A181629 -id:A181629 - cp's OEIS frontend

A346245 Numbers k for which A003415(k) > k*A003557(k).

30, 210, 330, 390, 462, 510, 546, 570, 690, 714, 798, 858, 870, 930, 966, 1110, 1218, 1230, 1290, 1302, 1410, 1554, 1590, 1722, 1770, 1830, 2010, 2130, 2190, 2310, 2370, 2490, 2670, 2730, 2910, 3030, 3090, 3210, 3270, 3390, 3570, 3810, 3930, 3990, 4110, 4170, 4290, 4470, 4530, 4710, 4830, 4890, 5010, 5190, 5370, 5430

Offset: 1

Antti Karttunen, Jul 19 2021

nonn

Numbers k such that A342001(k) > k.

Numbers k such that their arithmetic derivative (A003415(k)) is larger than A102631(k), k^2 / (squarefree kernel of k).

Positions of negative terms in A346244.
Cf. A003415, A003557, A102631, A342001.
Seems to have many common terms with A181629, A265501 and A286652.

A365687 a(n) = number of fractions m/n, 0 <= m < n, gcd(m,n) = 1 whose partial fraction decomposition has integer part 0.

Original entry on oeis.org

1, 1, 2, 2, 4, 1, 6, 4, 6, 2, 10, 2, 12, 3, 4, 8, 16, 3, 18, 4, 6, 5, 22, 4, 20, 6, 18, 6, 28, 0, 30, 16, 10, 8, 12, 6, 36, 9, 12, 8, 40, 1, 42, 10, 12, 11, 46, 8, 42, 10, 16, 12, 52, 9, 20, 12, 18, 14, 58, 2, 60, 15, 18, 32, 24, 1, 66, 16, 22, 2, 70, 12, 72, 18, 20

Offset: 1

William P. Orrick, Sep 15 2023

nonn

If n = p_1^r_1 p_2^r_2 ... p_k^r_k where p_1, ..., p_k are distinct primes, the partial fraction decomposition of m/n has the form z + Sum_{i=1..k} a_i / p_i^r_i = z + Sum_{i=1..k} Sum_{j=1..r_i} a_ij / p_i^j where 0 < a_i < p_i^r_i, gcd(a_i,p_i) = 1, and where 0 <= a_ij < p_i (a_ij > 0 when j = r_i). z is the integer part.
If 0 < m / n < 1 and gcd(m,n) = 1 then the integer part satisfies 1 - k <= z <= 0.
If n is a nonhyperbolic number, which means that Sum_{i=1..k} 1 / p_i^r_i >= 1, then the integer part is not zero for any m/n with 0 < m / n < 1, gcd(m,n) = 1.
Assuming gcd(m,n) = 1, if m/n has integer part z then (n - m)/n has integer part 1 - k - z. This means there are equally many reduced fractions with denominator n > 1 between 0 and 1 having integer part z and integer part 1 - k - z.

			a(10) = 2 because, of the four nonnegative reduced fractions less than 1 with denominator 10, two of them (7/10 and 9/10) have integer part 0:
   1/10 = -1 + 1/2 + 3/5
   3/10 = -1 + 1/2 + 4/5
   7/10 = 1/2 + 1/5
   9/10 = 1/2 + 2/5.

Cf. A070306, A181629, A028236.

def a(n):
    b = n
    fs = factor(b)
    # bzList will hold Bezout coefficients to express 1/n as combination
    # of the reciprocals of the prime power factors of n. ppList will
    # hold the prime power factors themselves.
    bzList = []
    bz0 = 1
    ppList = []
    for f in fs:
        q = f[0]^f[1]
        ppList.append(q)
        b = b / q
        bzThis = xgcd(q,b)
        bzList.append(bz0*bzThis[2])
        bz0 = bz0 * bzThis[1]
    ct = 0
    for j in n.coprime_integers(n):
        if sum(floor(j*bzList[i]/ppList[i])\
         for i in range(len(ppList))) == 0:
            ct = ct + 1
    return(ct)

a(n) = phi(n) if n is a prime power.
a(n) = phi(n) / 2 if n is the product of powers of two distinct primes.
a(n) = 0 if n is in A181629, that is, if n is a nonhyperbolic number.
a(n) = A070306(n) if n is a prime power or a product of powers of two distinct primes.

cp's OEIS Frontend

A346245 Numbers k for which A003415(k) > k*A003557(k).

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A365687 a(n) = number of fractions m/n, 0 <= m < n, gcd(m,n) = 1 whose partial fraction decomposition has integer part 0.

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