A110348
a(2) = 1 by definition; otherwise a(n) = A109347(n)/n.
Original entry on oeis.org
1, 1, 8, 45, 504, 5040, 86400, 1247400, 28828800, 544864320, 15850598400, 370507737600, 12996271411200, 362038989312000, 14867734494412800, 480878287553664000, 22629566473113600000, 833522365093017600000
Offset: 1
a(3) = 2*3*4/3 = 8, a(6)= 3*4*5*7*8*9/6 = 5040.
A121877
Numbers k such that (5^k - 3^k)/2 = A005059(k) is prime.
Original entry on oeis.org
13, 19, 23, 31, 47, 127, 223, 281, 2083, 5281, 7411, 7433, 19051, 27239, 35863, 70327, 128941, 147571, 182099, 866029
Offset: 1
-
Do[f=(5^n-3^n)/2;If[PrimeQ[f],Print[{n,f}]],{n,1,300}]
-
forprime(p=2,1e4,if(ispseudoprime((5^p-3^p)>>1),print1(p", "))) \\ Charles R Greathouse IV, Jun 16 2011
A122853
Numbers k such that (3^k + 5^k)/8 = A074606(k)/8 is a prime.
Original entry on oeis.org
3, 5, 7, 17, 19, 109, 509, 661, 709, 1231, 12889, 13043, 26723, 43963, 44789
Offset: 1
-
Do[f=5^n+3^n;If[PrimeQ[f/2^3],Print[{n,f/2^3}]],{n,1,1231}]
-
select(n->isprime((3^n + 5^n)/8), vector(2000,i,i)) \\ Charles R Greathouse IV, Feb 13 2011
A109325
Zsigmondy numbers for a = 3, b = 2: Zs(n, 3, 2) is the greatest divisor of 3^n - 2^n (A001047) that is relatively prime to 3^m - 2^m for all positive integers m < n.
Original entry on oeis.org
1, 5, 19, 13, 211, 7, 2059, 97, 1009, 11, 175099, 61, 1586131, 463, 3571, 6817, 129009091, 577, 1161737179, 4621, 267331, 35839, 94134790219, 5521, 4015426801, 320503, 397760329, 369181, 68629840493971, 7471, 617671248800299, 43112257
Offset: 1
Let n be 7; then the factorization of g(n) := 3^n-2^n is then g(7) = A(7) = 2059 since n is prime; let n be 3 then the factorization of g(3) = A(3) = 19 since n is prime; let n be 21, then the factorization is g(21) = A(3)*A(7)*A(21); and whether n is composite or not, with each n (at least) one new factor occurs besides the factors determined by the prime factors of n - so it is not purely multiplicative.
- N. Bliss, B. Fulan, S. Lovett, and J. Sommars, Strong Divisibility, Cyclotomic Polynomials, and Iterated Polynomials, Amer. Math. Monthly, 120 (2013), 519-536.
- Eric Weisstein's World of Mathematics, Zsigmondy's Theorem
-
f:=proc(a,M) local n,b,d,t1,t2;
b:=[];
for n from 1 to M do
t1:=divisors(n);
t2:=mul(a[d]^mobius(n/d), d in t1);
b:=[op(b),t2];
od;
b;
end; a:=[seq(3^n-2^n,n=1..50)];
f(a,50); # N. J. A. Sloane, Jun 07 2013
A109348
Zsigmondy numbers for a = 7, b = 3: Zs(n, 7, 3) is the greatest divisor of 7^n - 3^n that is relatively prime to 7^m - 3^m for all positive integers m < n.
Original entry on oeis.org
4, 5, 79, 29, 4141, 37, 205339, 1241, 127639, 341, 494287399, 2041, 24221854021, 82573, 3628081, 2885681, 58157596211761, 109117, 2849723505777919, 4871281, 8607961321, 197750389, 6842186811484434379, 5576881, 80962848274370701
Offset: 1
A109349
Zsigmondy numbers for a = 7, b = 5: Zs(n, 7, 5) is the greatest divisor of 7^n - 5^n that is relatively prime to 7^m - 5^m for all positive integers m < n.
Original entry on oeis.org
2, 3, 109, 37, 6841, 13, 372709, 1513, 176149, 1661, 964249309, 1801, 47834153641, 75139, 3162961, 3077713, 115933787267041, 30133, 5689910849522509, 3949201, 6868494361, 168846239, 13678413205562919109, 4654801, 97995219736887001
Offset: 1
A109254
New factors appearing in the factorization of 7^k - 2^k as k increases.
Original entry on oeis.org
5, 3, 67, 53, 11, 61, 13, 164683, 2417, 163, 739, 1871, 199, 1987261, 2221, 1301, 14894543, 71, 1289, 31, 136261, 17, 339121, 137, 443, 766606297, 19, 2017, 2279779036969771, 5329741, 43, 235448977, 23, 9552313, 47, 116462754638606501, 337, 16993, 101, 158305897173001
Offset: 1
a(1) = 5 because 7^1 - 2^1 = 5.
a(2) = 3 because, although 7^2 - 2^2 = 45 = 3^2 * 5 has prime factor 5, that has already appeared in this sequence, but the repeated prime factor of 3 is new.
a(3) = 67 because, although 7^3 - 2^3 = 335 = 5 * 67 has prime factor 5, that has already appeared in this sequence, but the prime factor of 67 is new.
a(4) = 53 because, although 7^4 - 2^4 = 2385 = 3^2 * 5 * 53, the prime factors of 3 and 5 have already appeared in this sequence, but the prime factor of 53 is new.
a(5) = 11 and a(6) = 61 because, although 7^5 - 2^5 = 16775 = 5^2 * 11 * 61, the prime factor of 5 has already appeared in this sequence, but the prime factors of 11 and 61 are new.
-
DeleteDuplicates[Flatten[FactorInteger[#][[All,1]]&/@Table[7^n-2^n,{n,50}]]] (* Harvey P. Dale, Apr 07 2022 *)
-
lista(nn) = {my(pf = []); for (k=1, nn, f = factor(7^k-2^k)[,1]; for (j=1, #f~, if (!vecsearch(pf, f[j]), print1(f[j], ", "); pf = vecsort(concat(pf, f[j])));););} \\ Michel Marcus, Nov 13 2016
A121938
Primes of the form (3^k + 5^k)/2^3 = A074606(k)/8.
Original entry on oeis.org
19, 421, 10039, 95383574161, 2384331073699, 1925929944387235853055979210606894889560480247048440342330377620014353281101
Offset: 1
-
Do[f=5^n+3^n;If[PrimeQ[f/2^3],Print[{n,f/2^3}]],{n,1,1231}] (* Alexander Adamchuk, Sep 14 2006 *)
A109291
New factors appearing in the factorization of 5^k - 2^k as k increases.
Original entry on oeis.org
3, 7, 13, 29, 1031, 19, 25999, 641, 5563, 11, 41, 1409, 11551, 541, 406898311, 1597, 31, 8161, 17, 22993, 82009, 3101039, 37, 397, 6357828601279, 61, 5521, 43, 1009, 3613, 23, 303293, 7591, 197479, 2650751, 380881, 151, 95801, 6660751, 53, 131, 25117, 1271899175923
Offset: 1
a(1) = 3 because 5^1 - 2^1 = 3.
a(2) = 7 because, although 5^2 - 2^2 = 21 = 3 * 7 has prime factor 3, that has already appeared in this sequence, but the factor of 7 is new.
a(3) = 13 because, although 5^3 - 2^3 = 117 = 3^2 * 13 has repeated prime factor 3, that has already appeared in this sequence, but the prime factor of 13 is new.
a(4) = 29 because, although 5^4 - 2^4 = 2385 = 609 = 3 * 7 * 29, the prime factors of 3 and 7 have already appeared in this sequence, but the prime factor of 29 is new.
a(5) = 1031 because, although 5^5 - 2^5 = 16775 = 3093 = 3 * 1031, the prime factor of 3 has already appeared in this sequence, but the prime factors of 1031 is new.
-
lista(nn) = {my(pf = []); for (k=1, nn, f = factor(5^k-2^k)[,1]; for (j=1, #f~, if (!vecsearch(pf, f[j]), print1(f[j], ", "); pf = vecsort(concat(pf, f[j])));););} \\ Michel Marcus, Nov 13 2016
Showing 1-9 of 9 results.
Comments