A347771
Unitary nontotient numbers: values not in range of unitary totient function uphi(n).
Original entry on oeis.org
5, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 33, 34, 35, 37, 38, 39, 41, 43, 45, 47, 49, 50, 51, 53, 55, 57, 59, 61, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 81, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98, 99, 101, 103, 105, 107, 109, 110, 111, 113, 114, 115, 117, 118, 119, 121, 122, 123, 125, 129, 131, 133, 134, 135
Offset: 1
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Select[Range[135], Length[invUPhi[#]] == 0 &] (* Amiram Eldar, Apr 01 2023, using the function invUPhi from A361966 *)
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A047994(n)=my(f=factor(n)~); prod(i=1, #f, f[1, i]^f[2, i]-1)
is(n)=for(k=1,n^2,if(A047994(k)==n,return(0)));1 \\ after A047994
A340521
List of possible orders of automorphism groups of finite groups.
Original entry on oeis.org
1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 28, 30, 32, 36, 40, 42, 44, 46, 48, 52, 54, 56, 58, 60
Offset: 1
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.
Original entry on oeis.org
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
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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]
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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
A308828
Number of sequences that include all residues modulo n starting with x_0 = 0 and then x_i given recursively by x_{i+1} = a * x_i + c (mod n) where a and c are integers in the interval [0..n-1].
Original entry on oeis.org
1, 1, 2, 2, 4, 2, 6, 8, 18, 4, 10, 4, 12, 6, 8, 32, 16, 18, 18, 8, 12, 10, 22, 16, 100, 12, 162, 12, 28, 8, 30, 128, 20, 16, 24, 36, 36, 18, 24, 32, 40, 12, 42, 20, 72, 22, 46, 64, 294, 100, 32, 24, 52, 162, 40, 48, 36, 28, 58, 16, 60, 30, 108, 512, 48, 20, 66, 32, 44, 24
Offset: 1
For n = 1:
a = 0, c = 0: [0];
#cycles = 1 -> a(1) = 1.
For n = 5:
a = 1, c = 1: [0, 1, 2, 3, 4];
a = 1, c = 2: [0, 2, 4, 1, 3];
a = 1, c = 3: [0, 3, 1, 4, 2];
a = 1, c = 4: [0, 4, 3, 2, 1];
#cycles = 4 -> a(5) = 4.
For n = 8:
a = 1, c = 1: [0, 1, 2, 3, 4, 5, 6, 7];
a = 1, c = 3: [0, 3, 6, 1, 4, 7, 2, 5];
a = 1, c = 5: [0, 5, 2, 7, 4, 1, 6, 3];
a = 1, c = 7: [0, 7, 6, 5, 4, 3, 2, 1];
a = 5, c = 1: [0, 1, 6, 7, 4, 5, 2, 3];
a = 5, c = 3: [0, 3, 2, 5, 4, 7, 6, 1];
a = 5, c = 5: [0, 5, 6, 3, 4, 1, 2, 7];
a = 5, c = 7: [0, 7, 2, 1, 4, 3, 6, 5];
#cycles = 8 -> a(8) = 8.
- D. E. Knuth, The Art of Computer Programming, Vol. 3, Random Numbers, Section 3.2.1.2, p. 16.
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checkFullSet(v,n)={my(v2=vector(n), unique=1); for(i=1, n, my(j=v[i]+1); if(v2[j]==1, unique=0; break, v2[j]=1;);); unique;};
doCycle(a,c,m)={my(v_=vector(m), x=c); v_[1]=c; for(i=1, m-1, v_[i+1]=(a*v_[i]+c)%m;); v_;};
getCycles(m)={my(L=List()); for(a=0, m-1, for(c=0, m-1, my(v1=doCycle(a,c,m)); if(checkFullSet(v1,m), listput(L, v1)););); Mat(Col(L))};
a(n)={my(M=getCycles(n)); matsize(M)[1]};
Showing 1-4 of 4 results.
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