A215800 -id:A215800 - cp's OEIS frontend

A215801 Prime numbers p such that (2^p + 1)/3 can be written in the form a^2 + 3*b^2.

Original entry on oeis.org

3, 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 109, 127, 139, 151, 199, 277, 313, 433, 457, 547, 613, 619, 643, 739, 967

Offset: 1

V. Raman, Aug 23 2012

nonn

These (2^p + 1)/3 numbers have no prime factors of the form 2 (mod 3) to an odd power.

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

Cf. A000051, A000978, A215800.

forprime(i=2, 100, a=factorint(2^i+1)~; has=0; for(j=1, #a, if(a[1, j]%3==2&&a[2, j]%2==1, has=1; break)); if(has==0, print(i" -\t"a[1, ])))

9 more terms from V. Raman, Aug 28 2012

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

A364614 Numbers not divisible by any prime of the form 3*k - 1.

Original entry on oeis.org

1, 3, 7, 9, 13, 19, 21, 27, 31, 37, 39, 43, 49, 57, 61, 63, 67, 73, 79, 81, 91, 93, 97, 103, 109, 111, 117, 127, 129, 133, 139, 147, 151, 157, 163, 169, 171, 181, 183, 189, 193, 199, 201, 211, 217, 219, 223, 229, 237, 241, 243, 247, 259, 271, 273, 277, 279

Offset: 1

Clark Kimberling, Aug 09 2023

nonn

The sequence is closed under multiplication.

Cf. A003627, A364832.

A215800 is a subsequence.

s = Select[Prime[Range[100]], Mod[#, 3] == 2 &] (* A003627 *)
Select[r = Range[Max[s]], Intersection[#/s, r] == {} &]

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A215801 Prime numbers p such that (2^p + 1)/3 can be written in the form a^2 + 3*b^2.

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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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A364614 Numbers not divisible by any prime of the form 3*k - 1.

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