A069484 -id:A069484 - cp's OEIS frontend

A069487 Areas of Pythagorean triangles (A069482, A069484, A069486).

30, 240, 840, 5544, 6864, 26520, 23256, 73416, 208104, 107880, 467976, 473304, 296184, 727560, 1494600, 2101344, 863760, 3138816, 2625864, 1492704, 5259504, 4248936, 7623384, 12845904, 7759224, 4244424

Offset: 1

Reinhard Zumkeller, Mar 29 2002

nonn

			prime(2)^3 * prime(1) - prime(1)^3 * prime(2) = 3^3 * 2 - 2^3 * 3 = 54 - 24 = 30 that is the area of the Pythagorean triangle (5, 12, 13), so a(1) = 30. - _Bernard Schott_, Sep 23 2019

Marius A. Burtea, Table of n, a(n) for n = 1..5000
César Aguilera, Two Prime Number Objects and The Velucchi Numbers, hal-02909691 [math.NT], 2020.

Cf. A069482, A069484, A069486.

Cf. A000040, A030078, A127917.

[NthPrime(n+1)^3*NthPrime(n)-NthPrime(n+1)*(NthPrime(n)^3):n in [1..26]]; // Marius A. Burtea, Sep 19 2019

a(n) = A030078(n+1)*A000040(n) - A000040(n+1)*A030078(n).
a(n) = A000040(n+1)^3*A000040(n) - A000040(n+1)*A000040(n)^3.
a(n) = A000040(n)*A127917(n+1) - A127917(n)*A000040(n+1). - César Aguilera, Sep 18 2019

A133529 Sum of squares of three consecutive primes.

Original entry on oeis.org

38, 83, 195, 339, 579, 819, 1179, 1731, 2331, 3171, 4011, 4899, 5739, 6867, 8499, 10011, 11691, 13251, 14859, 16611, 18459, 21051, 24219, 27531, 30219, 32259, 33939, 36099, 40779, 46059, 52059, 55251, 60291, 64323, 69651, 74019, 79107, 84387, 89859, 94731, 101283

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

It is easy to see that all terms > 83 are divisible by 3.

Likewise all terms except 38 are congruent to 3 (mod 8). - Franklin T. Adams-Watters, Jun 17 2015

			a(1)=38 because 2^2 + 3^2 + 5^2 = 38.

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

Cf. A034963, A133524-A133528, A133530-A133533.

[&+[ NthPrime(n+i)^2 :  i in [0..2]] : n in [1..20]]; // K. D. Bajpai, Jun 17 2015

a = 2; Table[Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a, {n, 1, 100}]
Total/@Partition[Prime[Range[50]]^2, 3, 1] (* Vincenzo Librandi, Jun 18 2015 *)

for( n= 1, 100,  k= sum(i=n, n+2, prime(i)^2) ; print1(k, ", ")) \\ K. D. Bajpai, Jun 17 2015

a(n) = A069484(n) + A001248(n+2). - Michel Marcus, Nov 08 2013

a(38)-a(41) from K. D. Bajpai, Jun 18 2015

A069482 a(n) = prime(n+1)^2 - prime(n)^2.

Original entry on oeis.org

5, 16, 24, 72, 48, 120, 72, 168, 312, 120, 408, 312, 168, 360, 600, 672, 240, 768, 552, 288, 912, 648, 1032, 1488, 792, 408, 840, 432, 888, 3360, 1032, 1608, 552, 2880, 600, 1848, 1920, 1320, 2040, 2112, 720, 3720, 768, 1560, 792, 4920, 5208, 1800, 912, 1848

Offset: 1

Reinhard Zumkeller, Mar 29 2002, Aug 05 2007

nonn

a(n) = A001248(n+1) - A001248(n) = A000040(n+1)^2 - A000040(n)^2 = (A000040(n+1) - A000040(n))*(A000040(n+1) + A000040(n)) = A001223(n)*A001043(n); together with A069484(n) and A069486(n) a Pythagorean triangle is formed with area = A069487(n).
For n>2: A078701(a(n)) = 3.
Except for the first two terms, these numbers are divisible by 24. Let p, q be consecutive primes. Then p > 3 = 3k+-1 and q = 3m+-1 and (3k+-1)^2 - (3m+-1)^2 is divisible by 3. Similarly, p = 4k+-1 and q=4m+-1 and (4k+-1)^2 - (4m+-1)^2 is divisible by 8. So 8 and 3 divide q^2 - p^2 => 24 divides q^2 - p^2. - Cino Hilliard, May 28 2009
Repetition of a(n) values occurs with decreasing frequency but increasing tallies (i.e., number of repetitions of a given value).
    Tally = 2, first a(n) value is        72, with first n=4, prime=7.
    Tally = 3, first a(n) value is      1848, with first n=36, prime=151.
    Tally = 4, first a(n) value is      4920, with first n=46, prime=199.
    Tally = 5, first a(n) value is 187117320, with first n=224752, prime 3118607.
Three a(n) values have a tally = 5, and none with tally > 5 for n<10,000,000. Note: Tallies for a given a(n) value are "confirmed" (i.e., not to be greater) only after examining a(n) values for all p(n) <= r/4-1, where r is the a(n) value in question, because twin primes provide the last chance for adding to the tally of any a(n) value. Tallies for the four a(n) values above are "confirmed" and all of them rely on twin primes for their last repetition. Thus r/4 +-1 is prime for the above four cases. However this is not true for all a(n) values that repeat.
Conjecture: The sum of prime factors with repetition (sopfr) applied to a(n), A001414(a(n)), covers all integers covered by sopfr, except 2,3,4,6,7,10,13,15. See A001414 for the sopfr sequence, which does not cover 0 and 1. - Richard R. Forberg, Feb 07 2015
Conjecture: There is no upper bound on the number of repetitions (i.e., size of a tally) that will occur for some a(n) values, because the number of possible ways of producing a value of a(n) grows with increasing n, despite decreasing prime density. This happens because there is increasing range in the size of prime gaps which increases the range of primes that can produce the same a(n) value much faster than the decrease in prime density which is decelerating with larger n. - Richard R. Forberg, Feb 17 2015

			A000040(10)=29, A000040(10+1)=31, A001248(10)=841, A001248(10+1)=961, a(10) = 961 - 841 = 120, A069486(10) = 2*31*29 = 1798, A069484(10) = 961 + 841 = 1802:
120^2 + 1798^2 = 14400 + 3232804 = 3247204 = 1802^2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A001248, A000040, A001223, A001043, A001414, A069483, A069484, A069486, A078701.

a069482 n = a069482_list !! (n-1)
a069482_list = zipWith (-) (tail a001248_list) a001248_list
-- Reinhard Zumkeller, Jun 08 2015

[NthPrime(n+1)^2 - NthPrime(n)^2: n in [1..40]]; // G. C. Greubel, May 19 2019

Table[Prime[n+1]^2 - Prime[n]^2, {n, 1, 40}] (* Vladimir Joseph Stephan Orlovsky, Mar 01 2009; modified by G. C. Greubel, May 19 2019 *)
#[[2]]-#[[1]]&/@Partition[Prime[Range[60]]^2,2,1] (* Harvey P. Dale, Jan 13 2011 *)
Differences[Prime[Range[100]]^2](* Waldemar Puszkarz, Feb 09 2015 *)

{a(n) = prime(n+1)^2 - prime(n)^2}; \\ G. C. Greubel, May 19 2019

from sympy import prime, primerange
def aupton(terms):
  p = list(primerange(1, prime(terms+1)+1))
  return [p[n+1]**2-p[n]**2 for n in range(terms)]
print(aupton(50)) # Michael S. Branicky, May 16 2021

[nth_prime(n+1)^2 - nth_prime(n)^2 for n in (1..40)] # G. C. Greubel, May 19 2019

A133538 Sum of seventh powers of two consecutive primes.

Original entry on oeis.org

2315, 80312, 901668, 20310714, 82235688, 473087190, 1304210412, 4298697186, 20654701756, 44762490420, 122444491244, 289686151014, 466572884988, 778441731570, 1681334260300, 3663362624656, 5631394320840, 9203454441344, 15155831763714, 20142518677488

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=2315 because 2^7 + 3^7 = 2315.

Muniru A Asiru, Table of n, a(n) for n = 1..4000

Cf. A034963, A133524, A133525, A133526, A133527, A133528, A133529, A133530, A133531, A133532, A069484, A133534, A133535, A133536, A133537.

[NthPrime(n)^7 + NthPrime(n+1)^7: n in [1..25]]; // Vincenzo Librandi, Aug 23 2018

seq(add(ithprime(n+k)^7,k=0..1),n=1..20); # Muniru A Asiru, Aug 22 2018

e = 7; Table[Prime[n]^e + Prime[n + 1]^e, {n, 1, 100}]
Total/@Partition[Prime[Range[20]]^7,2,1] (* Harvey P. Dale, Oct 16 2014 *)

a(n) = prime(n)^7 + prime(n+1)^7; \\ Michel Marcus, Aug 22 2018

a(n) = A092759(n) + A092759(n+1). - Michel Marcus, Nov 09 2013

A133534 Sum of third powers of two consecutive primes.

Original entry on oeis.org

35, 152, 468, 1674, 3528, 7110, 11772, 19026, 36556, 54180, 80444, 119574, 148428, 183330, 252700, 354256, 432360, 527744, 658674, 746928, 882056, 1064826, 1276756, 1617642, 1942974, 2123028, 2317770, 2520072, 2737926, 3491280

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=35 because 2^3+3^3=35.

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

Cf. A034963, A133524, A133525, A133526, A133527, A133528, A133529, A133530, A133531, A133532, A069484, A133535, A133536, A133537, A133538.

a = 3; Table[Prime[n]^a + Prime[n + 1]^a, {n, 1, 100}]
Total[#^3]&/@Partition[Prime[Range[50]],2,1] (* Harvey P. Dale, Jan 29 2021 *)

a(n) = A030078(n) + A030078(n+1). - Michel Marcus, Nov 09 2013

A133535 Sum of fourth powers of two consecutive primes.

Original entry on oeis.org

97, 706, 3026, 17042, 43202, 112082, 213842, 410162, 987122, 1630802, 2797682, 4699922, 6244562, 8298482, 12770162, 20007842, 25963202, 33996962, 45562802, 53809922, 67348322, 86408402, 110200562, 151271522, 192589682, 216611282

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=2^4+3^4=97.

Cf. A034963, A133524, A133525, A133526, A133527, A133528, A133529, A133530, A133531, A133532, A069484, A133534, A133536, A133537, A133538.

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

a(n) = A030514(n) + A030514(n+1). - Michel Marcus, Nov 09 2013

A133536 Sum of fifth powers of two consecutive primes.

Original entry on oeis.org

275, 3368, 19932, 177858, 532344, 1791150, 3895956, 8912442, 26947492, 49140300, 97973108, 185200158, 262864644, 376353450, 647540500, 1133119792, 1559520600, 2194721408, 3154354458, 3877300944, 5150127992, 7016097042, 9523100092

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=2^5+3^5=275.

Cf. A034963, A133524, A133525, A133526, A133527, A133528, A133529, A133530, A133531, A133532, A069484, A133534, A133535, A133537, A133538.

a = 5; Table[Prime[n]^a + Prime[n + 1]^a, {n, 1, 100}]

a(n) = A050997(n) + A050997(n+1). - Michel Marcus, Nov 09 2013

A133537 Sum of sixth powers of two consecutive primes.

Original entry on oeis.org

793, 16354, 133274, 1889210, 6598370, 28964378, 71183450, 195081770, 742859210, 1482327002, 3453230090, 7315830650, 11071467290, 17100578378, 32943576458, 64344894770, 93700908002, 141978756530, 218558666090, 279434510210

Offset: 1

Artur Jasinski, Sep 14 2007

nonn

			a(1)=793 because 2^6+3^6=793.

Cf. A034963, A133524, A133525, A133526, A133527, A133528, A133529, A133530, A133531, A133532, A069484, A133534, A133535, A133536, A133538.

a = 6; Table[Prime[n]^a + Prime[n + 1]^a, {n, 1, 100}]

a(n) = A030516(n) + A030516(n+1). - 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

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 *)

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A069487 Areas of Pythagorean triangles (A069482, A069484, A069486).

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A133529 Sum of squares of three consecutive primes.

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A069482 a(n) = prime(n+1)^2 - prime(n)^2.

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A133538 Sum of seventh powers of two consecutive primes.

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A133534 Sum of third powers of two consecutive primes.

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A133535 Sum of fourth powers of two consecutive primes.

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A133536 Sum of fifth powers of two consecutive primes.

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