A033205 -id:A033205 - cp's OEIS frontend

A121569 a(n) = Fibonacci((prime(n)+3)/2) - 1.

1, 2, 4, 12, 20, 54, 88, 232, 986, 1596, 6764, 17710, 28656, 75024, 317810, 1346268, 2178308, 9227464, 24157816, 39088168, 165580140, 433494436, 1836311902, 12586269024, 32951280098, 53316291172, 139583862444, 225851433716

Offset: 2

Alexander Adamchuk, Aug 08 2006

nonn

p = Prime[n] divides a(n) for p = {29,89,101,181,229,349,401,461,509,521,541,709,761,769,809,...} = A047650[n] Primes for which golden mean tau is a quadratic residue or Primes of the form x^2+20y^2.

Cf. A000045, A121567, A121568, A033205, A045468, A064739, A047650.

Table[Fibonacci[(Prime[n]+3)/2]-1,{n,2,50}]

a(n) = Fibonacci[ (Prime[n]+3)/2 ] - 1, n>1. a(n) = Sum[ Fibonacci[k], {k,1,(p-1)/2} ], p = Prime[n], n>1.

A139546 Numbers of form x^2+5y^2 (x>=0,y>=0) with repetition.

Original entry on oeis.org

0, 1, 4, 5, 6, 9, 9, 14, 16, 20, 21, 21, 24, 25, 29, 30, 36, 36, 41, 45, 45, 46, 49, 49, 54, 54, 56, 61, 64, 69, 69, 70, 80, 81, 81, 81, 84, 84, 86, 89, 94, 96, 100, 101, 105, 105, 109, 116, 120, 121, 125, 126, 126, 126, 129, 129, 134, 141, 141, 144, 144, 145, 149, 150

Offset: 1

Zak Seidov, Apr 26 2008

nonn

Primes in the sequence occur only once.

			0 = 0^2+5*0^2,
9 = 2^2+5*1^2 = 3^2+5*0^2,
81 = 1^2+5*4^2 = 6^2+5*3^2 = 9^2+5*0^2,
441: {x,y}={6,9},{11,8},{14,7},{19,4},{21,0}.

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.

Cf. A033205.

A215937 Numbers n such that 2^n + 1 can be written in the form a^2 + 5*b^2.

Original entry on oeis.org

2, 3, 7, 10, 11, 19, 23, 31, 43, 47, 50, 58, 71, 79, 82, 107, 127, 167, 178, 179, 191, 199, 250, 290, 298, 311, 347, 359, 410, 487, 563, 599, 683, 751, 802, 890, 907, 1051

Offset: 1

V. Raman, Aug 27 2012

nonn

These 2^n + 1 numbers can only have prime factors of the form 1 (mod 20) or 3 (mod 20) or 5 (mod 20) or 7 (mod 20) or 9 (mod 20) raised to an odd power, but their overall product 2^n+1 can only be 1 (mod 20) or 5 (mod 20) or 9 (mod 20). This statement is limited to odd numbers.
In general,
A number n can be written in the form a^2+5*b^2 if and only if n is 0,
or of the form 2^(2i) 5^j Prod_{p==1 or 9 mod 20} p^k Prod_{q==3 or 7 mod 20) q^(2m)
or of the form 2^(2i+1) 5^j Prod_{p==1 or 9 mod 20} p^k Prod_{q==3 or 7 mod 20) q^(2m+1),
for integers i,j,k,m, for primes p,q.

			3 is in the sequence because 2^3 + 1 = 9 can be written as 2^2 + 5 * 1^2 = 9.

Samuel S. Wagstaff, Jr., The Cunningham Project, Factorizations of 2^n-1, for odd n's < 1200

Cf. A000051, A000978, A215800, A215801.

Cf. A020669, A033205 (numbers and primes of the form x^2 + 5*y^2).

for(i=2, 500, a=factorint(2^i+1)~; has=0; for(j=1, #a, if(((a[1, j]%20>10)||(i%4<2))&&a[2, j]%2==1, has=1; break)); if(has==0, print(i",")))

for(i=2, 500, a=factorint(2^i+1)~; flag=0; flip=0; for(j=1, #a, if(((a[1, j]%20>10))&&a[2, j]%2==1, flag=1); if(((a[1, j]%20==2)||(a[1, j]%20==3)||(a[1, j]%20==7))&&a[2, j]%2==1, flip=flip+1)); if(flag==0&&flip%2==0, print(i",")))

Terms corrected by V. Raman, Sep 20 2012

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A121569 a(n) = Fibonacci((prime(n)+3)/2) - 1.

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A139546 Numbers of form x^2+5y^2 (x>=0,y>=0) with repetition.

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A215937 Numbers n such that 2^n + 1 can be written in the form a^2 + 5*b^2.

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