A064234 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.

1, 3, 28, 5, 66, 7, 232, 45, 190, 11, 276, 13, 1106, 0, 286, 17, 1854, 19, 3820, 891, 2752, 23, 1128, 595, 2046, 0, 532, 29, 1770, 31, 9952, 425, 1288, 0, 2486, 37, 8474, 0, 742, 41, 3486, 43, 7612, 5589, 2356, 47, 13584, 325, 9850, 0, 20554, 53, 5778, 0, 12926, 435, 13282, 59, 42540, 61

Offset: 1

Author

Robert G. Wilson v, Sep 22 2001

Keywords

• nonn

Comments

From Richard N. Smith, Jul 15 2019: (Start)

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).

The first counterexample of the comment is A063994(1541) = 484, which is an even nontotient, for the counterexamples <= 2^20, see the link.

Also A063994(1072871) = 68. (thus a(68) = 1072871).

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.

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)

Links

• Richard N. Smith, Table of n, a(n) for n = 1..187
• Richard N. Smith, Numbers n such that A063994(n) is an even nontotient (format: n,A063994(n))

Crossrefs

Cf. A063994, A005277, A002202, A049225, A254139.

Programs

Mathematica

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]

PARI

a063994(n)=my(f=factor(n)[, 1]); prod(i=1, #f, gcd(f[i]-1, n-1))
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

Extensions

a(54) - a(60) from Richard N. Smith, Jul 15 2019