A257643
Carmichael numbers k such that k-1 is squarefree.
Original entry on oeis.org
139952671, 74689102411, 121254376891, 187054437571, 231440115271, 236359158267, 303008129971, 306252926071, 380574791611, 426951670531, 556303918171, 639109148371, 660950414671, 1101375141511, 1483826843731, 1487491483171, 1861175569891, 2794268624071
Offset: 1
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t(n) = my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]>1||(n-1)%(f[i, 1]-1), return(0))); 1;
is(n) = n%2 && !isprime(n) && t(n) && n>1;
isok(n) = is(n) && issquarefree(n-1); \\ Altug Alkan, Nov 06 2015
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is(n) = my(f=factor(n)); for(i=1, #f~, if(f[i,1]%4<3 || f[i, 2]>1 || (n-1)%(f[i, 1]-1), return(0))); !isprime(n) && issquarefree(n-1)
is(n) = n%2 && !isprime(n) && t(n) && n>1 \\ Charles R Greathouse IV, Nov 09 2015
A329468
Carmichael numbers all of whose prime factors are congruent to 3 modulo 4.
Original entry on oeis.org
8911, 1024651, 1152271, 1773289, 5481451, 8830801, 9585541, 10267951, 14913991, 15888313, 26474581, 40917241, 45877861, 64377991, 67902031, 72108421, 72286501, 81926461, 94536001, 104852881, 111291181, 129762001, 139592101, 139952671, 178482151, 213835861, 368113411
Offset: 1
8911 = 7 * 19 * 67 is a term since it is a Carmichael number, and 7 == 19 == 67 == 3 (mod 4).
- Amiram Eldar, Table of n, a(n) for n = 1..10000
- Steven Galbraith, Jake Massimo and Kenneth G. Paterson, Safety in Numbers: On the Need for Robust Diffie-Hellman Parameter Validation, in: Dongdai Lin and Kazue Sako (eds.), Public-Key Cryptography - PKC 2019, 22nd IACR International Conference on Practice and Theory of Public-Key Cryptography, Beijing, China, April 14-17, 2019, Proceedings, Part II, Springer, 2019.
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aQ[n_] := CompositeQ[n] && Divisible[n - 1, CarmichaelLambda[n]] && AllTrue[ FactorInteger[n][[;;,1]], Mod[#, 4] == 3 &]; Select[Range[2*10^6], aQ]
A267462
Carmichael numbers that are not of the form x^2 + y^2 + z^2 where x, y and z are integers.
Original entry on oeis.org
8911, 1152271, 10267951, 14913991, 64377991, 67902031, 139952671, 178482151, 612816751, 652969351, 743404663, 2000436751, 2560600351, 3102234751, 3215031751, 5615659951, 5883081751, 7773873751, 8863329511, 9462932431, 10501586767, 11335174831, 12191597551, 13946829751, 16157879263, 21046047751
Offset: 1
Carmichael number 561 is not a term of this sequence because 561 = 2^2 + 14^2 + 19^2.
Carmichael number 8911 is a term because there is no integer values of x, y and z for the equation 8911 = x^2 + y^2 + z^2.
Carmichael number 10585 is not a term because 10585 = 0^2 + 37^2 + 96^2.
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filter:= proc(n)
local q;
if isprime(n) then return false fi;
if 2 &^ (n-1) mod n <> 1 then return false fi;
for q in ifactors(n)[2] do
if q[2] > 1 or (n-1) mod (q[1]-1) <> 0 then return false fi
od;
true
end proc:
select(filter, [seq(8*k+7, k=0..10^7)]); # Robert Israel, Jan 18 2016
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Select[8*Range[1,8000000]+7, CompositeQ[#] && Divisible[#-1, CarmichaelLambda[#]] &] (* Amiram Eldar, Jun 26 2019 *)
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isA004215(n) = { my(fouri, j) ; fouri=1 ; while( n >=7*fouri, if( n % fouri ==0, j= n/fouri -7 ; if( j % 8 ==0, return(1) ) ; ) ; fouri *= 4 ; ) ; return(0) ; } { for(n=1, 400, if(isA004215(n), print1(n, ", ") ; ) ; ) ; }
isA002997(n) = { my(f); bittest(n, 0) && !for(i=1, #f=factor(n)~, (f[2, i]==1 && n%(f[1, i]-1)==1)||return) && #f>1 }
for(n=0, 1e10, if(isA002997(n) && isA004215(n), print1(n, ", ")));
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isA002997(n) = { my(f); bittest(n, 0) && !for(i=1, #f=factor(n)~, (f[2, i]==1 && n%(f[1, i]-1)==1)||return) && #f>1 }
for(n=0, 1e10, if(isA002997(k=8*n+7), print1(k, ", ")));
A292021
Lucas-Carmichael numbers that are congruent to 1 (mod 4).
Original entry on oeis.org
20705, 80189, 120581, 1162349, 7274249, 8734109, 10260809, 14658349, 49412285, 90393029, 105818129, 110066669, 125532329, 256074029, 362868329, 366648281, 395032609, 434886605, 503733257, 558705449, 563601257, 574342145, 640057109, 939989609, 962529749
Offset: 1
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a=Select[Range[2, 10^6],!PrimeQ[#] && Union[Transpose[FactorInteger[#]][[2]]] == {1} && Union[Mod[# + 1, Transpose[FactorInteger[#]][[1]] + 1]]=={0} &] ;Select[a,Mod[#,4]==1 &] (* after Richard Pinch and Jeffrey Shallit at A006972 *)
Showing 1-4 of 4 results.
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