A068145 -id:A068145 - cp's OEIS frontend

A068146 Primes of the form a^a - b^b.

3, 23, 229, 3121, 776887, 16774091, 275311670611, 302875106545597, 437893890380859119, 808793517812627212561, 827240252970236315921, 1978419655651397488675723, 20880467999847900922348207352551

Offset: 1

Amarnath Murthy, Feb 23 2002

nonn

			229 = 4^4 - 3^3 is a prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..76

Cf. A068145.

Select[ Union[ Flatten[ Table[a^a - b^b, {a, 1, 40}, {b, 1, a - 1} ]]], PrimeQ ]

v=[]; for(a=2, 400, forstep(b=a-1, 1, -2, if(ispseudoprime(t=a^a-b^b), v=concat(v,t)))); v \\ Charles R Greathouse IV, Feb 14 2011

The sum of the reciprocals converges to 0.38150016336280165719931278557192073226416041392427864458688292865... - Cino Hilliard, Dec 15 2002

Edited and extended by Robert G. Wilson v and Sascha Kurz, Mar 01 2002

A066846 Numbers of the form a^a + b^b, a >= b > 0.

Original entry on oeis.org

2, 5, 8, 28, 31, 54, 257, 260, 283, 512, 3126, 3129, 3152, 3381, 6250, 46657, 46660, 46683, 46912, 49781, 93312, 823544, 823547, 823570, 823799, 826668, 870199, 1647086, 16777217, 16777220, 16777243, 16777472, 16780341, 16823872, 17600759, 33554432

Offset: 0

Leroy Quet, Jan 20 2002

			28 is included because 28 = 1^1 + 3^3.

Cf. A000312, A218346, A218347.

Cf. A068145: primes of the form a^a + b^b.

nn = 10; Select[Union[Flatten[Table[a^a + b^b, {a, nn}, {b, a, nn}]]], # <= nn^nn + 1 &] (* T. D. Noe, Nov 15 2012 *)

Name improved by Alex Ratushnyak, Oct 26 2012

A133664 Primes of the form a^a + b^b + c^c + d^d.

Original entry on oeis.org

7, 13, 59, 311, 337, 769, 3137, 3389, 9631, 46691, 49783, 49789, 139969, 143093, 823601, 826673, 826699, 870253, 916859, 16777729, 16780369, 16780601, 16823903, 16827001, 17600761, 17600813, 18427427, 33557561, 33604213, 34378231

Offset: 1

Jonathan Vos Post, Dec 28 2007

			a(1) = 7 = 2^2 + 1^1 + 1^1 + 1^1 = 4 + 1 + 1 + 1 = 7.
a(2) = 13 = 4 + 4 + 4 + 1.
a(3) = 59 = 27 + 27 + 4 + 1.
a(4) = 311 = 256 + 27 + 27 + 1.
a(5) = 337 = 256 + 27 + 27 + 27.
a(6) = 769 = 256 + 256 + 256 + 1.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A000312, A068145.

Select[Union[ Flatten[Table[ a^a + b^b + c^c + d^d, {a, 1, 20}, {b, 1, a}, {c, 1, b}, {d, 1, c}]]], PrimeQ]

v=[];for(a=1,100, for(b=1,a, for(c=1,b, for(d=1,c, if(ispseudoprime(t=a^a+b^b+c^c+d^d),v=concat(v,t)))))); v \\ Charles R Greathouse IV, Feb 15 2011

A000040 INTERSECTION {A000312(a) + A000312(b) + A000312(c) + A000312(d)}.

A218346 Numbers of the form a^a + b^b, with a > b > 0.

Original entry on oeis.org

5, 28, 31, 257, 260, 283, 3126, 3129, 3152, 3381, 46657, 46660, 46683, 46912, 49781, 823544, 823547, 823570, 823799, 826668, 870199, 16777217, 16777220, 16777243, 16777472, 16780341, 16823872, 17600759, 387420490, 387420493, 387420516, 387420745, 387423614, 387467145

Offset: 1

Alex Ratushnyak, Oct 26 2012

nonn

Subsequence of A066846.

			a(1) = 2^2 + 1^1 = 5,
a(2) = 3^3 + 1^1 = 28,
a(3) = 2^2 + 3^3 = 31.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000312, A066846, A218347.

Cf. A068145: primes of the form a^a + b^b.

N:= 10^12: # for terms <= N
S:= NULL:
for m from 1 do v:= m^m; if v > N then break fi; S:= S,v od:
sort(convert(select(`<=`,{seq(seq(S[i]+S[j],j=i+1..m-1),i=1..m-1)},N),list)); # Robert Israel, Aug 10 2020

nn = 10; Select[Union[Flatten[Table[a^a + b^b, {a, nn}, {b, a + 1, nn}]]], # <= nn^nn + 1 &] (* T. D. Noe, Nov 15 2012 *)

from itertools import count, takewhile
def aupto(lim):
  pows = list(takewhile(lambda x: x < lim, (i**i for i in count(1))))
  sums = (aa+bb for i, bb in enumerate(pows) for aa in pows[i+1:])
  return sorted(set(s for s in sums if s <= lim))
print(aupto(387467145))  # Michael S. Branicky, May 28 2021

A133663 Primes of the form a^a + b^b + c^c.

Original entry on oeis.org

3, 29, 3637, 6277, 46687, 826669, 16777499, 16780597, 404197709, 775664521, 10000003129, 10387420493, 285311673737, 305311670611, 8916100448513, 8916487869001, 8926101271799, 17832200896513, 17832200899637

Offset: 1

Jonathan Vos Post, Dec 28 2007

nonn

			a(1) = 3 = 1 + 1 + 1 is prime.
a(2) = 29 = 27 + 1 + 1 is prime.
a(3) = 3637 = 3125 + 256 + 256 is prime.
a(4) = 6277 = 3125 + 3125 + 27 is prime.
a(5) = 46687 = 46656 + 27 + 4 is prime.
a(6) = 826669 = 823543 + 3125 + 1 is prime.
a(7) = 16777499 = 16777216 + 256 + 27 is prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..7093

Cf. A000040, A000312, A068145.

Select[Union[ Flatten[Table[ a^a + b^b + c^c, {a, 1, 40}, {b, 1, a}, {c, 1, b}]]], PrimeQ]

v=[];for(a=1,386, for(b=1,a, for(c=1,b, if(ispseudoprime(t=a^a+b^b+c^c),v=concat(v,t))))); v \\ Charles R Greathouse IV, Feb 18 2011

A000040 INTERSECTION {A000312(i) + A000312(j) + A000312(k)}.

A136292 Primes of the form a^a + b^b + c^c + d^d + e^e.

Original entry on oeis.org

5, 11, 17, 31, 37, 43, 83, 89, 109, 263, 269, 521, 541, 547, 593, 773, 1051, 3181, 3187, 3413, 3691, 6763, 9377, 9403, 9887, 12527, 46663, 46993, 49787, 50549, 52937, 53189, 93851, 96697, 99563, 139999, 823547, 823553, 823573, 823651, 823831

Offset: 1

Jonathan Vos Post, Apr 11 2008

			a(1) = 5 = 1^1 + 1^1 + 1^1 + 1^1 + 1^1.
a(2) = 11 = 1^1 + 1^1 + 1^1 + 2^2 + 2^2.
a(3) = 17 = 1^1 + 2^2 + 2^2 + 2^2 + 2^2.
a(4) = 31 = 1^1 + 1^1 + 1^1 + 1^1 + 3^3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A000312, A068145, A133664.

Select[Union[ Flatten[Table[ a^a + b^b + c^c + d^d + e^e, {a, 1, 20}, {b, 1, a}, {c, 1, b}, {d, 1, c}, {e, 1, d}]]], PrimeQ]

v=[];for(a=1,50, for(b=1,a, for(c=1,b, for(d=1,c, for(e=1,d, if(ispseudoprime(t=a^a+b^b+c^c+d^d+e^e),v=concat(v,t))))))); v \\ Charles R Greathouse IV, Feb 15 2011

A000040 INTERSECTION {A000312(a) + A000312(b) + A000312(c) + A000312(d) + A000312(e)}.

A218347 Numbers of the form a^a + b^b, a>=b>=0.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 27, 28, 31, 54, 256, 257, 260, 283, 512, 3125, 3126, 3129, 3152, 3381, 6250, 46656, 46657, 46660, 46683, 46912, 49781, 93312, 823543, 823544, 823547, 823570, 823799, 826668, 870199, 1647086, 16777216, 16777217, 16777220, 16777243, 16777472, 16780341

Offset: 1

Alex Ratushnyak, Oct 26 2012

nonn

The subsequence of primes is A068145.

Cf. A000312, A066846, A068145, A218346.

A136294 Primes of the form a^a + b^b + c^c + d^d + e^e + f^f.

Original entry on oeis.org

41, 47, 61, 67, 113, 139, 293, 313, 571, 797, 823, 1307, 3191, 3391, 3463, 3643, 3947, 4153, 6257, 6263, 6793, 7019, 9433, 12757, 15629, 15881, 46687, 46919, 46997, 47681, 49811, 49843, 50069, 50321, 53419, 56039, 56543, 59183, 93319

Offset: 1

Jonathan Vos Post, Apr 11 2008

			a(1) = 41 = 1^1 + 1^1 + 2^2 + 2^2 + 2^2 + 3^3.
a(2) = 47 = 2^2 + 2^2 + 2^2 + 2^2 + 2^2 + 3^3.
a(3) = 61 = 1^1 + 1^1 + 1^1 + 2^2 + 3^3 + 3^3.
a(4) = 67 = 1^1 + 2^2 + 2^2 + 2^2 + 3^3 + 3^3.
a(5) = 113 = 1^1 + 2^2 + 3^3 + 3^3 + 3^3 + 3^3.

A136294 Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A000312, A068145, A133664, A136292

Select[Union[ Flatten[Table[ a^a + b^b + c^c + d^d + e^e + f^f, {a, 1, 20}, {b, 1, a}, {c, 1, b}, {d, 1, c}, {e, 1, d}, {f, 1, e}]]], PrimeQ]

v=[]; for(a=1, 30, for(b=1, a, for(c=1, b, for(d=1, c, for(e=1, d, for(f=1, e, if(ispseudoprime(t=a^a+b^b+c^c+d^d+e^e+f^f), v=concat(v, t)))))))); #v \\ Charles R Greathouse IV, Feb 15 2011

A000040 INTERSECTION {A000312(a) + A000312(b) + A000312(c) + A000312(d) + A000312(e) + A000312(f)}.

A140052 Indices m such that A114850(m)+A114850(k) is prime for some k

Original entry on oeis.org

6, 9, 9, 19, 20, 25, 33, 38, 40, 59, 69, 76, 99, 111, 126, 141, 147, 167, 188, 202, 211, 211, 220, 238, 263, 264, 279, 284, 297, 329, 336, 354, 407, 407, 407, 410, 426, 540, 568, 652, 683, 696, 769, 780, 948, 951

Offset: 1

Jonathan Vos Post, May 03 2008

The associated primes b(n), which grow too quickly for many to be given as a sequence themselves, are {primes of the form A114850(a) + A114850(b)} = {primes of the form A114850(a) + A114850(b)} and begin as follows. b(1) = 437893890380859631 = 256 + 437893890380859375 = 4^4 + 15^15 = semiprime(1)^semiprime(1) + semiprime(6)^semiprime(6).
b(2) = 88817841970012523233890533447265881 = 256 + 88817841970012523233890533447265625 = 4^4 + 25^25 = semiprime(1)^semiprime(1) + semiprime(9)^semiprime(9).
b(3) = 46656 + 88817841970012523233890533447265625 = 6^6 + 24^25 = semiprime(2)^semiprime(2) + semiprime(9)^semiprime(9). This is to A068145 "Primes of the form a^a + b^b" as A001358 semiprimes is to A000040 primes; and as A114850 "(n-th semiprime)^(n-th semiprime)" is to A051674 "(n-th prime)^(n-th prime)."
M. F. Hasler gave the present definition which allows us to list merely the indices, which in the 3 examples above, are [6, 1],[9, 1],[9, 2]. The first 13 [m,k] value pairs are (as found by M. F. Hasler as an extension) are [6, 1], [9, 1], [9, 2], [19, 5], [20, 8], [25, 7], [33, 11], [38, 6], [40, 33], [59, 14], [69, 62], [76, 57], [99, 22]. Hence our sequence begins a(1) = 6, a(2) = 9, a(3) = 9. For the sequence of corresponding k values {1, 1, 2, 5, 8, ...}, see A140053.

			a(1) = 6 because semiprime(6)^semiprime(6) + semiprime(1)^semiprime(1) = 15^15 + 4^4 = 437893890380859375 + 256 = 437893890380859631 is prime.

Cf. A000040, A001358, A051674, A068145, A114850, A137701, A140053.

? t=0;A001358=vector(100,i,until(bigomega(t++)==2,);t); ? for(i=1,#A001358, for(j=1,i-1, ispseudoprime(A001358[i]^A001358[i]+A001358[j]^A001358[j]) | next; print1([i,j]",")))

A001358(a(n))^A001358(a(n)) + A001358(A140053(n))^A001358(A140053(n)) is prime.

a(14)-a(46) from Donovan Johnson, Nov 11 2008

A140053 Indices k such that A114850(m)+A114850(k) is prime for some m>k.

Original entry on oeis.org

1, 1, 2, 5, 8, 7, 11, 6, 33, 14, 62, 57, 22, 7, 86, 61, 28, 70, 66, 134, 77, 131, 107, 58, 161, 252, 240, 52, 155, 32, 152, 322, 167, 200, 284, 258, 28, 173, 95, 563, 369, 57, 58, 126, 113, 369

Offset: 1

Jonathan Vos Post, May 03 2008

The associated primes b(n), which grow too quickly for many to be given as a sequence themselves, are {primes of the form A114850(a) + A114850(b)} = {primes of the form A114850(a) + A114850(b)} and begin as follows. b(1) = 437893890380859631 = 256 + 437893890380859375 = 4^4 + 15^15 = semiprime(1)^semiprime(1) + semiprime(6)^semiprime(6).
b(2) = 88817841970012523233890533447265881 = 256 + 88817841970012523233890533447265625 = 4^4 + 25^25 = semiprime(1)^semiprime(1) + semiprime(9)^semiprime(9).
b(3) = 46656 + 88817841970012523233890533447265625 = 6^6 + 24^25 = semiprime(2)^semiprime(2) + semiprime(9)^semiprime(9). This is to A068145 "Primes of the form a^a + b^b" as A001358 semiprimes is to A000040 primes; and as A114850 "(n-th semiprime)^(n-th semiprime)" is to A051674 "(n-th prime)^(n-th prime)."
M. F. Hasler gave the present definition which allows us to list merely the indices, which in the 3 examples above, are [6, 1],[9, 1],[9, 2]. The first 13 [m,k] value pairs are (as found by M. F. Hasler as an extension) are [6, 1], [9, 1], [9, 2], [19, 5], [20, 8], [25, 7], [33, 11], [38, 6], [40, 33], [59, 14], [69, 62], [76, 57], [99, 22]. Hence our sequence begins a(1) = 6, a(2) = 9, a(3) = 9. For the sequence of corresponding k values {6, 9, 9, 19, 20, ...}, see A140052.

			a(1) = 1 because semiprime(6)^semiprime(6) + semiprime(1)^semiprime(1) = 15^15 + 4^4 = 437893890380859375 + 256 = 437893890380859631 is prime.

Cf. A000040, A001358, A051674, A068145, A114850, A137701, A140052.

? t=0;A001358=vector(100,i,until(bigomega(t++)==2,);t); ? for(i=1,#A001358, for(j=1,i-1, ispseudoprime(A001358[i]^A001358[i]+A001358[j]^A001358[j]) | next; print1([i,j]",")))

A001358(a(n))^A001358(a(n)) + A001358(A140052(n))^A001358(A140052(n)) is prime.

a(14)-a(46) from Donovan Johnson, Nov 11 2008

cp's OEIS Frontend

A068146 Primes of the form a^a - b^b.

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A066846 Numbers of the form a^a + b^b, a >= b > 0.

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A133664 Primes of the form a^a + b^b + c^c + d^d.

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A218346 Numbers of the form a^a + b^b, with a > b > 0.

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A133663 Primes of the form a^a + b^b + c^c.

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A136292 Primes of the form a^a + b^b + c^c + d^d + e^e.

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A218347 Numbers of the form a^a + b^b, a>=b>=0.

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A136294 Primes of the form a^a + b^b + c^c + d^d + e^e + f^f.

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