A279767 Numbers m such that m and m+2 have the same prime signature.

3, 5, 11, 17, 18, 29, 33, 41, 50, 54, 55, 59, 71, 85, 91, 93, 101, 107, 137, 141, 143, 149, 159, 179, 183, 185, 191, 197, 201, 203, 213, 215, 217, 219, 227, 235, 239, 242, 247, 248, 265, 269, 281, 299, 301, 303, 306, 311, 319, 321, 327, 339, 340, 347, 348, 391, 393, 411, 413

Offset: 1

Author

Altug Alkan, Dec 18 2016

Keywords

• nonn
• easy

Comments

The sequence contains some terms such that m and m + 2k (k > 1) have the same prime signature. For some terms where m and m + 2k share the same prime signature this means that every alternate element between, and including m and m + 2k have the same prime signature. The first such example is where a(41951) = 402677, a(41953) = 402679, and a(41955) = 402681, share the same prime signature {1, 1}. Also the remaining alternate terms excluding endpoints share the same prime signature. Using the above example, a(41952) = 402678 and a(41954) = 402680 share the prime signature {1,1,3}. - Torlach Rush, Feb 25 2018

Examples

			18 is a term because 18 = 2 * 3^2 and 18 + 2 = 20 = 2^2 * 5.
19 is not a term because it is prime and 21 is the product of two primes, so the prime signatures are different.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5585 from Michel Marcus)
• Index to sequences related to prime signature.

Crossrefs

Cf. A001359, A052213, A052214.

Programs

Mathematica

primeSignature[n_] := Sort[Transpose[FactorInteger[n]][[2]]]; Select[ Range[2, 1000], primeSignature[#] == primeSignature[# + 2] &] (* Adapted from A052213 *)

PARI

isok(n) = vecsort(factor(n)[,2]) == vecsort(factor(n+2)[,2]); \\ Michel Marcus, Feb 25 2018