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A192577 Numbers n such that the arithmetic mean of the unitary divisors of n is a prime number. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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).

LINKS

Klaus Brockhaus, Table of n, a(n) for n = 1..10000

A. Roldan Martinez, Numeros y hoja de calculo

EXAMPLE

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

PROG

(MAGMA) 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

(PARI) 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

CROSSREFS

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

Sequence in context: A248881 A205534 A285377 * A236343 A168063 A039873

Adjacent sequences:  A192574 A192575 A192576 * A192578 A192579 A192580

KEYWORD

nonn

AUTHOR

Antonio Roldán, Jul 04 2011

STATUS

approved

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Last modified January 23 00:58 EST 2018. Contains 298093 sequences.