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

1,2

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.

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

Mathematica

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

Crossrefs

Cf. A002476, A003601, A107924.

Sequence in context: A038590 A218146 A129389 * A002476 A123365 A144921

Adjacent sequences: A107922 A107923 A107924 * A107926 A107927 A107928

Keyword

easy,nonn

Author

Zak Seidov, Jun 10 2005

Status

approved