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Number of primes p of the form 4m+1 with p <= n.
9

%I #15 Apr 28 2021 23:20:10

%S 0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,

%T 4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,

%U 8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,11,11,11

%N Number of primes p of the form 4m+1 with p <= n.

%C Asymptotic expansion: a(n) ~ pi(n)/2 ~ n/(2log(n)) (pi(n) is in sequence A000720).

%C Partial sums of A079260. - _Reinhard Zumkeller_, Feb 06 2014

%H T. D. Noe, <a href="/A066339/b066339.txt">Table of n, a(n) for n=1..10000</a>

%H R. Breusch, <a href="http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.mmj/1028999182">An Asymptotic Formula for Primes Of The Form 4n+1</a>

%F a(n) + A066490(n) = A000720(n) - 1 for n >= 2. - _Jianing Song_, Apr 28 2021

%t Table[ Length[ Select[ Union[ Table[ Prime[ PrimePi[i]], {i, 2, n}]], Mod[ #, 4] == 1 & ]], {n, 2, 100} ]

%o (PARI) for(n=1,200,print1(sum(i=1,n,if((i*isprime(i)-1)%4,0,1)),","))

%o (Haskell)

%o a066339 n = a066339_list !! (n-1)

%o a066339_list = scanl1 (+) $ map a079260 [1..]

%o -- _Reinhard Zumkeller_, Feb 06 2014

%Y Cf. A000720, A066490, A296021.

%K nonn

%O 1,13

%A Sharon Sela (sharonsela(AT)hotmail.com), Jan 01 2002

%E More terms from _Robert G. Wilson v_, Jan 03 2002