A324813 Numbers n for which A156552(n) is a square or a twice a square.

1, 2, 3, 5, 7, 10, 11, 13, 17, 19, 21, 23, 29, 31, 37, 41, 43, 47, 50, 53, 55, 59, 61, 67, 71, 73, 79, 83, 89, 91, 97, 98, 101, 103, 107, 109, 113, 127, 131, 137, 139, 147, 149, 151, 154, 157, 163, 167, 173, 179, 181, 187, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 247, 251, 257, 263, 266, 269, 271, 277, 281, 283, 293, 307

Offset: 1

Author

Antti Karttunen, Mar 16 2019

Keywords

• nonn
• base

Comments

Union of {1} and sequence A005940(1+A028982(n)), n >= 1, sorted into ascending order.

After the initial 1, gives the positions of ones in A324885.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to binary expansion of n
• Index entries for characteristic functions
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A005940, A028982, A156552, A324823 (characteristic function for terms > 1), A324825, A324885.

Cf. A000040, A324812 (subsequences), A324814.

Programs

Mathematica

Select[Range@ 320, AnyTrue[{#, #/2}, IntegerQ@ Sqrt@ # &] &@ Floor@ Total@ Flatten@ MapIndexed[#1 2^(#2 - 1) &, Flatten[Table[2^(PrimePi@ #1 - 1), {#2}] & @@@ FactorInteger@ #]] &] (* Michael De Vlieger, Mar 27 2019 *)

PARI

A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552 by David A. Corneth
A324823(n) = if(1==n,0, n=A156552(n); (issquare(n) || (!(n%2) && issquare(n/2))));
for(n=1, oo, if((1==n)||A324823(n), print1(n, ", ")));

PARI

is(n) = { n = A156552(n); n == 0 || n >>= (valuation(n, 2)%2); issquare(n); }; \\ David A. Corneth, Mar 16 2019