OFFSET
1,2
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.
EXAMPLE
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
KEYWORD
nonn
AUTHOR
Jon E. Schoenfield, Dec 24 2023
STATUS
approved