cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A084951 Primes in A075893: Primes of the form (p^2+q^2+r^2)/3, where p,q,r are 3 consecutive primes.

Original entry on oeis.org

113, 193, 577, 1913, 2833, 10753, 44617, 48593, 54617, 69193, 74177, 78593, 86729, 102673, 107873, 122273, 156577, 183497, 214993, 228233, 247697, 308809, 334513, 414313, 581177, 602753, 617369, 636353, 691697, 861857, 1408993, 1786097
Offset: 1

Views

Author

Hugo Pfoertner, Jun 14 2003

Keywords

Comments

With the exception of 2^2+3^2+5^2=38 and 3^2+5^2+7^2=83 all sums of squares of 3 consecutive primes are divisible by 3 because mod(p^2,3)=1 for all primes p>3.

Examples

			a(1)=113 because (7^2+11^2+13^2)/3=(49+121+169)/3=339/3=113 is prime.

Links

Crossrefs

Cf. A075893, A084952, A133529, A133940.

Programs

  • Mathematica
    b = {}; a = 2; Do[k = (Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a)/3; If[PrimeQ[k], AppendTo[b, n]], {n, 1, 200}]; b (* Artur Jasinski, Sep 30 2007 *)
  • PARI
    v=vector(10000);i=0;p=5;q=7; forprime(r=8,1e8,if(isprime(t=(p^2+q^2+r^2)/3), v[i++]=t; if(i==#v,return)); p=q; q=r) \\ Charles R Greathouse IV, Feb 14 2011

Extensions

Edited by N. J. A. Sloane, Jun 30 2008 at the suggestion of R. J. Mathar.

A133533 Sum of sixth powers of three consecutive primes.

Original entry on oeis.org

16418, 134003, 1904835, 6716019, 30735939, 76010259, 219219339, 789905091, 1630362891, 4048053411, 8203334331, 13637193699, 21850682619, 39264939507, 75124110099, 115865269131, 184159290171, 270079040451, 369892892379
Offset: 1

Views

Author

Artur Jasinski, Sep 14 2007

Keywords

Examples

			a(1)=16418 because 2^6+3^6+5^6=16418.

Links

Crossrefs

Cf. A034963, A133524, A133525, A133526, A133527, A133528, A133529, A133530, A133531, A133532.

Programs

  • Maple
    L:= [seq(ithprime(i)^6,i=1..100)]:
L[1..-3]+L[2..-2]+L[3..-1]; # Robert Israel, Jun 28 2018
  • Mathematica
    a = 6; Table[Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a, {n, 1, 100}]
Total/@(Partition[Prime[Range[25]],3,1]^6)  (* Harvey P. Dale, Mar 29 2011 *)

Formula

a(n) = A133537(n) + A030516(n+2). - Michel Marcus, Nov 09 2013

A133539 Sum of third powers of five consecutive primes.

Original entry on oeis.org

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

Views

Author

Artur Jasinski, Sep 14 2007

Keywords

Examples

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

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

Artur Jasinski, Sep 14 2007

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

A133559 Primes which have a partition as the sum of squares of five consecutive primes.

Original entry on oeis.org

373, 653, 5381, 6701, 8069, 19541, 24821, 53549, 56909, 69389, 93581, 107741, 131837, 184901, 196661, 237821, 252509, 344021, 370661, 395069, 498989, 609269, 783701, 1055429, 1174781, 1239341, 1492637, 1576229, 1713989, 1749149, 2024261
Offset: 1

Views

Author

Artur Jasinski, Sep 16 2007

Keywords

Comments

For sums of squares of two consecutive primes, only 2^2 + 3^2 = 13 is prime.
For sums of squares of three consecutive primes (A133529), it seems that only 83 belonging (checked for starting primes prime(k) for all k < 1000000).
Sums of squares of four (and all even numbers of) consecutive primes are even numbers with the exception of 2^2 + 3^2 + 5^2 + 7^2 = 87 = 3*29, which is not prime.

Examples

			a(1)=373 because prime(2)^2 + prime(3)^2 + prime(4)^2 + prime(5)^2 + prime(6)^2 = 3^2 + 5^2 + 7^2 + 11^2 + 13^2 = 373 is prime. [Corrected by _Jonathan Sondow_, Nov 04 2015]

Links

Crossrefs

Cf. A133529, A133538, A133560.

Programs

  • Mathematica
    b = {}; a = 2; Do[k = Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a + Prime[n + 3]^a + Prime[n + 4]^a; If[PrimeQ[k], AppendTo[b, k]], {n, 1, 100}]; b
Select[Total/@Partition[Prime[Range[200]]^2,5,1],PrimeQ] (* Harvey P. Dale, Apr 07 2015 *)

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

Views

Author

Artur Jasinski, Sep 14 2007

Keywords

Examples

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

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

Artur Jasinski, Sep 14 2007

Keywords

Examples

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

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

A075893 Average of three successive primes squared, (prime(n)^2+prime(n+1)^2+prime(n+2)^2)/3, n>=3.

Original entry on oeis.org

65, 113, 193, 273, 393, 577, 777, 1057, 1337, 1633, 1913, 2289, 2833, 3337, 3897, 4417, 4953, 5537, 6153, 7017, 8073, 9177, 10073, 10753, 11313, 12033, 13593, 15353, 17353, 18417, 20097, 21441, 23217, 24673, 26369, 28129, 29953, 31577, 33761
Offset: 3

Views

Author

Zak Seidov, Oct 17 2002

Keywords

Comments

Unlike the average of three successive primes, the average of three successive primes (greater than 3) squared is always integral.
A133529(n)/3, n >= 3. - Artur Jasinski, Sep 30 2007

Examples

			a(3)=65 because (prime(3)^2+prime(4)^2+prime(5)^2)/3=(5^2+7^2+11^2)/3=65.

Links

Crossrefs

Cf. A133529, A084951, A133940.

Programs

  • Magma
    [(NthPrime(n)^2+NthPrime(n+1)^2+NthPrime(n+2)^2)/3: n in [3..50]]; // Vincenzo Librandi, Aug 21 2018
  • Mathematica
    b = {}; a = 2; Do[k = (Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a)/3; AppendTo[b, k], {n, 3, 50}]; b (* Artur Jasinski, Sep 30 2007 *)
Mean[#]&/@Partition[Prime[Range[3,50]]^2,3,1] (* Harvey P. Dale, Jun 09 2013 *)

Formula

a(n) = (prime(n)^2+prime(n+1)^2+prime(n+2)^2)/3, n>=3.

Extensions

Edited by N. J. A. Sloane, Jun 30 2008 at the suggestion of R. J. Mathar

A133560 Primes which have a partition as the sum of squares of seven consecutive primes.

Original entry on oeis.org

1543, 3271, 4519, 7591, 9439, 11719, 23599, 39631, 45319, 51031, 56599, 90199, 151471, 173359, 210319, 222919, 235159, 261463, 313879, 367711, 402511, 459223, 478831, 499711, 610567, 634327, 732967, 760519, 819319, 883087, 939439, 968959
Offset: 1

Views

Author

Artur Jasinski, Sep 16 2007

Keywords

Comments

For sums of squares of two consecutive primes, only 2^2 + 3^2 = 13 is prime.
For sums of squares of three consecutive primes (see A133529), it seems that only 3^2 + 5^2 + 7^2 = 83 is prime.
Sums of squares of four (and all even numbers of) consecutive primes are even numbers except for 2^2 + 3^2 + 5^2 + 7^2 = 87 = 3*29, which is not prime.
For sums of squares of five of consecutive primes see A133559.
For every prime p > 3, p^2 mod 3 = 1, so the sum of the squares of any 3 such primes will be divisible by 3. - Jon E. Schoenfield, Sep 04 2023

Examples

			a(3)=4519 because 13^2 + 17^2 + 19^2 + 23^2 + 29^2 + 31^2 + 37^2 = 4519 is prime.

Links

Crossrefs

Cf. A133529, A133538, A133559, A133561.

Programs

  • Maple
    select(isprime,[seq(add(ithprime(n+k)^2,k=0..6),n=1..80)]); # Muniru A Asiru, Jul 19 2018
  • Mathematica
    b = {}; a = 2; Do[k = Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a + Prime[n + 3]^a + Prime[n + 4]^a + Prime[n + 5]^a + Prime[n + 6]^a; If[PrimeQ[k], AppendTo[b, k]], {n, 1, 100}]; b
(* Second program: *)
Select[Map[Total, Partition[Prime@ Range@ 80, 7, 1]^2], PrimeQ] (* Michael De Vlieger, Jul 20 2018 *)

A133561 Numbers n for which sum of squares of seven consecutive primes(n,n+1,n+2,n+3,n+4,n+5,n+6) is prime.

Original entry on oeis.org

3, 5, 6, 8, 9, 10, 14, 18, 19, 20, 21, 26, 32, 34, 37, 38, 39, 41, 44, 47, 49, 52, 53, 54, 59, 60, 63, 64, 66, 68, 70, 71, 75, 83, 88, 89, 91, 92, 97, 100, 107, 108, 110, 112, 113, 117, 122, 125, 128, 129, 131, 135, 141, 142, 150, 151, 157, 158, 165, 168, 169, 178, 183
Offset: 1

Views

Author

Artur Jasinski, Sep 16 2007

Keywords

Comments

For sum of squares of two consecutive primes only 2^2+3^2=13 is prime.
For sum of squares of three consecutive primes A133529 seems that only 83 belonging (checked for all n<1000000).
Sums of squares of four (and all even number) of consecutive primes are even numbers with exception n=1 but 2^2+3^2+5^2+7^2=87=3*29 is not prime.
Sums of squares of five of consecutive primes A133559.
Sums of squares of seven of consecutive primes A133562.

Examples

			a(3)=6 because prime(6)^2+prime(7)^2+prime(8)^2+prime(9)^2+prime(10)^2+prime(11)^2+prime(12)^2 = 13^2+17^2+19^2+23^2+29^2+31^2+37^2=4519 is prime.

Links

Crossrefs

Cf. A133538, A133559, A133562.

Programs

  • Maple
    select(n->isprime(add(ithprime(n+k)^2,k=0..6)),[$1..200]); # Muniru A Asiru, Jul 28 2018
  • Mathematica
    b = {}; a = 2; Do[k = Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a + Prime[n + 3]^a + Prime[n + 4]^a + Prime[n + 5]^a + Prime[n + 6]^a; If[PrimeQ[k], AppendTo[b, n]], {n, 1, 100}]; b
  • PARI
    is(n) = ispseudoprime(sum(i=0, 6, prime(n+i)^2)) \\ Felix Fröhlich, Jul 28 2018
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