A327164 -id:A327164 - cp's OEIS frontend

A327158 Unitary multiply-perfect numbers: n divides usigma(n), where usigma is the sum of unitary divisors of n (A034448).

1, 6, 60, 90, 87360

Offset: 1

Antti Karttunen, Aug 28 2019

10^13 < a(6) <= 146361946186458562560000. - Giovanni Resta, Aug 29 2019

Peter Hagis, Jr., Lower bounds for unitary multiperfect numbers, Fibonacci Quarterly, Vol. 22, No. 2 (1984), pp. 140-143.

Fixed points of A323166, positions of zeros in A327164.
Cf. A002827 (a subsequence), A034448, A327163.
Cf. also A007691.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
isA327158(n) = (gcd(n,A034448(n))==n);

A327163 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = gcd(n,usigma(n)) * (-1)^[gcd(n,usigma(n))==n], and usigma is the sum of unitary divisors of n (A034448).

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 2, 2, 4, 2, 5, 2, 4, 6, 2, 2, 7, 2, 8, 2, 4, 2, 9, 2, 4, 2, 5, 2, 7, 2, 2, 6, 4, 2, 4, 2, 4, 2, 4, 2, 7, 2, 5, 10, 4, 2, 5, 2, 4, 6, 4, 2, 7, 2, 11, 2, 4, 2, 12, 2, 4, 2, 2, 2, 7, 2, 4, 6, 4, 2, 13, 2, 4, 2, 5, 2, 7, 2, 4, 2, 4, 2, 5, 2, 4, 6, 5, 2, 14, 15, 5, 2, 4, 16, 9, 2, 4, 6, 8, 2, 7, 2, 4, 6

Offset: 1

Antti Karttunen, Aug 28 2019

nonn

Restricted growth sequence transform of function f, defined as f(n) = -A323166(n) = -n when n is one of unitary multiply-perfect numbers (A327158), otherwise f(n) = A323166(n) = gcd(n,A034448(n))
For all i, j:
  A305800(i) = A305800(j) => a(i) = a(j) => A327164(i) = A327164(j).

Antti Karttunen, Table of n, a(n) for n = 1..87360

Cf. A034448, A305800, A323166, A327158, A327164.

Cf. also A326193.

up_to = 87360;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A323166(n) = gcd(n, A034448(n));
Aux327163(n) = { my(u=A323166(n)); u*((-1)^(u==n)); };
v327163 = rgs_transform(vector(up_to, n, Aux327163(n)));
A327163(n) = v327163[n];

cp's OEIS Frontend

A327158 Unitary multiply-perfect numbers: n divides usigma(n), where usigma is the sum of unitary divisors of n (A034448).

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