A340624 Numbers k such that A340388(k) > A018782(k).

16, 32, 64, 128, 256, 512, 768, 1024, 1536, 2048, 2304, 3072, 4096, 4608, 6144, 8192, 9216, 12288, 16384, 18432, 24576, 32768, 36864, 49152, 65536, 73728, 98304, 110592, 131072, 147456, 165888, 196608, 221184, 248832, 262144, 294912, 331776, 373248, 393216, 442368

Offset: 1

Author

Jianing Song, Apr 25 2021

Keywords

• nonn
• hard

Comments

Numbers k such that A340388(k) is not the smallest number whose prime factors are all congruent to 1 modulo 4 and with exactly k divisors.

Despite being an analog of A072066, this sequence seems to be considerably sparser than A072066. What's the reason for that?

All powers of 2 that are greater than or equal to 16 are here. All numbers of the form 3 * 2^e with e >= 8 are here.

All powers of 3 that are greater than or equal to 3^15 = 14348907 are here. For example, we have A340388(3^15) = (5 * 13 * 17 * 29 * ... * 113 * 137)^2, while a(3^15) <= (5^4 * 13 * 17 * 29 * .. * 113)^2, so 3^15 is a term. Apparently 3^15 is the smallest odd term in this sequence.

Similarly, let q be a prime, then all powers of q that are greater than or equal to q^(N+1) are here, where N is the number of primes congruent to 1 modulo 4 below 5^q. It seems that q^(N+1) is the smallest q-rough term in this sequence.

Examples

			16 is a term since A340388(16) = 5 * 13 * 17 * 29 > A018782(16) = 5^3 * 13 * 17.
		

Crossrefs

Cf. A018782, A340388, A072066.

Programs

PARI

isA340624(n) = (A340388(n) > A018782(n)) \\ See A018782 and A340388 for their programs