A073499
Numbers k such that k^(k+1) + (k+1)^k is prime.
Original entry on oeis.org
1, 2, 80, 342, 848, 1194, 2658, 4790, 9376
Offset: 1
1^2 + 2^1 = 3 and 2^3 + 3^2 = 17 are the primes corresponding to the first two terms. The next five terms correspond to primes of 155, 870, 2487, 3678 and 9106 decimal digits.
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Do[ If[ PrimeQ[n^(n + 1) + (n + 1)^n], Print[n]], {n, 1, 1650}]
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for(n=1,1650, if(isprime((n^(n+1))+((n+1)^n)), print1(n,",")))
A085682
Numbers k such that k^k - (k-1)^k is prime.
Original entry on oeis.org
2, 3, 7, 43, 79, 463, 1277
Offset: 1
7 is in the sequence because 7^7 - 6^7 = 543607 is prime.
A099498
Semiprimes of the form A007925(n) = n^(n+1)-(n+1)^n.
Original entry on oeis.org
7849, 3667649, 91171007, 2395420006033, 11877172892329028459041, 604107995057426434824791, 107174878415004743976428761769, 424678439961073471604787362241217, 1983672219242345491970468171243171249, 10788746499945827829225142589096882612369, 42855626937384013751014398588294858582343260060671
Offset: 1
a(1)=7849 because 5^6-6^5=7849=47*167 is a semiprime.
Cf.
A007925 n^(n+1)-(n+1)^n,
A072179 n^(n+1)-(n+1)^n is prime,
A099499 primes of the form n^(n+1)-(n+1)^n,
A099497 n^(n+1)-(n+1)^n is a semiprime.
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IsSemiprime:=func; [s: n in [3..30] | IsSemiprime(s) where s is n^(n+1)-(n+1)^n]; // Vincenzo Librandi, Sep 21 2012
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Select[Table[n^(n + 1) - (n + 1)^n, {n, 30}], PrimeOmega[#] == 2&] (* Vincenzo Librandi, Sep 21 2012 *)
A099497
Numbers k such that A007925(k) = k^(k+1) - (k+1)^k is a semiprime.
Original entry on oeis.org
5, 7, 8, 11, 17, 18, 21, 23, 25, 27, 32, 47, 51, 56, 59, 165
Offset: 1
a(1) = 5 because 5^6 - 6^5 = 7849 = 47*167 is a semiprime.
a(1) = 5 because 5^6 - 6^5 = 47*167
a(2) = 7 because 7^8 - 8^7 = 23*159463
a(3) = 8 because 8^9 - 9^8 = 257*354751
a(4) = 11 because 11^12 - 12^11 = 33479*71549927
a(5) = 17 because 17^18 - 18^17 = 443881*26757560905578361
a(6) = 18 because 18^19 - 19^18 = 100417*6015993258685545623
a(7) = 21 because 21^22 - 22^21 = 10745792197529*9973660056412561
a(8) = 23 because 23^24 - 24^23 = 92798617729*4576344458074395243073
a(9) = 25 because 25^26 - 26^25 = 1627*1219220786258356172077730898121187
a(10) = 27 because 27^28 - 28^27 = 12298336501553*877252504725615101634783073
a(11) = 32 because 32^33 - 33^32 = 3506869732968391733353*12220478717670771804763962407
a(12) = 47 because 47^48 - 48^47 = 11*15621013371424880252957237277868559270462038147831682437840584991339231377934499
a(13) = 51 because 51^52 - 52^51 = 10562756058978342869988055703171*5575962824795589360993690554534422732411612977322491058843
a(14) = 56 because 56^57 - 57^56 = 5*843980334169667457302970806376511482920948635540290643213973523914715036518308339240201775858865907
a(15) = 59 because 59^60 - 60^59 = 1994803969065168661575061125592557043358338451845483*8529249434913526091880095870250840825853220069057672947
a(16) = 165 because 165^166 - 166^165 = 7633959407*16307690786821361595026621717879347561301150483781862339651556401266189322630373265190696672506475741217560239791446654891805648807872536646884416611251422684856600732984767987061831649144878649678190762809385448362714901584206533854093359279076584767352259587745683931159999248465944943129517543272252180930134912057221968601458271001580745436226192252814407
Cf.
A007925 (n^(n+1)-(n+1)^n),
A072179 (k^(k+1)-(k+1)^k is prime),
A099498 (semiprimes of the form k^(k+1)-(k+1)^k).
165 from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 12 2008
A084852
Numbers k such that k^(k+1) + (k+1)^k - k(k+1) is prime.
Original entry on oeis.org
2, 5, 6, 7, 8, 9, 14, 37, 81, 143, 302
Offset: 1
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Do[If[PrimeQ[n^(n+1)+(n+1)^n-n(n+1)], Print[n]], {n, 1000}]
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is(n)=ispseudoprime(n^(n+1)+(n+1)^n-n*(n+1)) \\ Charles R Greathouse IV, Jun 12 2017
A099499
Primes of the form A007925(n)=n^(n+1)-(n+1)^n.
Original entry on oeis.org
17, 162287, 2486784401, 83695120256591, 84721522804414816904952398305908708011513455440403306207160333176150520399
Offset: 1
a(2)=162287 because A007925(A072179(2))=6^7-7^6=162287 is prime.
Cf.
A007925 n^(n+1)-(n+1)^n,
A072179 n^(n+1)-(n+1)^n is prime,
A099497 n^(n+1)-(n+1)^n is a semiprime,
A099498 semiprimes of the form n^(n+1)-(n+1)^n.
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[a: n in [0..50] | IsPrime(a) where a is n^(n+1)-(n+1)^n ]; // Vincenzo Librandi, Jul 18 2012
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Select[Table[n^(n+1)-(n+1)^n,{n,0,1000}],PrimeQ] (* Vincenzo Librandi, Jul 18 2012 *)
A084853
Numbers k such that k^(k+1) + (k+1)^k + k(k+1) is prime.
Original entry on oeis.org
1, 2, 3, 4, 5, 27
Offset: 1
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Do[If[PrimeQ[n^(n+1)+(n+1)^n+n(n+1)], Print[n]], {n, 0, 1000}]
Select[Range[30],PrimeQ[#^(#+1)+(#+1)^#+#(#+1)]&] (* Harvey P. Dale, May 28 2016 *)
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is(n)=ispseudoprime(n^(n+1)+(n+1)^n+n*(n+1)) \\ Charles R Greathouse IV, Jun 13 2017
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