Search: a366609 -id:a366609
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A366608
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a(n) = phi(4^n+1), where phi is Euler's totient function (A000010).
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+10
12
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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
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OFFSET
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0,2
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LINKS
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FORMULA
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MATHEMATICA
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EulerPhi[4^Range[0, 30]+1] (* Paolo Xausa, Oct 14 2023 *)
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PROG
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(PARI) {a(n) = eulerphi(4^n+1)}
(Python)
from sympy import totient
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A366607
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Sum of the divisors of 4^n+1.
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+10
11
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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
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OFFSET
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0,1
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LINKS
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FORMULA
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EXAMPLE
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a(3)=84 because 4^3+1 has divisors {1, 5, 13, 65}.
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MAPLE
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a:=n->numtheory[sigma](4^n+1):
seq(a(n), n=0..100);
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MATHEMATICA
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DivisorSigma[1, 4^Range[0, 30]+1] (* Paolo Xausa, Oct 14 2023 *)
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PROG
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(Python)
from sympy import divisor_sigma
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A366605
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Number of distinct prime divisors of 4^n + 1.
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+10
10
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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
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OFFSET
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0,4
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LINKS
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FORMULA
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MATHEMATICA
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PrimeNu[4^Range[0, 100]+1] (* Paolo Xausa, Oct 14 2023 *)
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PROG
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(PARI) for(n = 0, 100, print1(omega(4^n + 1), ", "))
(Python)
from sympy import primenu
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A366606
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Number of divisors of 4^n+1.
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+10
10
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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
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OFFSET
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0,1
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LINKS
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FORMULA
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EXAMPLE
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a(3)=4 because 4^3+1 has divisors {1, 5, 13, 65}.
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MAPLE
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a:=n->numtheory[tau](4^n+1):
seq(a(n), n=0..100);
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MATHEMATICA
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DivisorSigma[0, 4^Range[0, 100]+1] (* Paolo Xausa, Oct 14 2023 *)
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PROG
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(PARI) a(n) = numdiv(4^n+1);
(Python)
from sympy import divisor_count
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A366670
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Smallest prime dividing 6^n + 1.
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+10
6
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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
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OFFSET
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0,1
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LINKS
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FORMULA
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MATHEMATICA
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A366671
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Smallest prime dividing 8^n + 1.
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+10
5
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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
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OFFSET
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0,1
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COMMENTS
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a(n) = 3 if n is odd. a(n) = 5 if n == 2 (mod 4). - Robert Israel, Nov 20 2023
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LINKS
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FORMULA
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MAPLE
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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:
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MATHEMATICA
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PROG
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(Python)
from sympy import primefactors
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A372867
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Distinct terms in A242017, listed in the order of their appearance.
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+10
0
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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
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OFFSET
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1,1
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COMMENTS
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LINKS
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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