A133530 -id:A133530 - cp's OEIS frontend

A133539 Sum of third powers of five consecutive primes.

1834, 4023, 8909, 15643, 27467, 50525, 78119, 123859, 185921, 253261, 332695, 451781, 606507, 764567, 985823, 1239911, 1480051, 1767711, 2112517, 2516723, 3071485, 3712769, 4312457, 4965713, 5555773, 6085997, 7104079, 8259443

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=1834 because 2^3+3^3+5^3+7^3+11^3=1834.

Cf. A034963, A133524, A133525, A133526, A133527, A133528, A133529, A133530, A133531, A133532, A069484, A133534, A133535, A133536, A133537, A034964, A133538, A133540, A133541, A133542, A133543.

a = 3; Table[Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a + Prime[n + 3]^a + Prime[n + 4]^a, {n, 1, 100}]
Total[#^3]&/@Partition[Prime[Range[40]],5,1] (* Harvey P. Dale, May 01 2013 *)

a(n) = A133525(n) + A030078(n+4). - Michel Marcus, Nov 09 2013

A133543 Sum of seventh powers of five consecutive primes.

Original entry on oeis.org

20391154, 83139543, 493476029, 1387269643, 4791271547, 22021660685, 49471526279, 143993064739, 337853466881, 606267252541, 1095640496695, 2242839022421, 4636558630107, 7584547192247, 13373440186463

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=20391154 because 2^7+3^7+5^7+7^7+11^7=20391154

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A034963, A133524, A133525, A133526, A133527, A133528, A133529, A133530, A133531, A133532, A069484, A133534, A133535, A133536, A133537, A034964, A133538, A133539, A133540, A133541, A133542.

a = 7; Table[Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a + Prime[n + 3]^a + Prime[n + 4]^a, {n, 1, 100}]
Total/@Partition[Prime[Range[20]]^7,5,1] (* Harvey P. Dale, Mar 05 2022 *)

A133541 Sum of fifth powers of five consecutive primes.

Original entry on oeis.org

181258, 552519, 1972133, 4445107, 10864643, 31214741, 59472599, 127396699, 240776801, 381348901, 590182759, 979749101, 1625329443, 2354069543, 3557186207, 5132070551, 6786946651, 9149078751, 12243523093, 16477457435

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=181258 because 2^5+3^5+5^5+7^5+11^5=181258.

Cf. A034963, A133524, A133525, A133526, A133527, A133528, A133529, A133530, A133531, A133532, A069484, A133534, A133535, A133536, A133537, A034964, A133538, A133539, A133540, A133542, A133543.

a = 5; Table[Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a + Prime[n + 3]^a + Prime[n + 4]^a, {n, 1, 100}]
Total/@Partition[Prime[Range[30]]^5,5,1] (* Harvey P. Dale, Dec 02 2017 *)

a(n) = A133527(n) + A050997(n+4). - Michel Marcus, Nov 09 2013

A133542 Sum of sixth powers of five consecutive primes.

Original entry on oeis.org

1905628, 6732373, 30869213, 77899469, 225817709, 818869469, 1701546341, 4243135181, 8946193541, 15119520701, 25303912709, 46580770157, 86195577389, 132965847509, 217102866629, 334423935221, 463593800381, 664500722261

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=1905628 because 2^6+3^6+5^6+7^6+11^6=1905628.

Cf. A034963, A133524, A133525, A133526, A133527, A133528, A133529, A133530, A133531, A133532, A069484, A133534, A133535, A133536, A133537, A034964, A133538, A133539, A133540, A133541, A133543.

a = 6; Table[Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a + Prime[n + 3]^a + Prime[n + 4]^a, {n, 1, 100}]
Total/@(Partition[Prime[Range[30]],5,1]^6)  (* Harvey P. Dale, Mar 13 2011 *)

a(n) = A133528(n) + A030516(n+4). - Michel Marcus, Nov 09 2013

A164129 Primes that are the sums of cubes of three consecutive primes.

Original entry on oeis.org

66347, 199081, 332207, 581237, 733123, 1047691, 2647943, 3612799, 7505063, 10620793, 22715029, 32180581, 36355409, 60621553, 76753387, 98784001, 116319367, 147594259, 162516943, 177616529, 216596449, 252725563, 343774313

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 10 2009

nonn

			23^3+29^3+31^3=66347, 37^3+41^3+43^3=199081, 43^3+47^3+53^3=332207,..

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A133530, A034962

lst={};Do[p=Prime[n]^3+Prime[n+1]^3+Prime[n+2]^3;If[PrimeQ[p],AppendTo[lst,p]],{n,6!}];lst
Select[Total[#^3]&/@Partition[Prime[Range[100]],3,1],PrimeQ] (* Harvey P. Dale, Nov 04 2015 *)

Edited by Charles R Greathouse IV, Oct 12 2009

A259772 Primes p such that p^3 + q^2 + r is also prime, where p,q,r are consecutive primes.

Original entry on oeis.org

3, 17, 19, 43, 53, 89, 107, 149, 293, 401, 439, 449, 659, 809, 821, 937, 1009, 1031, 1091, 1097, 1123, 1163, 1181, 1259, 1277, 1367, 1427, 1657, 1721, 1777, 1789, 1811, 1987, 2027, 2063, 2207, 2333, 2417, 2503, 2657, 2713, 3067, 3079, 3083, 3251, 3389, 3491, 3527

Offset: 1

K. D. Bajpai, Jul 05 2015

nonn

			a(2) = 17 is prime: 17^3 + 19^2 + 23 = 5297 which is also prime.
a(3) = 19 is prime: 19^3 + 23^2 + 29 = 7417 which is also prime.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000040, A034962, A133529, A133530, A258269, A304292.

[p: p in PrimesUpTo (3000) | IsPrime(k) where k is (p^3 + NextPrime(p)^2 + NextPrime(NextPrime(p)))];

select(n -> isprime(n) and isprime((n)^3+nextprime(n)^2+nextprime(nextprime((n)))), [seq(n, n=1..10000)]);

Select[Prime[Range[1000]], PrimeQ[#^3 + NextPrime[#]^2 + NextPrime[NextPrime[#]]]&]
Select[Partition[Prime[Range[500]],3,1],PrimeQ[#[[1]]^3+ #[[2]]^2+ #[[3]]]&][[All,1]] (* Harvey P. Dale, Dec 23 2021 *)

forprime(p=1, 3000, q=nextprime(p+1); r=nextprime(q+1); k=(p^3 + q^2 + r); if(isprime(k), print1(p,", ")))

A131686 Sum of squares of five consecutive primes.

Original entry on oeis.org

208, 373, 653, 989, 1469, 2189, 2981, 4061, 5381, 6701, 8069, 9917, 12029, 14069, 16709, 19541, 22061, 24821, 27989, 31421, 35789, 40661, 45029, 49589, 53549, 56909, 62837, 69389, 76709, 84149, 93581, 100253, 107741, 115541, 124109, 131837

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=208 because 2^2+3^2+5^2+7^2+11^2=208

Cf. A034963, A133524, A133525, A133526, A133527, A133528, A133529, A133530, A133531, A133532, A069484, A133534, A133535, A133536, A133537, A034964, A133539, A133540, A133541, A133542, A133543.

a = 2; Table[Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a + Prime[n + 3]^a + Prime[n + 4]^a, {n, 1, 100}]

A258269 Primes of the form p^3 + q^2 + r, where p, q, r are consecutive primes.

Original entry on oeis.org

59, 5297, 7417, 81769, 152419, 714479, 1237037, 3330907, 25248317, 64648901, 84801217, 90728159, 286628773, 530133671, 554065817, 823543381, 1028270917, 1096980919, 1299792317, 1321357391, 1417523659, 1574410169, 1648622903, 1997248987, 2084078057, 2556384373

Offset: 1

K. D. Bajpai, May 25 2015

nonn

			a(1) = 59 is prime of the form 3^3 + 5^2 + 7.
a(2) = 5297 is prime of the form 17^3 + 19^2 + 23.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040, A034962, A133529, A133530.

[k: p in PrimesUpTo (3000) | IsPrime(k) where k is (p^3 + NextPrime(p)^2 + NextPrime(NextPrime(p)))];

A258269:= n-> (ithprime(n)^3+ithprime(n+1)^2+ithprime(n+2)): select(isprime, [seq((A258269(n), n=1..5000))]);

Select[Table[p = Prime[n]; q = NextPrime[p]; r = NextPrime[q]; p^3 + q^2 + r, {n, 5000}], PrimeQ]

forprime(p=1, 5000, q=nextprime(p+1); r=nextprime(q+1);  k=(p^3 + q^2 + r); if(isprime(k), print1(k,", ")))

A164130 Sums s of squares of three consecutive primes, such that s-+2 are primes.

Original entry on oeis.org

195, 5739, 18459, 32259, 33939, 60291, 74019, 169491, 187131, 244899, 276819, 388179, 783531, 902139, 3588339, 5041491, 5145819, 5193051, 8687091, 9637491, 10227291, 10910019, 11341491, 11757339, 14834379, 15354651, 16115091

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 10 2009

nonn

			5^2 + 7^2 + 11^2 = 195 is a sum of the squared consecutive primes 5, 7 and 11, and 193 and 197 are primes, so 195 is a member of the sequence.

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

Cf. A075893, A133529, A133530, A034962, A164129

q:= 2: r:= 3: R:= NULL: count:= 0:
while count < 100 do
  p:= q; q:= r; r:= nextprime(r);
  s:= p^2+q^2+r^2;
  if isprime(s-2) and isprime(s+2) then
    count:= count+1; R:= R,s;
  fi;
od:
R; # Robert Israel, Apr 21 2023

lst={};Do[p=Prime[n]^2+Prime[n+1]^2+Prime[n+2]^2;If[PrimeQ[p-2]&&PrimeQ[p+2], AppendTo[lst,p]],{n,8!}];lst

A133529 INTERSECT A087679. - R. J. Mathar, Aug 27 2009

Comment turned into example by R. J. Mathar, Aug 27 2009

A176613 Smallest prime p of three consecutive primes such that the sum of their n-th powers is prime, or 0 if such a prime does not exist.

Original entry on oeis.org

2, 5, 3, 23, 0, 11, 0, 5, 0, 23, 3, 137, 0, 5, 3, 89, 0, 71, 0, 17, 0, 23, 0, 23, 3, 131, 3, 419, 0, 31, 0, 859, 0, 31, 0, 127, 0, 11, 0, 359, 0, 31, 0, 347, 0, 509, 0, 137, 0, 193, 0, 769, 0, 23, 0, 17

Offset: 0

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 21 2010

nonn

Let p = prime(i), q = prime(i+1), r = prime(i+2).
(*) p^n + q^n + r^n has to be a prime.
When n is even and p > 3, then (*) is composite because primes greater than 3 are either of form 6k-1 or 6k+1 for some k. Hence, squares (or any even power) of such a prime has the form 6k+1. Adding three such even powers will produce a number of the form 6k+3, which is divisible by 3.
When n is even and p = 3, sequence A160773 gives the even n for which 3^n + 5^n + 7^n is prime.

			5 + 7 + 11 = 23 = prime(9); 3^2 + 5^2 + 7^2 = 83 = prime(23); 23^3 + 29^3 + 31^3 = 66347 = prime(6616).

Robert Israel, Table of n, a(n) for n = 0..500

Cf. A133530, A133531, A133532, A133533, A160773.

f:= proc(n) local p,q,r;
  if n::even then
    if isprime(3^n+5^n+7^n) then return 3
    else return 0
    fi
  fi;
  p:= 2: q:= 3: r:= 5:
  while not isprime(p^n + q^n + r^n) do
    p:= q; q:= r; r:= nextprime(r)
  od;
  p
end proc:
f(0):= 2:
map(f, [$0..100]);

a(0) term added by T. D. Noe, Nov 23 2010

cp's OEIS Frontend

A133539 Sum of third powers of five consecutive primes.

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A133543 Sum of seventh powers of five consecutive primes.

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A133541 Sum of fifth powers of five consecutive primes.

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A133542 Sum of sixth powers of five consecutive primes.

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A164129 Primes that are the sums of cubes of three consecutive primes.

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A259772 Primes p such that p^3 + q^2 + r is also prime, where p,q,r are consecutive primes.

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A131686 Sum of squares of five consecutive primes.

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A258269 Primes of the form p^3 + q^2 + r, where p, q, r are consecutive primes.

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A164130 Sums s of squares of three consecutive primes, such that s-+2 are primes.