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 A104657 Positive integers n such that n^19 + 1 is semiprime (A001358). 12
 2, 10, 28, 106, 190, 292, 556, 756, 858, 906, 1012, 1030, 1032, 1060, 1372, 1450, 1488, 1720, 1722, 1758, 1782, 1822, 1972, 2356, 2436, 2446, 2620, 2748, 2788, 2998, 3186, 3300, 3318, 3360, 3466, 3510, 3822, 3852, 4138, 4326, 4506, 4908, 5236, 5518, 5782 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS We have the polynomial factorization: n^19 + 1 = (n + 1) * (n^18 - n^17 + n^16 - n^15 + n^14 - n^13 + n^12 - n^11 + n^10 - n^9 + n^8 - n^7 + n^6 - n^5 + n^4 - n^3 + n^2 - n + 1). Hence after the initial n=1 prime the binomial can never be prime. It can be semiprime iff n+1 is prime and (n^18 - n^17 + n^16 - n^15 + n^14 - n^13 + n^12 - n^11 + n^10 - n^9 + n^8 - n^7 + n^6 - n^5 + n^4 - n^3 + n^2 - n + 1) is prime. LINKS Robert Price, Table of n, a(n) for n = 1..1000 FORMULA a(n)^19 + 1 is semiprime (A001358). EXAMPLE 2^19 + 1 = 524289 = 3 * 174763, 10^19 + 1 = 10000000000000000001 = 11 * 909090909090909091, 1012^19 + 1 = 125438178100868833265294241234853844232270960601988910249 = 1013 * 1238284087866424810121364671617510801898035149081825373. MATHEMATICA Select[Range[1000000], PrimeQ[# + 1] && PrimeQ[(#^19 + 1)/(# + 1)] &] (* Robert Price, Mar 10 2015 *) Select[Range[5800], PrimeOmega[#^19+1]==2&] (* Harvey P. Dale, Feb 15 2019 *) PROG (MAGMA) IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [1..1100]|IsSemiprime(n^19+1)]; // Vincenzo Librandi, Mar 10 2015 CROSSREFS Cf. A001358, A006313, A103854, A104238, A104335, A105041, A105066, A105078, A105122, A105142, A105237, A104479, A104494. Sequence in context: A296849 A296380 A291053 * A000900 A124023 A127921 Adjacent sequences:  A104654 A104655 A104656 * A104658 A104659 A104660 KEYWORD easy,nonn AUTHOR Jonathan Vos Post, Apr 21 2005 EXTENSIONS a(12)-a(45) from Robert Price, Mar 09 2015 STATUS approved

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Last modified April 11 16:42 EDT 2021. Contains 342888 sequences. (Running on oeis4.)