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%I A069482 #63 Apr 28 2023 03:09:53
%S A069482 5,16,24,72,48,120,72,168,312,120,408,312,168,360,600,672,240,768,552,
%T A069482 288,912,648,1032,1488,792,408,840,432,888,3360,1032,1608,552,2880,
%U A069482 600,1848,1920,1320,2040,2112,720,3720,768,1560,792,4920,5208,1800,912,1848
%N A069482 a(n) = prime(n+1)^2 - prime(n)^2.
%C A069482 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).
%C A069482 For n>2: A078701(a(n)) = 3.
%C A069482 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
%C A069482 Repetition of a(n) values occurs with decreasing frequency but increasing tallies (i.e., number of repetitions of a given value).
%C A069482     Tally = 2, first a(n) value is        72, with first n=4, prime=7.
%C A069482     Tally = 3, first a(n) value is      1848, with first n=36, prime=151.
%C A069482     Tally = 4, first a(n) value is      4920, with first n=46, prime=199.
%C A069482     Tally = 5, first a(n) value is 187117320, with first n=224752, prime 3118607.
%C A069482 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.
%C A069482 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
%C A069482 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
%H A069482 Reinhard Zumkeller, <a href="/A069482/b069482.txt">Table of n, a(n) for n = 1..1000</a>
%H A069482 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>
%e A069482 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:
%e A069482 120^2 + 1798^2 = 14400 + 3232804 = 3247204 = 1802^2.
%t A069482 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 *)
%t A069482 #[[2]]-#[[1]]&/@Partition[Prime[Range[60]]^2,2,1] (* _Harvey P. Dale_, Jan 13 2011 *)
%t A069482 Differences[Prime[Range[100]]^2](* _Waldemar Puszkarz_, Feb 09 2015 *)
%o A069482 (Haskell)
%o A069482 a069482 n = a069482_list !! (n-1)
%o A069482 a069482_list = zipWith (-) (tail a001248_list) a001248_list
%o A069482 -- _Reinhard Zumkeller_, Jun 08 2015
%o A069482 (PARI) {a(n) = prime(n+1)^2 - prime(n)^2}; \\ _G. C. Greubel_, May 19 2019
%o A069482 (Magma) [NthPrime(n+1)^2 - NthPrime(n)^2: n in [1..40]]; // _G. C. Greubel_, May 19 2019
%o A069482 (Sage) [nth_prime(n+1)^2 - nth_prime(n)^2 for n in (1..40)] # _G. C. Greubel_, May 19 2019
%o A069482 (Python)
%o A069482 from sympy import prime, primerange
%o A069482 def aupton(terms):
%o A069482   p = list(primerange(1, prime(terms+1)+1))
%o A069482   return [p[n+1]**2-p[n]**2 for n in range(terms)]
%o A069482 print(aupton(50)) # _Michael S. Branicky_, May 16 2021
%Y A069482 Cf. A001248, A000040, A001223, A001043, A001414, A069483, A069484, A069486, A078701.
%K A069482 nonn
%O A069482 1,1
%A A069482 _Reinhard Zumkeller_, Mar 29 2002, Aug 05 2007

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