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A192577
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Numbers n such that the arithmetic mean of the unitary divisors of n is a prime number.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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).
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LINKS
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EXAMPLE
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48 has unitary divisors 1, 3, 16, 48 and (1+3+16+48)/4 = 17 is prime, therefore 48 is in the sequence.
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PROG
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(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, ", ")))) }
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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