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%I #45 Jul 17 2019 09:27:26
%S 1,3,28,5,66,7,232,45,190,11,276,13,1106,0,286,17,1854,19,3820,891,
%T 2752,23,1128,595,2046,0,532,29,1770,31,9952,425,1288,0,2486,37,8474,
%U 0,742,41,3486,43,7612,5589,2356,47,13584,325,9850,0,20554,53,5778,0,12926,435,13282,59,42540,61
%N The least k such that A063994(k) = Product_{primes p dividing k} gcd(p-1, k-1) = n, or 0 if there's no such k.
%C From _Richard N. Smith_, Jul 15 2019: (Start)
%C The comment in the "Mathematica" section is not true: A063994(65513) = 76 (thus a(76) = 65513 instead of 0), but 76 is an even nontotient (in the sequence A005277).
%C The first counterexample of the comment is A063994(1541) = 484, which is an even nontotient, for the counterexamples <= 2^20, see the link.
%C Also A063994(1072871) = 68. (thus a(68) = 1072871).
%C Conjecture: a(n) = 0 iff n == 2 mod 4 and n+1 is composite, if this conjecture is true, then a(54), a(110), a(294), etc. would be 0.
%C Another conjecture: If A063994(k) = n and n == 2 mod 4, then n+1 is prime and k is a power of n+1. (End)
%H Richard N. Smith, <a href="/A064234/b064234.txt">Table of n, a(n) for n = 1..187</a>
%H Richard N. Smith, <a href="/A064234/a064234.txt">Numbers n such that A063994(n) is an even nontotient (format: n,A063994(n))</a>
%t f[ n_ ] := If[ n == 1, 1, Apply[ Times, GCD[ n - 1, Transpose[ FactorInteger[ n ] ] [ [ 1 ] ] - 1 ] ] ]; a = Table[ 0, {100} ]; Do[ m = f[ n ]; If[ m < 101 && a[ [ m ] ] == 0, a[ [ m ] ] = n ], {n, 1, 10^7} ]; a a(54) > 2*10^7. The zeros appear at positions that are the values in the sequence A005277, the nontotients: even n such that phi(m) = n has no solution. [Warning: This is wrong, see the "comment" section]
%o (PARI) a063994(n)=my(f=factor(n)[, 1]); prod(i=1, #f, gcd(f[i]-1, n-1))
%o a(n)=if(n%4==2 && !isprime(n+1), 0, for(k=1, 2^30, if(a063994(k)==n,return(k)))) \\ _Richard N. Smith_, Jul 15 2019, after _Charles R Greathouse IV_ in A063994
%Y Cf. A063994, A005277, A002202, A049225, A254139.
%K nonn
%O 1,2
%A _Robert G. Wilson v_, Sep 22 2001
%E a(54) - a(60) from _Richard N. Smith_, Jul 15 2019
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