cp's OEIS Frontend

Numbers k such that A235137(k) == 3 (mod k).

A positive integer k is a 3-Sondow number if satisfies any of the following equivalent properties:

1) p^s divides k/p + 3 for every prime power divisor p^s of k.

2) 3/k + Sum_{prime p|k} 1/p is an integer.

3) 3 + Sum_{prime p|k} k/p == 0 (mod k).

4) Sum_{i=1..k} i^phi(k) == 3 (mod k).

Numbers k such that A235137(k) == 4 (mod k).

A positive integer k is a 4-Sondow number if satisfies any of the following equivalent properties:

1) p^s divides k/p + 4 for every prime power divisor p^s of k.

2) 4/k + Sum_{prime p|k} 1/p is an integer.

3) 4 + Sum_{prime p|k} k/p == 0 (mod k).

4) Sum_{i=1..k} i^phi(k) == 4 (mod k).

Other numbers in the sequence: 8858009688, 209981586408, 33961686334238753642827085044344

Numbers k such that A235137(k) == 5 (mod k).

A positive integer k is a 5-Sondow number if satisfies any of the following equivalent properties:

1) p^s divides k/p + 5 for every prime power divisor p^s of k.

2) 5/k + Sum_{prime p|k} 1/p is an integer.

3) 5 + Sum_{prime p|k} k/p == 0 (mod k).

4) Sum_{i=1..k} i^phi(k) == 5 (mod k).

Numbers k such that A235137(k) == 6 (mod k).

A positive integer k is a 6-Sondow number if satisfies any of the following equivalent properties:

1) p^s divides k/p + 6 for every prime power divisor p^s of k.

2) 6/k + Sum_{prime p|k} 1/p is an integer.

3) 6 + Sum_{prime p|k} k/p == 0 (mod k).

4) Sum_{i=1..k} i^phi(k) == 6 (mod k).

Other numbers in the sequence: 13287014532, 314972379612, 50942529501358130464240627566516

Numbers k such that A235137(k) == 7 (mod k).

A positive integer k is a 7-Sondow number if satisfies any of the following equivalent properties:

1) p^s divides k/p + 7 for every prime power divisor p^s of k.

2) 7/k + Sum_{prime p|k} 1/p is an integer.

3) 7 + Sum_{prime p|k} k/p == 0 (mod k).

4) Sum_{i=1..k} i^phi(k) == 7 (mod k).

Numbers k such that A235137(k) == 8 (mod k).

A positive integer k is a 8-Sondow number if satisfies any of the following equivalent properties:

1) p^s divides k/p + 8 for every prime power divisor p^s of k.

2) 8/k + Sum_{prime p|k} 1/p is an integer.

3) 8 + Sum_{prime p|k} k/p == 0 (mod k).

4) Sum_{i=1..k} i^phi(k) == 8 (mod k).

Other numbers in the sequence: 17716019376, 419963172816, 67923372668477507285654170088688

These are the weak primary pseudoperfect numbers mentioned in Grau-Oller-Sondow (2013).

Includes the primary pseudoperfect numbers (A054377). Any weak primary pseudoperfect number which is not a primary pseudoperfect number must have more than 58 distinct prime factors, and therefore must be greater than 10^110; none are known.

A positive integer j is a k-Sondow number if satisfies any of the following equivalent properties:

1) p^s divides j/p + k for every prime power divisor p^s of j.

2) k/j + Sum_{prime p|j} 1/p is an integer.

3) k + Sum_{prime p|j} j/p == 0 (mod j).

4) Sum_{i=1..j} i^A000010(j) == k (mod j).

Numbers m such that A235137(m) == 1 (mod m).