A264866 -id:A264866 - cp's OEIS frontend

A264865 Primes of the form 2^x + y (x >= 0 and 0 <= y < 2^x) such that all the numbers 2^(x-a) + (y+a) (0 < a <= x) are composite.

5, 7, 13, 19, 31, 47, 61, 71, 101, 211, 239, 241, 271, 281, 311, 331, 379, 421, 449, 491, 617, 619, 631, 751, 797, 827, 853, 863, 883, 971, 991, 1009, 1051, 1117, 1171, 1217, 1277, 1291, 1297, 1301, 1321, 1327, 1429, 1453, 1471, 1483, 1487, 1531, 1567, 1607, 1627, 1637, 1667, 1669, 1697, 1709, 1723, 1747, 1801, 1847

Offset: 1

Zhi-Wei Sun, Nov 26 2015

nonn

Conjecture: The sequence has infinitely many terms.
This is motivated by part (i) of the conjecture in A231201.
See also A264866 for a related conjecture.

			a(1) = 5 since 5 = 2^2 + 1 is a prime with 1 < 2^2, and 2^0 + 3 = 2^1 + 2 = 4 is composite.
a(3) = 13 since 13 = 2^3 + 5 is a prime with 5 < 2^3, and 2^0 + 8 = 2^1 + 7 = 9 and 2^2 + 6 = 10 are both composite.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Write n = k + m with 2^k + m prime, a message to Number Theory List, Nov. 16, 2013.
Z.-W. Sun, On a^n+ bn modulo m, arXiv:1312.1166 [math.NT], 2013-2014.
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2015.

Cf. A000040, A000079, A231201, A231557, A264866.

p[n_]:=p[n]=Prime[n]
x[n_]:=x[n]=Floor[Log[2,p[n]]]
y[n_]:=y[n]=p[n]-2^(x[n])
n=0;Do[Do[If[PrimeQ[2^(x[k]-a)+y[k]+a],Goto[aa]],{a,1,x[k]}];n=n+1;Print[n," ",p[k]];Label[aa];Continue,{k,1,283}]

A264904 Primes of the form x^2 + y^2 with 0 < x < y such that all the numbers (x-a)^2 + (y+a)^2 (a = 1,...,x) are composite.

Original entry on oeis.org

5, 17, 37, 53, 101, 109, 197, 257, 293, 401, 409, 577, 677, 701, 733, 857, 1093, 1297, 1373, 1601, 1609, 1697, 2029, 2141, 2213, 2417, 2917, 3137, 3253, 3373, 3389, 3853, 4261, 4357, 4493, 4909, 5209, 5477, 5641, 5801, 6257, 7057, 7229, 7573, 7937, 8101, 8837, 9029, 9413, 9613, 10009, 10429, 10453, 10613, 12101, 12109, 12553, 13457, 13693, 14177

Offset: 1

Zhi-Wei Sun, Nov 28 2015

nonn

Note that the sequence contains all primes of the form n^2 + 1 with n > 1. A conjecture of Landau states that there are infinitely many primes of the form n^2 + 1.
Conjecture: For any prime p > 5 of the form x^2 + y^2 (0 < x < y), there is a prime q not equal to p of the form u^2 + v^2 (0 < u < v) with u + v = x + y.
A subsequence of A002313. - Altug Alkan, Dec 18 2015
Conjecture: each odd number m > 1 is a unique sum m = x + y with 0 < x < y, where x^2 + y^2 is in the sequence. - Thomas Ordowski, Jan 16 2017

			a(1) = 5 since 5 = 1^2 + 2^2 is a prime with 0 < 1 < 2, and 0^2 + 3^2 = 9 is composite.
a(4) = 53 since 53 = 2^2 + 7^2 is a prime with 0 < 2 < 7, and 0^2 + 9^2 = 81 and 1^2 + 8^2 = 65 are both composite.

Zhi-Wei Sun, Table of n, a(n) for n = 1..3500

Cf. A000040, A000290, A002144, A002313, A002496, A264865, A264866.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Y[n_]:=Y[n]=Sum[If[SQ[n-4*y^2],2y,0],{y,0,Sqrt[n/4]}]
X[n_]:=X[n]=Sqrt[n-Y[n]^2]
p[n_]:=p[n]=Prime[n]
x[n_]:=x[n]=X[p[n]]
y[n_]:=y[n]=Y[p[n]]
n=0;Do[If[Mod[p[k]-1,4]==0,Do[If[PrimeQ[a^2+(x[k]+y[k]-a)^2],Goto[aa]],{a,0,Min[x[k],y[k]]-1}];n=n+1;Print[n," ",p[k]]];Label[aa];Continue,{k,2,1669}]

cp's OEIS Frontend

A264865 Primes of the form 2^x + y (x >= 0 and 0 <= y < 2^x) such that all the numbers 2^(x-a) + (y+a) (0 < a <= x) are composite.

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