

A107925


Odd numbers k such that k^2 is an arithmetic number.


5



1, 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 91, 97, 103, 109, 121, 127, 133, 139, 151, 157, 163, 181, 193, 199, 211, 217, 223, 229, 241, 247, 259, 271, 277, 283, 301, 307, 313, 331, 337, 349, 367, 373, 379, 397, 403, 409, 421, 427, 433, 439, 457, 463, 469, 481
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OFFSET

1,2


COMMENTS

Also, odd numbers k such that the arithmetic and geometric means of the divisors of k^2 are both integer.
Even numbers with this property are more rare and given by A107924.
For p prime, p^(k1) is a term in A003601 if and only if (p^k1)/(p1) is divisible by k. So p^e is a term here if and only if k  (p^k1)/(p1) for k = 2*e+1. (Note that p cannot be equal to 2 if k  (p^k1)/(p1).)
If a,b are both here, gcd(a,b) = 1, then a*b is also here. If a is A107924 and b is here, gcd(a,b) = 1, then a*b is also in A107924.
Let r >= 1, p_1, p_2, ..., p_r be distinct primes, k_1, k_2, ..., k_r be odd numbers such that Product_{i=1..r} (p_i)^(k_i) is an arithmetic number. Then there exists a number i in 1..r such that (p_i)^(k_i) is an arithmetic number. See my link for a proof. (End)


LINKS



MATHEMATICA

Select[Range[1, 500, 2], Mod[DivisorSigma[1, #^2], DivisorSigma[0, #^2]]==0&]


CROSSREFS



KEYWORD

easy,nonn


AUTHOR



STATUS

approved



