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A265237
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Carmichael numbers (A002997) that are the sum of two squares.
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4
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1105, 2465, 10585, 29341, 46657, 115921, 162401, 252601, 278545, 294409, 314821, 410041, 488881, 530881, 552721, 1461241, 1909001, 2433601, 3224065, 3581761, 4335241, 5148001, 5310721, 5444489, 5632705, 6054985, 6189121, 7207201, 7519441, 8134561, 8355841
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OFFSET
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1,1
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COMMENTS
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Carmichael numbers that are the sum of two distinct nonzero squares.
29341 is the first term for which neither of the squares can be the square of a prime.
Carmichael numbers that are not the sum of two squares start 561, 1729, 2821, 6601, 8911, 15841, ...
A Carmichael number m is a sum of two squares if and only if p == 1 (mod m) for every prime p|m. Observation, numerically checked by Amiram Eldar: the first 13 terms of this sequence are odd composites m such that m | EulerNumber(m-1) (A122045). - Thomas Ordowski, Mar 01 2020
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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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1105 is a term because 1105 = 23^2 + 24^2.
2465 is a term because 2465 = 41^2 + 28^2.
10585 is a term because 10585 = 37^2 + 96^2.
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MATHEMATICA
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PROG
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(PARI) is(n)=if(n<5, return(0)); my(f=factor(n)%4); if(vecmin(f[, 1])>1, return(0)); for(i=1, #f[, 1], if(f[i, 1]==3 && f[i, 2]%2, return(0))); 1
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, 1e7, if(is(n)&&is_c(n), print1(n, ", ")))
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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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