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A368448
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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.
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0
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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