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This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 108 = 4*27, where a(108) = 35, although a(4) = 7 and a(27) = 10. See A344702.
Antti Karttunen, <a href="/indexA344696/Si#SIGMANb344696.txt">Index entries Table of n, a(n) for sequences related to sigma(n) = 1..10404</a>
<a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
Antti Karttunen, <a href="/A344696/a344696.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
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Cf. also A344756.
This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 108, = 4*27, where a(4*27) = a(108) = 35, although a(4) = 7 and a(27) = 10.