A192618 -id:A192618 - cp's OEIS frontend

A192577 Numbers n such that the arithmetic mean of the unitary divisors of n is a prime number.

3, 5, 6, 9, 12, 13, 25, 37, 48, 61, 73, 81, 121, 157, 193, 277, 313, 361, 397, 421, 457, 541, 613, 625, 661, 673, 733, 757, 768, 841, 877, 997, 1093, 1153, 1201, 1213, 1237, 1321, 1381, 1453, 1621, 1657, 1753, 1873, 1933, 1993, 2017, 2137, 2341, 2401, 2473

Offset: 1

Antonio Roldán, Jul 04 2011

nonn

Subsequence of A103826.
Similar to A187073, but considering unitary divisors, not prime divisors.
The odd terms of the sequence are: (1) the terms of A005383 (numbers n such that both n and (n+1)/2 are primes) and (2) the terms of A192618 (prime powers p^k with even exponents k>0 such that (1+p^k)/2 is prime).
[Note that A034448(n) and A034444(n) are multiplicative, so the arithmetic mean A034448(n)/A034444(n) is multiplicative with a(p^e) = (1+p^e)/2.]
The even terms of the sequence are 6, 12, 48, 768, 196608,... (no others < 10^10) with formula n = 3*2^(2^(k-1)) and averages 3, 5, 17, 257, 65537, ... (Fermat numbers, A000215).

			48 has unitary divisors 1, 3, 16, 48 and (1+3+16+48)/4 = 17 is prime, therefore 48 is in the sequence.

Klaus Brockhaus, Table of n, a(n) for n = 1..10000
A. Roldan Martinez, Numeros y hoja de calculo

Cf. A103826, A187073, A005383, A192618, A056798, A000215.

UnitaryDivisors:=func< n | [ d: d in Divisors(n) | Gcd(d, n div d) eq 1 ] >; [ n: n in [1..2500] | IsPrime(k) and s mod #U eq 0 where k is s div #U where s is &+U where U is UnitaryDivisors(n) ]; // Klaus Brockhaus, Jul 09 2011

usigma(n)= {local(f, u=1); f=factor(n); for(i=1, matsize(f)[1], u*=(1+ f[i, 1]^f[i, 2])); return(u)}
ud(n)= {local (f, u); f=factor(n); u=2^(matsize(f)[1]); return(u) }
{  for (n=2, 10^4, c=usigma(n)/ud(n); if (c==truncate(c),if(isprime(c), print1(n, ", ")))) }
\\ Antonio Roldán, Oct 08 2012

cp's OEIS Frontend

A192577 Numbers n such that the arithmetic mean of the unitary divisors of n is a prime number.

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