A107925 Odd numbers k such that k^2 is an arithmetic number.

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

Offset: 1

Author

Zak Seidov, Jun 10 2005

Keywords

• easy
• nonn

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.

Contains A002476 as a subsequence.

From Jianing Song, Apr 25 2022: (Start)

For p prime, p^(k-1) is a term in A003601 if and only if (p^k-1)/(p-1) is divisible by k. So p^e is a term here if and only if k | (p^k-1)/(p-1) for k = 2*e+1. (Note that p cannot be equal to 2 if k | (p^k-1)/(p-1).)

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

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Jianing Song, Notes on A107925

Crossrefs

Cf. A002476, A003601, A107924.

Programs

Mathematica

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