A command line calculator which supports mathematical expression with scientific functions is very useful for most developers.
Computations with the Mertens function. To accompany a preprint manuscript (http://arxiv.org/abs/2102.05842).
The least n with rad(n) | f(n) whose covering digraph has girth k, for eleven arithmetic functions and k = 2..10: 52 values proved minimal, 38 of them computed here for the first time, with a surgery theorem on the cycle and a certificate deciding m_f(k+1) < m_f(k) without computing it.
For S(f) = { n : rad(n) divides f(n) }, entire congruence classes of primes divide no element at all: none of S(sigma*_2) is divisible by a prime 3 mod 4, none of S(Phi_3) by a prime 2 mod 3. Also: exactly 1/4 of triples of primes 3 mod 4 are realizable in S(sigma*), against no pair.
Squarefree n dividing the product of (p+b) over its own prime divisors. Membership depends only on b mod n, and S_b is finite for every b by a bound that is really a statement about arithmetic progressions of primes of difference b. Its maximum is the product of an effective universe of measured size ~ b^0.36.
Basic Android Studio calculator that perform basic arithmetic functions
A prime having an incoming arrow in the covering digraph of f is necessary but not sufficient for it to divide an element of S(f) = { n : rad(n) | f(n) }: for f(q^e) = q^(2e) - q^e + 1 the prime 3 is covered by 511 primes below 8000 and divides nothing. Such failures occur only at primes ramified for F.
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