A093106 -id:A093106 - cp's OEIS frontend

A093107 Numbers n such that the Zsigmondy number Zs(n,3,1) differs from the n-th cyclotomic polynomial evaluated at 3.

2, 4, 8, 16, 20, 32, 39, 42, 55, 64, 100, 128, 253, 256, 272, 294, 328, 342, 500, 507, 512, 605, 610, 666, 812, 876, 930, 1024, 1081, 1474, 1711, 1806, 2048, 2058, 2485, 2500, 2756, 2943, 3088, 3403, 3502, 4096, 4624, 4656, 5671, 5819, 6162, 6498, 6591, 6655

Offset: 1

Ralf Stephan, Mar 20 2004

nonn

A019321(n) does not equal A064079(n).

Cf. A093106, A093108, A093109.

Select[Range[10000], GCD[#, Cyclotomic[#, 3]]!=1 &] (* Emmanuel Vantieghem, Nov 13 2016 *)

More terms from Vladeta Jovovic, Apr 02 2004

Definition corrected by Jerry Metzger, Nov 04 2009

A093108 Numbers n such that the Zsigmondy number Zs(n,4,1) differs from the n-th cyclotomic polynomial evaluated at 4.

Original entry on oeis.org

3, 9, 10, 21, 27, 50, 55, 68, 78, 81, 147, 155, 171, 243, 250, 253, 301, 406, 410, 605, 657, 666, 729, 889, 979, 1014, 1029, 1081, 1156, 1250, 1378, 1582, 1711, 1830, 1962, 2056, 2187, 2211, 2265, 2328, 2485, 2892, 3081, 3249, 3403, 4082, 4658, 4805, 4965

Offset: 1

Ralf Stephan, Mar 20 2004

nonn

A019322(n) does not equal A064080(n).

Cf. A093107, A093106, A093109.

Select[Range[10000], GCD[#, Cyclotomic[#, 4]]!=1 &] (* Emmanuel Vantieghem, Nov 13 2016 *)

More terms from Vladeta Jovovic, Apr 02 2004

Definition corrected by Jerry Metzger, Nov 04 2009

A093109 Numbers n such that the Zsigmondy number Zs(n,5,1) differs from the n-th cyclotomic polynomial evaluated at 5.

Original entry on oeis.org

2, 4, 6, 8, 16, 18, 32, 42, 52, 54, 55, 64, 93, 128, 162, 171, 256, 272, 294, 355, 406, 486, 506, 512, 605, 676, 820, 1024, 1332, 1458, 1474, 1711, 1806, 1830, 2048, 2058, 2162, 2504, 2525, 2715, 2756, 2883, 2943, 3081, 3249, 3629, 3916, 4096, 4374, 4624, 5210

Offset: 1

Ralf Stephan, Mar 20 2004

nonn

Numbers n such that A019323(n) does not equal A064081(n).
Vladeta Jovovic points out that the sequence seems to contain the powers of two as well as the numbers of the form 2*3^k.
Numbers of the form ord(5,p)*p^k where prime p <> 5 and k > 0. Also numbers n > 0 such that A019323(n) =/= 1 (mod n). Also A019323(n) mod n = gcd(n, A019323(n)) = p. - Thomas Ordowski, Oct 22 2017

Cf. A093106, A093107, A093108, A019323, A064081.

Select[Range[10000], GCD[#, Cyclotomic[#, 5]]!=1 &] (* Emmanuel Vantieghem, Nov 13 2016 *)

More terms from Vladeta Jovovic, Apr 02 2004

Definition corrected by Jerry Metzger, Nov 04 2009

A250208 Ratio of the primitive part of 2^n-1 to the product of primitive prime factors of 2^n-1.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Eric Chen, Mar 02 2015

nonn

As with A178764, it can be shown that all terms are either 1 or prime.
a(2*3^n)   =  3 (n>=1).
a(4*5^n)   =  5 (n>=1).
a(3*7^n)   =  7 (n>=1).
a(10*11^n) = 11 (n>=1).
a(12*13^n) = 13 (n>=1).
a(8*17^n)  = 17 (n>=1).
a(18*19^n) = 19 (n>=1).
...
a(A014664(k)*prime(k)^n) = prime(k).
For other n (while Phi_n(2) is squarefree), a(n) = 1.
a(n) != 1 for n = {6, 18, 20, 21, 54, 100, 110, 136, 147, 155, 156, 162, ...}.
At least, a(A049093(n)) = 1. (In fact, since Phi_n(2) is not completely factored for n = 991, 1207, 1213, 1217, 1219, 1229, 1231, 1237, 1243, 1249, ..., so it is unknown whether they are squarefree or not, but it is likely that Phi_n(2) is squarefree for all n except 364 and 1755 (because it is likely 1093 and 3511 are the only two Wieferich primes), so a(991), a(1207), a(1213), ..., are likely to be 1.)

			a(11) = 1 since Phi_11(2) = (2^11-1)/(2-1) = 2047, and the primitive prime factors of 2^11-1 are 23 and 89, so a(11) = 2047/(23*89) = 1.
a(18) = 3 since Phi_18(2) = 2^6 - 2^3 + 1 = 57, and the only primitive prime factor of 2^18-1 is 19, so a(18) = 57/19 = 3.

Eric Chen, Table of n, a(n) for n = 1..990
Eric Chen, Table of n, a(n) for n = 1..1280 status (with unknown terms but all are conjectured to be 1)

Cf. A178764.

Cf. A001220, A237043, A093106, A049094, A049093, A014491, A072226, A161508.

a250208[n_] = If[n == 364, 1093, If[n == 1755, 3511, GCD[Cyclotomic[n, 2], n]]]; Table[a250208[n], {n, 0, 200}]

a(n) = if (n==364, 1093, if (n==1755, 3511, gcd(polcyclo(n, 2), n)));

isprimitive(p, n) = {for (r=1, n-1, if (((2^r-1) % p) == 0, return (0)); ); return (1); }
ppf(n) = {my(pf = factor(2^n-1)[,1]); prod(k=1,#pf, if (isprimitive(pf[k], n), pf[k], 1));}
a(n) = if (issquarefree(m=polcyclo(n,2)), gcd(m, n), m/ppf(n)); \\ Michel Marcus, Mar 06 2015

a(n) = A019320(n) / A064078(n) while Phi_n(2) is squarefree.
a(n) = GCD(Phi_n(2), n) while Phi_n(2) is squarefree.
Notice: a(364) = 1093, a(1755) = 3511. (See A001220.)

A333973 Numbers k such that A019320(k) is greater than A064078(k) and the latter is a prime or a prime power.

Original entry on oeis.org

18, 20, 21, 54, 147, 342, 602, 889, 258121

Offset: 1

Jeppe Stig Nielsen, Sep 22 2020

The unique prime factor of A064078(k) is then a unique prime to base 2 (see A161509), but not a cyclotomic number.
Subsequence of A161508. In fact, subsequence of the set difference A161508 \ A072226.
In all known examples, A064078(k) is a prime. If A064078(k) was a prime power p^j with j>1, then p would be both a Wieferich prime (A001220) and a unique prime to base 2.
Subsequence of A093106 (the characterization of A093106 can be useful when searching for more terms).
Should this sequence be infinite?

Henri Lifchitz and Renaud Lifchitz, Phi(258121,2)/719.
Wikipedia, Unique prime, section Binary unique primes.

Cf. A019320, A064078, A093106, A072226, A144755, A161508, A161509, A247071.

for(n=1,+oo,c=polcyclo(n,2); c % n < 2 && next(); c/=(c%n); ispseudoprime(if(ispower(c,,&b),b,c))&&print1(n, ", "))

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A093107 Numbers n such that the Zsigmondy number Zs(n,3,1) differs from the n-th cyclotomic polynomial evaluated at 3.

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A093108 Numbers n such that the Zsigmondy number Zs(n,4,1) differs from the n-th cyclotomic polynomial evaluated at 4.

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A093109 Numbers n such that the Zsigmondy number Zs(n,5,1) differs from the n-th cyclotomic polynomial evaluated at 5.

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A250208 Ratio of the primitive part of 2^n-1 to the product of primitive prime factors of 2^n-1.

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A333973 Numbers k such that A019320(k) is greater than A064078(k) and the latter is a prime or a prime power.

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