A069484 -id:A069484 - cp's OEIS frontend

A133541 Sum of fifth powers of five consecutive primes.

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

A069486 a(n) = 2prime(n)prime(n+1).

Original entry on oeis.org

12, 30, 70, 154, 286, 442, 646, 874, 1334, 1798, 2294, 3034, 3526, 4042, 4982, 6254, 7198, 8174, 9514, 10366, 11534, 13114, 14774, 17266, 19594, 20806, 22042, 23326, 24634, 28702, 33274, 35894, 38086, 41422, 44998

Offset: 1

Reinhard Zumkeller, Mar 29 2002

nonn

a(n) = 2*A006094(n);

together with A069482(n) and A069484(n) a Pythagorean triangle is formed with area = A069487(n).

Cf. A006094, A069482, A069484, A069487.

2Times@@#&/@Partition[Prime[Range[40]],2,1] (* Harvey P. Dale, Dec 17 2012 *)

A216432 Semiprimes that are sums of squares of two consecutive primes.

Original entry on oeis.org

34, 74, 458, 4058, 28898, 45002, 51218, 57818, 64802, 84122, 115202, 145802, 233978, 352802, 363002, 530522, 609458, 649802, 756458, 924818, 994082, 1391162, 1609418, 2179922, 2599442, 2832218, 3328202, 3484802, 3864362, 3942482, 5425418, 5746058, 6125018

Offset: 1

Zak Seidov, Sep 09 2012

nonn

Semiprimes in A069484. - Zak Seidov, Apr 11 2014

			34 = 2*17 = 3^2+5^2, 74 = 2*37 = 5^2+7^2, 458 = 2*229 = 13^2+17^2.

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

Subsequence of A100484.

Cf. A000040, A001358, A069484.

Select[(Total/@Partition[Prime[Range[500]]^2,2,1]),PrimeOmega[#]==2&] (* Harvey P. Dale, Sep 23 2012 *)

v=List();p=3;forprime(q=5,1e4,if(isprime((p^2+q^2)\2),listput(v,p^2+q^2));p=q);Vec(v) \\ Charles R Greathouse IV, Sep 23 2012

A048851 Length of hypotenuse squared in right triangle formed by a prime spiral plotted in Cartesian coordinates.

Original entry on oeis.org

8, 13, 34, 74, 170, 290, 458, 650, 890, 1370, 1802, 2330, 3050, 3530, 4058, 5018, 6290, 7202, 8210, 9530, 10370, 11570, 13130, 14810, 17330, 19610, 20810, 22058, 23330, 24650, 28898, 33290, 35930, 38090, 41522, 45002, 47450, 51218, 54458

Offset: 1

Enoch Haga

			a(2) = 13 because c^2 = a^2 + b^2 = 4 + 9 = 13.

H. E. Huntley, The Divine Proportion, A Study in Mathematical Beauty. New York: Dover, 1970. See Chapter 13, Spira Mirabilis, especially Fig. 13-5, page 173.

Cf. A006094.

To begin prime spiral, plot (2, 0), (0, 2). Square of hypotenuse is c^2 = a^2 + b^2, or 8 = 4 + 4, so a(1) = 8.

a(n) = A069484(n-1), n >= 2. - Mamuka Jibladze, Mar 24 2017

A069485 Greatest prime factor of prime(n+1)^2 + prime(n)^2.

Original entry on oeis.org

13, 17, 37, 17, 29, 229, 13, 89, 137, 53, 233, 61, 353, 2029, 193, 37, 277, 821, 953, 61, 89, 101, 1481, 1733, 53, 2081, 269, 2333, 29, 14449, 3329, 3593, 293, 1597, 22501, 73, 25609, 373, 28909, 6197, 32401, 389, 101, 2237, 7841, 42061, 29, 257, 281, 821

Offset: 1

Reinhard Zumkeller, Mar 29 2002

nonn

How small can members of this sequence be? For example, a(52837) = 97 since 650107^2 + 650099^2 = 2 * 5^4 * 29 * 37 * 73 * 89 * 97. - Charles R Greathouse IV, May 14 2014

			A069482(10) = A000040(11)^2 + A000040(10)^2 = 29^2 + 31^2 = 841 + 961 = 1802 = 2*17*53, therefore a(10) = 53.

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

Cf. A069483.

seq(max(map2(op,1,ifactors(ithprime(i+1)^2 + ithprime(i)^2)[2])), i=1..1000); # Robert Israel, May 18 2014

Table[ FactorInteger[ Prime[n + 1]^2 + Prime[n]^2] [[ -1, 1]], {n, 1, 50} ]
FactorInteger[#][[-1,1]]&/@Total/@Partition[Prime[Range[60]]^2,2,1] (* Harvey P. Dale, Jul 08 2019 *)

gpf(n)=my(f=factor(n)[,1]); f[#f]
a(n)=my(p=prime(n)); gpf(nextprime(p+1)^2 + p^2) \\ Charles R Greathouse IV, May 14 2014

a(n) = A006530(A069484(n)).

Edited and extended by Robert G. Wilson v, Apr 18 2002

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}]

A160054 Primes prime(k) such that prime(k)^2 + prime(k+1)^2 - 1 is a perfect square.

Original entry on oeis.org

7, 11, 23, 109, 211, 307, 1021, 4583, 42967, 297779, 1022443, 1459811, 10781809, 125211211, 11673806759, 3019843939831, 40047392632801, 88212019638251209, 444190204424015227, 57852556614292865039, 9801250757169593701501, 64747502900142088755541, 619216322498658374863033

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 01 2009

nonn

An infinite number of solutions exists for a^2 + b^2 - 1 = c^2 over the set of natural numbers a, b, c.
If we constrain these to b=a+2, i.e., 2a^2 + 4a + 3 = c^2, the solutions are with a = 1, 11, 69, 407, 2377, ... (The twin prime 11 is also in this sequence here. The solutions can be generated recursively from a(0)=1, m(0)=3 and a(k+1) = 3*a(k) + 2*m(k) + 2, m(k+1) = 4*a(k) + 3*m(k) + 4.)
Filtering these solutions for prime pairs a(k) and b(k) would generate the subset of lower twin primes in the sequence.
The equivalent procedure can be carried out for other prime gaps 2*d such that prime(k)=a, prime(k+1)=a+2*d, 2*a^2 + 4*a*d + 4*d^2 - 1 = m^2. This decomposes the sequence into classes according to the gap 2*d.
a(17) > 5*10^12. - Donovan Johnson, May 17 2010

			7^2 + 11^2 - 1 = 13^2.
11^2 + 13^2 - 1 = 17^2.
23^2 + 29^2 - 1 = 37^2.
109^2 + 113^2 - 1 = 157^2.
211^2 + 223^2 - 1 = 307^2.
307^2 + 311^2 - 1 = 19^2*23^2.
1021^2 + 1031^2 - 1 = 1451^2.
4583^2 + 4591^2 - 1 = 13^2*499^2.

Cf. A050791, A129288, A271050.

[n: n in [0..2*10^7] | IsSquare(n^2+NextPrime(n+1)^2-1) and IsPrime(n)]; // Vincenzo Librandi, Aug 02 2015

lst = {}; p = q = 2; While[p < 4000000000, q = NextPrime@ p; If[ IntegerQ[ Sqrt[p^2 + q^2 - 1]], AppendTo[lst, p]; Print@ p]; p = q]; lst (* Robert G. Wilson v, May 31 2009 *)

p=2;forprime(q=3,1e6,if(issquare(q^2+p^2-1),print1(p", "));p=q) \\ Charles R Greathouse IV, Nov 06 2014

is(n)=issquare(n^2+nextprime(n+1)^2-1)&&isprime(n) \\ Charles R Greathouse IV, Nov 29 2014

{A000040(k): A069484(k)-1 in A000290}.

Edited and 4 more terms from R. J. Mathar, May 08 2009
a(13) from Robert G. Wilson v, May 31 2009
a(15)-a(16) from Donovan Johnson, May 17 2010
More terms from Jinyuan Wang, Jan 09 2021

A240749 Numbers n such that prime(n)^2 + prime(n+1)^2 is a semiprime.

Original entry on oeis.org

2, 3, 6, 14, 30, 35, 37, 39, 41, 46, 52, 57, 68, 81, 82, 97, 101, 104, 112, 123, 126, 145, 154, 175, 189, 195, 209, 215, 221, 222, 259, 264, 272, 276, 308, 312, 314, 343, 357, 367, 370, 373, 389, 398, 399, 403, 411, 416, 418, 425, 432, 436, 447, 456, 462, 471, 473, 477, 485, 487, 489, 499, 509, 520, 538, 547

Offset: 1

Zak Seidov, Apr 11 2014

nonn

a(n) = position of A216432(n) in A069484.

			a(1) = 2: prime (2)^2 + prime (3)^2  = 3^2 + 5^2 = 34 = A069484(2) = A216432 (1).
a(2) = 3: prime (3)^2 + prime (4)^2  = 5^2 + 7^2 = 74 = A069484(3)  = A216432 (2).
a(3) = 6: prime (6)^2 + prime (7)^2  = 13^2 + 17^2 = 458 = A069484(6)  = A216432 (3).

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A069484, A216432.

with(numtheory):
isok := n -> evalb(bigomega(ithprime(n)^2 + ithprime(n+1)^2) = 2);
A240749_list := n -> select(isok, [$1..n]); A240749_list(555); # Peter Luschny, Apr 12 2014

Position[Total/@Partition[Prime[Range[600]]^2,2,1],?(PrimeOmega[#] == 2&)]// Flatten (* _Harvey P. Dale, Apr 12 2017 *)

isok(n) = bigomega(prime(n)^2  + prime(n+1)^2) == 2;
lista(nn) = {for(n=1, nn, if (isok(n), print1(n, ", ")));} \\ Michel Marcus, Apr 12 2014

s=[]; for(n=2, 600, if(isprime((prime(n)^2+prime(n+1)^2)/2), s=concat(s, n))); s \\ Colin Barker, Apr 12 2014

A287922 a(n) = prime(1)^2 + prime(n)^2.

Original entry on oeis.org

8, 13, 29, 53, 125, 173, 293, 365, 533, 845, 965, 1373, 1685, 1853, 2213, 2813, 3485, 3725, 4493, 5045, 5333, 6245, 6893, 7925, 9413, 10205, 10613, 11453, 11885, 12773, 16133, 17165, 18773, 19325, 22205, 22805, 24653, 26573, 27893, 29933, 32045, 32765, 36485

Offset: 1

XU Pingya, Jun 02 2017

Cf. A001248, A069484.

Table[4+Prime[n]^2, {n, 43}]
Prime[Range[50]]^2+4 (* Harvey P. Dale, Feb 16 2020 *)

from sympy import prime, primerange
def aupton(terms): return [4 + p*p for p in primerange(2, prime(terms)+1)]
print(aupton(43)) # Michael S. Branicky, Aug 13 2021

a(n) = 4 + prime(n)^2.

cp's OEIS Frontend

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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A069486 a(n) = 2*prime(n)*prime(n+1).

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A216432 Semiprimes that are sums of squares of two consecutive primes.

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A048851 Length of hypotenuse squared in right triangle formed by a prime spiral plotted in Cartesian coordinates.

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A069485 Greatest prime factor of prime(n+1)^2 + prime(n)^2.

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Maple

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A131686 Sum of squares of five consecutive primes.

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A160054 Primes prime(k) such that prime(k)^2 + prime(k+1)^2 - 1 is a perfect square.

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A069486 a(n) = 2prime(n)prime(n+1).