A359260 Numbers m such that the arithmetic mean of the first k divisors of m is an integer for all k in 1..d(m), where d(m) = A000005(m).

1, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 37, 41, 43, 47, 49, 51, 53, 59, 61, 67, 69, 71, 73, 79, 83, 87, 89, 91, 97, 101, 103, 107, 109, 113, 123, 127, 131, 133, 137, 139, 141, 149, 151, 157, 159, 163, 167, 169, 173, 177, 179, 181, 191, 193, 197, 199, 211

Offset

1,2

Comments

All the terms are arithmetic numbers (A003601).

All the terms are odd numbers.

All the odd primes are terms.

There are infinitely many composite numbers in this sequence. For example, if p is a prime of the form 6*k-1 (A007528), then 3*p is a term. Also, if p is a prime of the form 6*k + 1 (A002476), then p^2 is a term.

prime(n)^k is a term for k = 0..A359262(n).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

15 is a term since its divisors are {1, 3, 5, 15}, 1/1 =1, (1 + 3)/2 = 2, (1 + 3 + 5)/3 = 3, and (1 + 3 + 5 + 15)/4 = 6 are all integers.

Mathematica

q[n_] := AllTrue[Accumulate[(d = Divisors[n])]/Range[Length[d]], IntegerQ]; Select[Range[1, 200, 2], q]

Prog

(PARI) is(n) = {my(s = k = 0); fordiv(n, d, k++; s += d; if(s%k, return(0))); 1; }

Crossrefs

Subsequence of A003601.

Subsequences: A065091, A343022 \ {81}.

Cf. A000005, A002476, A007528, A065091, A359261, A359262.

Sequence in context: A103796 A368100 A302568 * A294674 A340077 A179458

Adjacent sequences: A359257 A359258 A359259 * A359261 A359262 A359263

Keyword

nonn,easy

Author

Amiram Eldar, Dec 23 2022

Status

approved