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A275750
Prime numbers of the form 4^n - 27.
3
37, 229, 997, 1048549, 4194277, 67108837, 1125899906842597, 72057594037927909, 288230376151711717, 1361129467683753853853498429727072845797, 1393796574908163946345982392040522594123749, 1725436586697640946858688965569256363112777243042596638790631055949797
OFFSET
1,1
COMMENTS
Values of the exponent n are given in A274519. If the exponent is odd, then the rightmost digit of a(n) will be 7. If the exponent is even, then the rightmost digit of a(n) will be 9.
As a result of the recent extensions to A274519 by Vincenzo Librandi,
a(13) = 4^305 - 27 > 4.2491 * 10^183
a(14) = 4^515 - 27 > 1.1505 * 10^310
a(15) = 4^2029 - 27 > 3.7994 * 10^1221
a(16) = 4^2393 - 27 > 5.3648 * 10^1440
a(17) = 4^2605 - 27 > 2.3242 * 10^1568
a(18) = 4^3530 - 27 > 1.8696 * 10^2125
a(19) = 4^4036 - 27 > 8.2058 * 10^2429
a(20) = 4^4750 - 27 > 6.0947 * 10^2859
a(21) > 4^5000 - 27 > 1.9950 * 10^3010.
These primes a(m) can be used to generate numbers having abundance 26. The formula a(m)*(a(m)+27)/2 produces some of the terms in A275701.
FORMULA
a(m) = 4^A274519(m) - 27.
EXAMPLE
a(1) = 4^A274519(1) - 27 = 4^3 - 27 = 64 - 27 = 37.
a(2) = 4^A274519(2) - 27 = 4^4 - 27 = 256 - 27 = 229.
a(3) = 4^A274519(3) - 27 = 4^5 - 27 = 1024 - 27 = 997.
a(4) = 4^A274519(4) - 27 = 4^10 - 27 = 1048576 - 27 = 1048549.
a(5) = 4^A274519(5) - 27 = 4^11 - 27 = 4194304 - 27 = 4194277.
a(6) = 4^A274519(6) - 27 = 4^13 - 27 = 67108864 - 27 = 67108837.
MATHEMATICA
Select[4^Range[3, 120] - 27, PrimeQ] (* Michael De Vlieger, Aug 08 2016 *)
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Timothy L. Tiffin, Aug 07 2016
STATUS
approved