A133531 -id:A133531 - cp's OEIS frontend

A133543 Sum of seventh powers of five consecutive primes.

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 *)

A133540 Sum of fourth powers of five consecutive primes.

Original entry on oeis.org

17764, 46309, 129749, 259445, 536885, 1229525, 2124485, 3915125, 6610565, 9749525, 13921925, 20888885, 31132085, 42152165, 58884485, 79416485, 99924245, 126756965, 160369445, 202960565, 266078165, 341740325, 415341125, 498962405

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=17764 because 2^4+3^4+5^4+7^4+11^4=17764.

Cf. A034964, A133526, A133531, A133535, A133539, A133541, A133542, A133543.

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

a(n) = A133526(n) + A030514(n+4). - Michel Marcus, Nov 09 2013

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

A306214 Numbers that are the sum of fourth powers of three distinct positive integers in arithmetic progression.

Original entry on oeis.org

98, 353, 707, 962, 1568, 2177, 2658, 3107, 4322, 4737, 5648, 7187, 7793, 7938, 9587, 11312, 12657, 13058, 15392, 15938, 17123, 19362, 20657, 23153, 23603, 25088, 28593, 30963, 31202, 32738, 34832, 35747, 40962, 42528, 45233, 45377, 49712, 49763, 54722, 57153, 57267, 61250, 63938, 67667, 69152

Offset: 1

Antonio Roldán, Jan 29 2019

nonn

The remainder of a(n) divided by 16 is less than 5. - Jinyuan Wang, Feb 03 2019

			353 = 2^4 + 3^4 + 4^4, with 3 - 2 = 4 - 3 = 1;
7187 = 1^4 + 5^4 + 9^4, with 5 - 1 = 9 - 5 = 4.

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

Cf. A000583, A126657, A133531, A190176, A306212, A306213.

N:= 10^5: # for all terms <= N
Res:= NULL:
for a from 1 to floor((N/3)^(1/4)) do
  for d from 1 do
    v:= a^4 + (a+d)^4 + (a+2*d)^4;
    if v > N then break fi;
    Res:= Res, v
  od
od:
sort(convert({Res},list)); # Robert Israel, Feb 17 2019

for(n=1, 70000, k=(n/3)^(1/4); a=2; v=0; while(a<=k&&v==0, d=sqrt(sqrt(2*n+30*a^4)/2-3*a^2); if(d==truncate(d)&&d>=1&&d<=a-1, v=1; print1(n,", ")); a+=1))

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}]

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

A133543 Sum of seventh powers of five consecutive primes.

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A133540 Sum of fourth 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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A306214 Numbers that are the sum of fourth powers of three distinct positive integers in arithmetic progression.

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A131686 Sum of squares of five consecutive primes.

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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.

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