A368448 Positive integers k such that there is no m different from k where both s(k) = s(m) and s(k+1) = s(m+1), where s(k) is the prime signature of k.

1, 2, 3, 4, 7, 8, 15, 16, 24, 26, 27, 31, 32, 35, 48, 63, 64, 80, 99, 124, 127, 128, 224, 242, 243, 255, 256, 288, 343, 511, 512, 528, 575, 624, 675, 728, 783, 960, 999, 1023, 1024, 1088, 1295, 1331, 2047, 2048, 2186, 2187, 2208, 2303, 2400, 3375, 3968, 4095, 4096

Offset: 1

Author

Jon E. Schoenfield, Dec 24 2023

Keywords

• nonn

Comments

In other words, numbers k that are uniquely identified by the values of the ordered pair (s(k), s(k+1)), where s(k) is the prime signature of k.

Other than the first two terms, every term <= 4096 is either a proper power (a number of the form b^e with e > 1) or one less than a proper power.

For the analogous sequence using the number of divisors rather than the prime signature, see A161460.

Examples

			The prime factorizations of k = 15 and k+1 = 16 are 3 * 5 and 2^4, respectively, so their prime signatures can be represented as [1,1] and [4], respectively. If any ordered pair of consecutive integers m and m+1 has this same ordered pair of prime signatures, then m+1 = p^4 for some prime p, so m = p^4 - 1 = (p-1)*(p+1)*(p^2+1), which is a multiple of 16 for any odd prime p, so the prime signature of m cannot be [1,1] unless the prime p is even, i.e., p = 2, so m = 2^4 - 1 = 15; there is no m other than k = 15 that yields the same pair of prime signatures, so k = 15 is a term of the sequence.
k = 125 is not a term of the sequence: 125 = 5^3 and 126 = 2 * 3^2 * 7, and the same pair of prime signatures occurs for m and m+1 at m = 67^3 = 300763; m+1 = 300764 = 2^2 * 17 * 4423.
		

Crossrefs

Cf. A124832 (prime signatures), A161460.