A265113 Primes p such that p and p^2 have the same number of 1's in their binary representations.

2, 3, 7, 31, 79, 127, 157, 317, 379, 751, 1087, 1151, 1277, 1279, 1531, 1789, 1951, 2297, 2557, 2927, 3067, 3259, 3319, 3581, 4253, 4349, 5119, 5231, 5503, 5807, 5821, 6271, 6653, 6871, 8191, 8447, 8689, 9209, 10079, 10837, 11597, 11903, 12799, 13309, 13591

Offset: 1

Author

Robert Israel, Dec 01 2015

Keywords

• nonn
• base

Comments

Primes p such that p^2 is in A089042.

Primes p such that A000120(p) = A000120(p^2).

Contains all terms > 43 in A079361.

Subset of A077436.

Examples

			7 is in the sequence because 7 and 7^2 = 49 have binary representations 111 and 110001 which both have three 1's.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000120, A077436, A079361, A089042.

Programs

Magma

[NthPrime(n): n in [1..2000] | Multiplicity({* z: z in Intseq(NthPrime(n)^2, 2) *}, 1) eq &+Intseq(NthPrime(n), 2)]; // Vincenzo Librandi, Dec 02 2015

Maple

f:= proc(n) isprime(n) and (convert(convert(n,base,2),`+`) = convert(convert(n^2,base,2),`+`)) end proc:
select(f, [2,seq(i,i=3..10^5,2)]);

Mathematica

Select[ Prime@ Range@ 1700, DigitCount[n, 2, 1] == DigitCount[n^2, 2, 1],  &] (* Robert G. Wilson v, Dec 01 2015 *)

PARI

c(k, d, b) = {my(c=0, f); while (k>b-1, f=k-b*(k\b); if (f==d, c++); k\=b); if (k==d, c++); return(c)}
forprime(p=2, 1e5, if(c(p, 1, 2) == c(p^2, 1, 2), print1(p, ", "))) \\ Altug Alkan, Dec 02 2015