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A181662
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a(n) is the smallest positive integral multiple of 2^n not in the range of the Euler phi function.
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2
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3, 14, 68, 152, 304, 608, 1984, 3968, 12032, 24064, 48128, 96256, 192512, 385024, 770048, 1540096, 3080192, 6160384, 12320768, 24641536, 49283072, 98566144, 197132288, 394264576, 788529152, 1577058304, 3154116608, 6308233216, 12616466432, 25232932864, 50465865728, 100931731456
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OFFSET
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0,1
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COMMENTS
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Let a(n) = 2^n * k, then k must be odd, otherwise a(n)/2 is a totient number, which implies that a(n) is a totient.
Note that 271129 * 2^m is a nontotient for all m (see A058887), so k <= 271129. In fact, let p be smallest prime such that 2^e*p + 1 is composite for all 0 <= e <= n, then k <= p (since 2^n*p is a nontotient).
Actually, k is equal to p. To verify this, it suffices to show that k cannot be an odd composite number < 271129; that is to say, if 2^n * k is a nontotient for an odd composite number < 271129, then there exists k' < k such that 2^n * k' is a nontotient.
The case k < 383 can be easily checked. Let k be an odd composite number in the range (383, 271129), k * 2^n is a nontotient implies n < 2554 unless k = 98431 or 248959 (see the a-file below), then 383 * 2^n is a nontotient (the least n such that 383 * 2^n + 1 is prime is n = 6393). For k = 98431 or 248959, k * 2^n is a nontotient implies n < 7062, then 2897 * 2^n is a nontotient (the least n such that 2897 * 2^n + 1 is prime is n = 9715. (End)
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REFERENCES
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David Harden, Posting to Sequence Fans Mailing List, Sep 19 2010.
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LINKS
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FORMULA
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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