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A344703
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Numbers k for which sigma(k^2) and psi(k^2) share a factor, where sigma is the sum of divisors, A000203, and psi is the Dedekind psi function, A001615.
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3
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14, 21, 26, 28, 35, 38, 39, 42, 52, 56, 57, 62, 63, 65, 70, 74, 76, 77, 78, 82, 84, 86, 93, 95, 98, 99, 104, 105, 111, 112, 114, 117, 119, 122, 124, 126, 129, 130, 133, 134, 140, 143, 146, 148, 152, 154, 155, 156, 158, 161, 166, 168, 171, 172, 175, 182, 183, 185, 186, 189, 190, 194, 195, 198, 201, 203, 206, 208, 209
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OFFSET
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1,1
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COMMENTS
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Numbers k for which A344695(k^2) > 1.
It can be shown that sigma(m) and psi(m) share a factor if m is nonsquare. (See A344695 for more detail.) So here we consider only square numbers, m = k^2.
For prime p, sigma(p^2) and psi(p^2) are coprime, since sigma(p^2) = p^2 + p + 1 = psi(p^2) + 1. So all terms are composite. We can say more, since for prime p and positive integer e, psi(p^(2*e)) = p^(2*e-1) * (p+1), whereas sigma(p^(2*e)) can be shown to be congruent to 1 modulo p and to 1 modulo p+1, so shares no factors with p^(2*e-1) * (p+1). So all terms are divisible by more than one prime.
If k is in the sequence, m*k is also present for any positive integer m coprime to k.
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LINKS
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EXAMPLE
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Sigma (A000203) and the Dedekind psi function (A001615) are both multiplicative, so we gain insight by showing the values of these functions using their multiplicative properties:-
sigma(14^2) = sigma(2^2) * sigma(7^2) = 7 * 57 = 7 * (3*19).
psi(14^2) = psi(2^2) * psi(7^2) = 6 * 56 = (2*3) * (2^3*7).
So sigma(14^2) and psi(14^2) share factors 3 and 7, so 14 is in the sequence.
Looking in particular at the shared factor 3, we see it is present in sigma(7^2) and psi(2^2). For prime p, sigma(p^2) and psi(p^2) equate to polynomials in p, so we deduce 3 divides sigma(p^2) for p congruent to 7 modulo 3, and divides psi(p^2) for p congruent to 2 modulo 3. From this we see all products of a prime of the form 3m+1 and a prime of the form 3m+2 are in the sequence; for instance (3*4+1) * (3*1+2) = 13 * 5 = 65.
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MAPLE
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filter:= proc(k) local n, F, sig, psi, t;
n:= k^2;
F:= map(t -> [t[1], 2*t[2]], ifactors(k)[2]);
sig:= mul((t[1]^(1+t[2])-1)/(t[1]-1), t=F);
psi:= n*mul(1+1/t[1], t=F);
igcd(sig, psi) > 1
end proc:
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PROG
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(PARI)
A001615(n) = if(1==n, n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
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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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