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A265285
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Carmichael numbers (A002997) k such that k-1 is a square.
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3
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OFFSET
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1,1
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COMMENTS
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This sequence contains all Carmichael numbers n such that for all primes p dividing n, p-1 divides n-1 and furthermore, n-1 is a square.
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LINKS
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G. Tarry, I. Franel, A. Korselt, and G. Vacca, Problème chinois, L'intermédiaire des mathématiciens 6 (1899), pp. 142-144.
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EXAMPLE
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46657 is a term because 46657 - 1 = 46656 = 216^2.
2433601 is a term because 2433601 - 1 = 2433600 = 1560^2.
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MAPLE
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isA002997:= proc(n) local F, p;
if n::even or isprime(n) then return false fi;
F:= ifactors(n)[2];
if max(seq(f[2], f=F)) > 1 then return false fi;
andmap(f -> (n-1) mod (f[1]-1) = 0, F)
end proc:
select(isA002997, [seq(4*i^2+1, i=1..10^6)]); # Robert Israel, Dec 08 2015
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PROG
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(PARI) is_c(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=1, 1e10, if(is_c(n) && issquare(n-1), print1(n, ", ")))
(PARI) lista(kmax) = {my(m); for(k = 2, kmax, m = k^2 + 1; if(!isprime(m), f = factor(k); for(i = 1, #f~, f[i, 2] *= 2); fordiv(f, d, if(!(m % (d+1)) && isprime(d+1), m /= (d+1))); if(m == 1, print1(k^2 + 1, ", ")))); } \\ Amiram Eldar, May 02 2024
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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