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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.

%I A368448 #10 Dec 25 2023 13:46:15
%S A368448 1,2,3,4,7,8,15,16,24,26,27,31,32,35,48,63,64,80,99,124,127,128,224,
%T A368448 242,243,255,256,288,343,511,512,528,575,624,675,728,783,960,999,1023,
%U A368448 1024,1088,1295,1331,2047,2048,2186,2187,2208,2303,2400,3375,3968,4095,4096
%N 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.
%C A368448 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.
%C A368448 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.
%C A368448 For the analogous sequence using the number of divisors rather than the prime signature, see A161460.
%e A368448 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.
%e A368448 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.
%Y A368448 Cf. A124832 (prime signatures), A161460.
%K A368448 nonn
%O A368448 1,2
%A A368448 _Jon E. Schoenfield_, Dec 24 2023