Displaying 1-10 of 24 results found.
Numbers k such that A019320(k) is greater than A064078(k) and the latter is a prime or a prime power.
+20
1
18, 20, 21, 54, 147, 342, 602, 889, 258121
COMMENTS
The unique prime factor of A064078(k) is then a unique prime to base 2 (see A161509), but not a cyclotomic number. 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?
PROG
(PARI) for(n=1, +oo, c=polcyclo(n, 2); c % n < 2 && next(); c/=(c%n); ispseudoprime(if(ispower(c, , &b), b, c))&&print1(n, ", "))
a(n) is the least prime such that the multiplicative order of 2 mod a(n) equals n, or a(n)=1 if no such prime exists.
+10
18
1, 3, 7, 5, 31, 1, 127, 17, 73, 11, 23, 13, 8191, 43, 151, 257, 131071, 19, 524287, 41, 337, 683, 47, 241, 601, 2731, 262657, 29, 233, 331, 2147483647, 65537, 599479, 43691, 71, 37, 223, 174763, 79, 61681, 13367, 5419, 431, 397, 631, 2796203, 2351, 97, 4432676798593, 251, 103, 53, 6361, 87211
COMMENTS
If a(n) differs from 1, then a(n) is the minimal prime divisor of A064078(n); a(n)=n+1 iff n+1 is prime from A001122; a(n)=2n+1 iff 2n+1 is prime from A115591. If a(n) > 1 then a(n) is the index where n occurs first in A014664. - M. F. Hasler, Feb 21 2016 Bang's theorem (special case of Zsigmondy's theorem, see links): a(n)>1 for all n>6. - Jeppe Stig Nielsen, Aug 31 2020
PROG
(PARI) A112927(n, f=factor(2^n-1)[, 1])=!for(i=1, #f, znorder(Mod(2, f[i]))==n&&return(f[i])) \\ Use the optional 2nd arg to give a list of pseudoprimes to try when factoring of 2^n-1 is too slow. You may try factor(2^n-1, 0)[, 1]. - M. F. Hasler, Feb 21 2016
Number of prime factors of cyclotomic(n,2), which is A019320(n), the value of the n-th cyclotomic polynomial evaluated at x=2.
+10
12
0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 1, 2, 3, 3, 3, 2, 3, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 3, 1, 2, 3, 2, 3, 2, 2, 3, 1, 1, 3, 1, 3, 2, 2, 2, 1, 1, 2, 2, 1, 1, 3, 4, 1, 2, 3, 2, 2, 1, 3, 4
COMMENTS
The Mobius transform of this sequence yields A046051, the number of prime factors of Mersenne number 2^n-1. The number of prime factors in the primitive part of 2^n-1. - T. D. Noe, Jul 19 2008
EXAMPLE
a(11) = 2 because cyclotomic(11,2) = 2047, which has two factors: 23 and 89.
MATHEMATICA
Join[{0}, Table[Plus@@Transpose[FactorInteger[Cyclotomic[n, 2]]][[2]], {n, 2, 100}]]
PROG
(PARI) a(n) = #factor(polcyclo(n, 2))~; \\ Michel Marcus, Mar 06 2015
Zsigmondy numbers for a = 4, b = 1: Zs(n, 4, 1) is the greatest divisor of 4^n - 1^n ( A024036) that is relatively prime to 4^m - 1^m for all positive integers m < n.
+10
11
3, 5, 7, 17, 341, 13, 5461, 257, 1387, 41, 1398101, 241, 22369621, 3277, 49981, 65537, 5726623061, 4033, 91625968981, 61681, 1826203, 838861, 23456248059221, 65281, 1100586419201, 13421773, 22906579627, 15790321, 96076792050570581
COMMENTS
By Zsigmondy's theorem, the n-th Zsigmondy number for bases a and b is not 1 except in the three cases (1) a = 2, b = 1, n = 1, (2) a = 2, b = 1, n = 6, (3) n = 2 and a+b is a power of 2.
Number of primitive prime factors of 2^n - 1.
+10
11
0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 1, 2, 3, 3, 3, 1, 3, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 3, 1, 2, 3, 2, 3, 2, 2, 3, 1, 1, 3, 1, 3, 2, 2, 2, 1, 1, 2, 2, 1, 1, 3, 4, 1, 2, 3, 2, 2, 1, 3, 3, 2, 3, 2, 2, 3
COMMENTS
A prime factor of 2^n - 1 is called primitive if it does not divide 2^r - 1 for any r < n. Equivalently, p is a primitive prime factor of 2^n - 1 if ord(2,p) = n. Zsigmondy's theorem says that there is at least one primitive prime factor for n > 1, except for n=6. See A086252 for those n that have a record number of primitive prime factors. The prime factors are not counted with multiplicity, which matters for a(364)=4 and a(1755)=6. - Jeppe Stig Nielsen, Sep 01 2020
FORMULA
a(n) = Sum{d|n} mu(n/d) A046800(d), inverse Mobius transform of A046800.
EXAMPLE
a(11) = 2 because 2^11 - 1 = 23*89 and both 23 and 89 have order 11.
MATHEMATICA
Join[{0}, Table[cnt=0; f=Transpose[FactorInteger[2^n-1]][[1]]; Do[If[MultiplicativeOrder[2, f[[i]]]==n, cnt++ ], {i, Length[f]}]; cnt, {n, 2, 200}]]
PROG
(PARI) a(n) = sumdiv(n, d, moebius(n/d)*omega(2^d-1)); \\ Michel Marcus, Sep 12 2017
CROSSREFS
Cf. A046800, A046051 (number of prime factors, with repetition, of 2^n-1), A086252, A002588, A005420, A002184, A046801, A049093, A049094, A059499, A085021, A097406, A112927, A237043.
EXTENSIONS
Terms to a(500) in b-file from T. D. Noe, Nov 11 2010
Table T(n,k) in which n-th row lists prime factors of 2^n - 1 (n >= 2), without repetition.
+10
10
0, 1, 3, 7, 3, 5, 31, 3, 7, 127, 3, 5, 17, 7, 73, 3, 11, 31, 23, 89, 3, 5, 7, 13, 8191, 3, 43, 127, 7, 31, 151, 3, 5, 17, 257, 131071, 3, 7, 19, 73, 524287, 3, 5, 11, 31, 41, 7, 127, 337, 3, 23, 89, 683, 47, 178481, 3, 5, 7, 13, 17, 241
REFERENCES
J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
LINKS
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
EXAMPLE
The irregular triangle T(n,k) begins for n >= 2:
n\k 1 2 3 4 5
2: 3
3: 7
4: 3 5
5: 31
6: 3 7
7: 127
8: 3 5 17
9: 7 73
10: 3 11 31
11: 23 89
12: 3 5 7 13
13: 8191
14: 3 43 127
15: 7 31 151
16: 3 5 17 257
17: 131071
18: 3 7 19 73
19: 524287
20: 3 5 11 31 41
... (End)
MATHEMATICA
Array[FactorInteger[2^# - 1][[All, 1]] &, 25, 0] (* Paolo Xausa, Apr 18 2024 *)
Zsigmondy numbers for a = 7, b = 1: Zs(n, 7, 1) is the greatest divisor of 7^n - 1^n ( A024075) that is relatively prime to 7^m - 1^m for all positive integers m < n.
+10
10
6, 1, 19, 25, 2801, 43, 137257, 1201, 39331, 2101, 329554457, 2353, 16148168401, 102943, 4956001, 2882401, 38771752331201, 117307, 1899815864228857, 1129901, 11898664849, 247165843, 4561457890013486057, 5762401, 79797014141614001
COMMENTS
By Zsigmondy's theorem, the n-th Zsigmondy number for bases a and b is not 1 except in the three cases (1) a = 2, b = 1, n = 1, (2) a = 2, b = 1, n = 6, (3) n = 2 and a+b is a power of 2.
Number of divisors of 2^n - 1 that are relatively prime to 2^m - 1 for all 0 < m < n.
+10
9
1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 4, 2, 2, 4, 8, 2, 2, 2, 2, 2, 4, 4, 4, 2, 4, 2, 4, 2, 8, 4, 4, 2, 8, 4, 2, 4, 8, 8, 8, 2, 8, 2, 4, 4, 4, 4, 2, 2, 4, 4, 2, 4, 4, 8, 2, 4, 8, 4, 8, 4, 4, 8, 2, 2, 8, 2, 8, 4, 4, 4, 2, 2, 4, 4, 2, 2, 8, 16, 2, 4, 8, 4, 4, 2, 8, 8
COMMENTS
a(364) = 24 is the first term not a power of 2. - Jianing Song, Apr 29 2018
EXAMPLE
Divisors of 2^8-1 are {1, 3, 5, 15, 17, 51, 85, 255}, but only 1 and 17 are relatively prime to 2^m - 1 for all m < 8, thus a(8)=2.
MATHEMATICA
a = {1}; Do[ d = Divisors[2^n - 1]; l = Length[d]; c = 0; k = 1; While[ k < l + 1, If[ Union[ GCD[a, d[[k]] ]] == {1}, c++ ]; k++ ]; Print[c]; a = Union[ Flatten[ Append[a, Transpose[ FactorInteger[2^n - 1]][[ 1]] ]]], {n, 1, 100} ]
PROG
(Haskell)
a063982 n = a063982_list !! (n-1)
a063982_list = f [] $ tail a000225_list where
f us (v:vs) = (length ds) : f (v:us) vs where
ds = [d | d <- a027750_row v, all ((== 1). (gcd d)) us]
(PARI) a(n) = {my(v = vector(n-1, k, 2^k-1), na = 0, nb); fordiv(2^n-1, d, nb = 0; for (k=1, n-1, if (gcd(d, v[k]) == 1, nb++, break); ); if (nb == n-1, na++); ); return (na); } \\ Michel Marcus, Apr 30 2018
Zsigmondy numbers for a = 3, b = 1: Zs(n, 3, 1) is the greatest divisor of 3^n - 1^n ( A024023) that is relatively prime to 3^m - 1^m for all positive integers m < n.
+10
9
2, 1, 13, 5, 121, 7, 1093, 41, 757, 61, 88573, 73, 797161, 547, 4561, 3281, 64570081, 703, 581130733, 1181, 368089, 44287, 47071589413, 6481, 3501192601, 398581, 387440173, 478297, 34315188682441, 8401, 308836698141973, 21523361
COMMENTS
By Zsigmondy's theorem, the n-th Zsigmondy number for bases a and b is not 1 except in the three cases (1) a = 2, b = 1, n = 1, (2) a = 2, b = 1, n = 6, (3) n = 2 and a+b is a power of 2.
Zsigmondy numbers for a = 5, b = 1: Zs(n, 5, 1) is the greatest divisor of 5^n - 1^n ( A024049) that is relatively prime to 5^m - 1^m for all positive integers m < n.
+10
9
4, 3, 31, 13, 781, 7, 19531, 313, 15751, 521, 12207031, 601, 305175781, 13021, 315121, 195313, 190734863281, 5167, 4768371582031, 375601, 196890121, 8138021, 2980232238769531, 390001, 95397958987501, 203450521
COMMENTS
By Zsigmondy's theorem, the n-th Zsigmondy number for bases a and b is not 1 except in the three cases (1) a = 2, b = 1, n = 1, (2) a = 2, b = 1, n = 6, (3) n = 2 and a+b is a power of 2.
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