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Search: a366609 -id:a366609
Displaying 1-7 of 7 results found. page 1
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A366608 a(n) = phi(4^n+1), where phi is Euler's totient function (A000010). +10
12
1, 4, 16, 48, 256, 800, 3840, 12544, 65536, 186624, 986880, 3345408, 16515072, 52306176, 252645120, 760320000, 4288266240, 13628740608, 64258375680, 218462552064, 1095233372160, 3105655160832, 16510446886912, 56000724240384, 280012271910912, 869940000000000 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
Max Alekseyev, Table of n, a(n) for n = 0..561
FORMULA
a(n) = A053285(2*n). - Max Alekseyev, Jan 08 2024
MATHEMATICA
EulerPhi[4^Range[0, 30]+1] (* Paolo Xausa, Oct 14 2023 *)
PROG
(PARI) {a(n) = eulerphi(4^n+1)}
(Python)
from sympy import totient
def A366608(n): return totient((1<<(n<<1))+1) # Chai Wah Wu, Oct 14 2023
CROSSREFS
Cf. A000010, A052539, A057940, A274903, A295501, A366605, A366606, A366607, A366609.
Cf. A053285, A366579, A366618, A366630, A366639, A366658, A366667, A366669, A366690, A366716.
KEYWORD
nonn
AUTHOR
Sean A. Irvine, Oct 14 2023
STATUS
approved
A366607 Sum of the divisors of 4^n+1. +10
11
3, 6, 18, 84, 258, 1302, 4356, 20520, 65538, 351120, 1110276, 5048232, 17041416, 82623888, 284225796, 1494039792, 4301668356, 20788904016, 73234343952, 332019460560, 1103789883396, 5936210280000, 18679788287496, 84884999116320, 282937726148616 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
Max Alekseyev, Table of n, a(n) for n = 0..561
FORMULA
a(n) = sigma(4^n+1) = A000203(A052539(n)).
a(n) = A069061(2*n). - Max Alekseyev, Jan 08 2024
EXAMPLE
a(3)=84 because 4^3+1 has divisors {1, 5, 13, 65}.
MAPLE
a:=n->numtheory[sigma](4^n+1):
seq(a(n), n=0..100);
MATHEMATICA
DivisorSigma[1, 4^Range[0, 30]+1] (* Paolo Xausa, Oct 14 2023 *)
PROG
(Python)
from sympy import divisor_sigma
def A366607(n): return divisor_sigma((1<<(n<<1))+1) # Chai Wah Wu, Oct 14 2023
CROSSREFS
Cf. A000203, A052539, A057940, A274903, A366603, A366605, A366606, A366608, A366609.
Cf. A069061, A366578, A366617, A366629, A366638, A366657, A366666, A366668, A366689, A366715.
KEYWORD
nonn
AUTHOR
Sean A. Irvine, Oct 14 2023
STATUS
approved
A366605 Number of distinct prime divisors of 4^n + 1. +10
10
1, 1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 3, 3, 4, 2, 5, 2, 4, 4, 4, 2, 6, 3, 5, 3, 5, 3, 6, 3, 3, 4, 5, 2, 6, 3, 6, 5, 5, 4, 9, 3, 5, 5, 5, 4, 10, 2, 4, 3, 6, 6, 9, 2, 4, 6, 6, 5, 8, 3, 7, 6, 6, 4, 10, 2, 9, 7, 6, 4, 8, 4, 6, 7, 5, 2, 12, 4, 9, 5, 4, 4, 10, 4, 6, 8, 10 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Max Alekseyev, Table of n, a(n) for n = 0..561
FORMULA
a(n) = omega(4^n+1) = A001221(A052539).
a(n) = A046799(2*n). - Max Alekseyev, Jan 08 2024
MATHEMATICA
PrimeNu[4^Range[0, 100]+1] (* Paolo Xausa, Oct 14 2023 *)
PROG
(PARI) for(n = 0, 100, print1(omega(4^n + 1), ", "))
(Python)
from sympy import primenu
def A366605(n): return primenu((1<<(n<<1))+1) # Chai Wah Wu, Oct 14 2023
CROSSREFS
Cf. A001221, A052539, A057940, A274903, A366604, A366606, A366607, A366608, A366609.
Cf. A046799, A366580, A366615, A366627, A366636, A366655, A366664, A119704, A366686, A366712.
KEYWORD
nonn
AUTHOR
Sean A. Irvine, Oct 14 2023
STATUS
approved
A366606 Number of divisors of 4^n+1. +10
10
2, 2, 2, 4, 2, 6, 4, 8, 2, 16, 4, 8, 8, 16, 4, 48, 4, 16, 16, 16, 4, 64, 8, 32, 8, 64, 8, 64, 8, 8, 16, 32, 4, 64, 12, 96, 32, 32, 16, 768, 8, 32, 32, 32, 16, 1536, 4, 16, 8, 64, 64, 512, 4, 16, 64, 96, 32, 256, 8, 128, 64, 64, 16, 1024, 4, 768, 128, 64, 16 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
Max Alekseyev, Table of n, a(n) for n = 0..561
FORMULA
a(n) = sigma0(4^n+1) = A000005(A052539(n)).
a(n) = A046798(2*n). - Max Alekseyev, Jan 08 2024
EXAMPLE
a(3)=4 because 4^3+1 has divisors {1, 5, 13, 65}.
MAPLE
a:=n->numtheory[tau](4^n+1):
seq(a(n), n=0..100);
MATHEMATICA
DivisorSigma[0, 4^Range[0, 100]+1] (* Paolo Xausa, Oct 14 2023 *)
PROG
(PARI) a(n) = numdiv(4^n+1);
(Python)
from sympy import divisor_count
def A366606(n): return divisor_count((1<<(n<<1))+1) # Chai Wah Wu, Oct 14 2023
CROSSREFS
Cf. A000005, A052539, A057940, A274903, A366602, A366605, A366607, A366608, A366609.
Cf. A046798, A366577, A366616, A366628, A366637, A366656, A366665, A344897, A366688, A366714.
KEYWORD
nonn
AUTHOR
Sean A. Irvine, Oct 14 2023
STATUS
approved
A366670 Smallest prime dividing 6^n + 1. +10
6
2, 7, 37, 7, 1297, 7, 13, 7, 17, 7, 37, 7, 1297, 7, 37, 7, 353, 7, 13, 7, 41, 7, 37, 7, 17, 7, 37, 7, 281, 7, 13, 7, 2753, 7, 37, 7, 577, 7, 37, 7, 17, 7, 13, 7, 89, 7, 37, 7, 193, 7, 37, 7, 1297, 7, 13, 7, 17, 7, 37, 7, 41, 7, 37, 7, 4926056449, 7, 13, 7, 137 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
Max Alekseyev, Table of n, a(n) for n = 0..4095
FORMULA
a(n) = A020639(A062394(n)). - Paul F. Marrero Romero, Oct 17 2023
MATHEMATICA
Table[FactorInteger[6^n + 1][[1, 1]], {n, 0, 68}] (* Paul F. Marrero Romero, Oct 17 2023 *)
CROSSREFS
Cf. A020639, A062394, A274904, A366630.
Cf. A002586, A366609, A366671, A038371, A366719.
KEYWORD
nonn
AUTHOR
Sean A. Irvine, Oct 15 2023
STATUS
approved
A366671 Smallest prime dividing 8^n + 1. +10
5
2, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 193, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 641, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 193, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5, 3, 769, 3, 5, 3, 17, 3, 5, 3, 97, 3, 5, 3, 17, 3, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
a(n) = 3 if n is odd. a(n) = 5 if n == 2 (mod 4). - Robert Israel, Nov 20 2023
LINKS
Max Alekseyev, Table of n, a(n) for n = 0..16383
FORMULA
a(n) = A020639(A062395(n)). - Paul F. Marrero Romero, Oct 20 2023
a(n) = A002586(3*n) for n >= 1. - Robert Israel, Nov 20 2023
MAPLE
P1000:= mul(ithprime(i), i= 4..1000):
f:= proc(n) local t;
if n::odd then return 3 elif n mod 4 = 2 then return 5 fi;
t:= igcd(8^n+1, P1000);
if t <> 1 then min(numtheory:-factorset(t)) else min(numtheory:-factorset(8^n+1)) fi
end proc:
map(f, [$0..100]); # Robert Israel, Nov 20 2023
MATHEMATICA
Table[FactorInteger[8^n + 1][[1, 1]], {n, 0, 78}] (* Paul F. Marrero Romero, Oct 20 2023 *)
PROG
(Python)
from sympy import primefactors
def A366671(n): return min(primefactors((1<<3*n)+1)) # Chai Wah Wu, Oct 16 2023
CROSSREFS
Cf. A002586, A366609, A052536, A366655, A274905, A366670, A038371, A366719.
KEYWORD
nonn
AUTHOR
Sean A. Irvine, Oct 15 2023
STATUS
approved
A372867 Distinct terms in A242017, listed in the order of their appearance. +10
0
3, 5, 17, 97, 641, 257, 193, 274177, 65537, 449, 59649589127497217, 769, 1238926361552897, 5441, 5953, 2424833, 7873, 2753, 3329, 10753, 45592577, 18433, 4673, 15937, 444929, 11777, 12161, 21698561, 6977 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Conjecture: every term except 3 belongs to A366609. - Bill McEachen, Jun 12 2024
LINKS
Table of n, a(n) for n=1..29.
CROSSREFS
Cf. A023394, A070592, A093179, A242017.
KEYWORD
nonn,more
AUTHOR
Jean-Marc Rebert, May 15 2024
STATUS
approved
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Last modified August 18 19:26 EDT 2024. Contains 375273 sequences. (Running on oeis4.)