3037, 3217, 5147, 6047, 7457, 12527, 13757, 14657, 20357, 21017, 23747, 32057, 35027, 47417, 57047, 84137, 115727, 125627, 127247, 136337, 147137, 149027, 212057, 219937, 225257, 230017, 240047, 242357, 264137, 284117, 304127
Offset: 1
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.
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.
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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
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]
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