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A058887
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Smallest prime p such that (2^n)*p is a nontotient number.
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6
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3, 7, 17, 19, 19, 19, 31, 31, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47, 47
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OFFSET
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0,1
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COMMENTS
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For n=8,9,...,582, a(n) = 47. Note that A040076(47)=583.
For n=583,584,...,6392, a(n) = 383. Note that A040076(383)=6393.
Subsequent primes are 2897, 3061, 5297, and 7013 (cf. A057192 and A071628). [These are primes p such that the least e such that 2^e*p + 1 is prime sets a new record. - Jianing Song, Dec 14 2021]
Starting with some large N, a(n)=p for all n >= N. This prime p will likely be the first prime Sierpiński number, which is conjectured to be 271129.
In particular, a(n) <= 271129 for all n.
a(n) is the smallest prime p such that 2^e*p + 1 is composite for all 0 <= e <= n. A proof is given in the a-file below.
a(n) is also the smallest number k such that 2^n*k is a nontotient number (see A181662). (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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D. Bressoud, CNT.m Computational Number Theory Mathematica package.
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FORMULA
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Min{p|p is prime and card(invphi((2^n)*p))=0}.
a(0) = 3;
a(1) = 7;
a(2) = 17;
a(3..5) = 19;
a(6..7) = 31;
a(8..582) = 47;
a(583..6392) = 383;
a(6393..9714) = 2897;
a(9715..33287) = 3061;
a(33288..50010) = 5297;
a(50011..126112) = 7013;
a(126113..31172164) = 10223.
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EXAMPLE
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For n=1, the initial segment of {2p} sequence is nops(invphi({2p}))={4, 4, 2, 0, 2, 0, 0, 0, 2, 2, ...}, where the position of the first 0 is 4, corresponding to p(4)=7, so a(1)=7.
For n=8 the same initial segment is: {11, 32, 23, 18, 24, 10, 11, 4, 9, 21, 2, 16, 9, 12, 0, 14, 5, 6, 12, ...}, where the first 0 is the 15th, corresponding to p(15)=47, thus a(8)=47.
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MATHEMATICA
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Needs["CNT`"]; Table[p=3; While[PhiInverse[p*2^n] != {}, p=NextPrime[p]]; p, {n, 0, 20}]
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PROG
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(PARI) a(n) = my(p=2); while(istotient(2^n*p), p=nextprime(p+1)); p; \\ Michel Marcus, May 14 2020
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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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