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A078728
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a(n) is the smallest m such that m < 10^n, 10^n + m is prime and if the natural number k is such that 1 < k < 10 and 3 doesn't divide k*10^n + m then k*10^n+m is prime.
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0
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3, 57, 297, 177, 237, 25111, 231339, 67419, 273817, 345111, 2001753, 912277, 5236153, 9228627, 10599391, 2835261, 60120003, 14054037, 27923629, 41783347, 24590943, 112161513, 230484021, 11446969, 205242589, 583389307, 873650007
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OFFSET
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1,1
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COMMENTS
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a(n) is the smallest m such that m < 10^n and all six numbers 10^n + m, (Mod[m, 3]+2)*10^n + m, 4*10^n + m, (Mod[m, 3]+5)*10^n + m, 7*10^n + m & (Mod[m, 3]+8)*10^n + m are primes.
Carlos Rivera in Puzzle 245 of www.primepuzzles.net wrote "if the Faride's results ( a(n) for n=1,...,24 ) are plotted in Excel and a trend 'potential' function is asked, we obtain that a(n) is approximately equal to 0.5*n^6; this means that for n=999 a(n)=5*10^17, approximately." Since 10^n+a(n) is prime, for each n a(n)=0 (mod 3) or a(n)=1 (mod 3).
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LINKS
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FORMULA
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a[n_] := (For[m=1, !PrimeQ[10^n+2m-1]||!PrimeQ[(Mod[2m-1, 3]+2)10^n+2m-1]||! PrimeQ[4*10^n+2m-1]||!PrimeQ[(Mod[2m-1, 3]+5)10^n+2m-1]||!PrimeQ [7*10^n+2m-1]||!PrimeQ[(Mod[2m-1, 3]+8)10^n+2m-1], m++ ];2m-1)
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EXAMPLE
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a(6)=25111 because all the six numbers 1025111, 3025111, 4025111, 6025111, 7025111, 9025111 are primes and 25111 is the smallest number with this property.
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MATHEMATICA
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a[n_] := (For[m=1, !PrimeQ[10^n+2m-1]||!PrimeQ[(Mod[2m-1, 3]+2)10^n+2m-1]||! PrimeQ[4*10^n+2m-1]||!PrimeQ[(Mod[2m-1, 3]+5)10^n+2m-1]||!PrimeQ [7*10^n+2m-1]||!PrimeQ[(Mod[2m-1, 3]+8)10^n+2m-1], m++ ]; 2m-1); Do[Print[a[n]], {n, 32}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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