A069486 -id:A069486 - 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

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

A069484 a(n) = prime(n+1)^2 + prime(n)^2.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Mar 29 2002

Together with A069482(n) and A069486(n) a Pythagorean triangle is formed with area = A069487(n).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Janyarak Tongsomporn, Saeree Wananiyaku, and Jörn Steuding, Sums of consecutive prime squares, Integers (2022) Vol. 22, #A9.

Cf. A069485, A075892, A069482, A069486.

Cf. A001248, A000040, A048851.

seq(ithprime(n)^2+ithprime(n+1)^2, n = 1 .. 100); # Stefano Spezia, Dec 21 2018

Table[Prime[n]^2+Prime[n+1]^2,{n,5!}] (* Vladimir Joseph Stephan Orlovsky, Apr 12 2010 *)
Total[#^2]&/@Partition[Prime[Range[50]],2,1] (* Harvey P. Dale, May 26 2012 *)

v=primes(101);vector(#v-1,i,v[i]^2+v[i+1]^2) \\ Charles R Greathouse IV, Aug 21 2011

from sympy import prime
for n in range(1,101): print(n, prime(n)**2+prime(n+1)**2) # Stefano Spezia, Dec 21 2018

a(n) = A001248(n+1) + A001248(n) = A000040(n+1)^2 + A000040(n)^2.
a(n) = A048851(n+1).
a(n) = 2 * A075892(n) for n > 1.

A332821 One part of a 3-way classification of the positive integers. Numbers n for which A048675(n) == 1 (mod 3).

Original entry on oeis.org

2, 5, 9, 11, 12, 16, 17, 21, 23, 28, 30, 31, 39, 40, 41, 47, 49, 52, 54, 57, 59, 66, 67, 70, 72, 73, 75, 76, 83, 87, 88, 91, 96, 97, 100, 102, 103, 109, 111, 116, 126, 127, 128, 129, 130, 133, 135, 136, 137, 138, 148, 149, 154, 157, 159, 165, 167, 168, 169, 172, 175, 179, 180, 183, 184, 186, 190, 191, 197, 203, 211, 212

Offset: 1

Antti Karttunen and Peter Munn, Feb 25 2020

nonn

The positive integers are partitioned between A332820, this sequence and A332822.
For each prime p, the terms include exactly one of p and p^2. The primes alternate between this sequence and A332822. This sequence has the primes with odd indexes, those in A031368.
The terms are the even numbers in A332822 halved. The terms are also the numbers m such that 5m is in A332822, and so on for alternate primes: 11, 17, 23 etc. Likewise, the terms are the numbers m such that 3m is in A332820, and so on for alternate primes: 7, 13, 19 etc.
The numbers that are half of the even terms of this sequence are in A332820, which consists exactly of those numbers. The numbers that are one third of the terms that are multiples of 3 are in A332822, which consists exactly of those numbers. For larger primes, an alternating pattern applies as described in the previous paragraph.
If we take each odd term of this sequence and replace each prime in its factorization by the next smaller prime, the resulting number is in A332822, which consists entirely of those numbers.
The product of any 2 terms of this sequence is in A332822, the product of any 3 terms is in A332820, and the product of a term of A332820 and a term of this sequence is in this sequence. So if a number k is present, k^2 is in A332822, k^3 is in A332820, and k^4 is in this sequence.
If k is an even number, exactly one of {k/2, k, 2k} is in the sequence (cf. A191257 / A067368 / A213258); and generally if k is a multiple of a prime p, exactly one of {k/p, k, k*p} is in the sequence.

Positions of ones in A332823; equivalently, numbers in row 3k+1 of A277905 for some k >= 0.
Cf. A048675, A332820, A332822.
Comparable 2 or 3-way classifications: A000379/A000028, A001969/A000069, A003159/A036554, A005843/A005408, A028260/A026424, A191257/A067368/A213258, A325431/A325432, A329609/A329604/A332812.
Subsequences: intersection of A026478 and A066208, A031368 (prime terms), A033431\{0}, A052934\{1}, A069486, A099800, A167747\{1}, A244725\{0}, A244728\{0}, A338911 (semiprime terms).

Select[Range@ 212, Mod[Total@ #, 3] == 1 &@ Map[#[[-1]]*2^(PrimePi@ #[[1]] - 1) &, FactorInteger[#]] &] (* Michael De Vlieger, Mar 15 2020 *)

isA332821(n) =  { my(f = factor(n)); (1==((sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3)); };

{a(n) : n >= 1} = {2 * A332820(k) : k >= 1} U {A003961(A332822(k)) : k >= 1}.

{a(n) : n >= 1} = {A332822(k)^2 : k >= 1} U {A331590(2, A332820(k)) : k >= 1}.

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

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

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A069484 a(n) = prime(n+1)^2 + prime(n)^2.

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A332821 One part of a 3-way classification of the positive integers. Numbers n for which A048675(n) == 1 (mod 3).

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