A322691 Five-column table read by rows: Primitive distinct quintuples that have the same value of phi, sigma, and tau.

15132960, 15870624, 15966240, 15975036, 16854684, 15175160, 15572856, 16579134, 16629354, 17492046, 17671392, 18346968, 18644448, 20598318, 20608038, 26382240, 27668256, 27843360, 27850284, 28026540, 28020384, 29474016, 29563296, 29667924, 31301556, 30743000, 31130008, 31356440, 34531750

Offset: 1

Author

Jud McCranie, Dec 30 2018

Keywords

• nonn
• tabf

Comments

The terms are consecutive quintuples, ordered so that (A) a(5i-4) < a(5i-3) < ... < a(5i) for i > 0, and (B) a(5i+1) < a(5i+6) for i >= 0. This sequence has primitive terms only. If k is relatively prime to all of the terms in a primitive quintuple, then multiplying the terms in that quintuple by k gives another solution - see A322681.

From David A. Corneth, Feb 15 2019: (Start)

Some numbers occur in more than one quintuple, for example 1773744050 is in the quintuples [1579877800, 1652932372, 1653851276, 1663815260, 1773744050] and [1652932372, 1653851276, 1663815260, 1773744050, 1774581050].

The 4693 distinct terms in the first 5000 terms have only 111 distinct prime factors, the largest being 22751. All of these primes differ 1 from a 29-smooth number. (End)

From David A. Corneth, Feb 17 2019: (Start)

A quintuple (e1, e2, e3, e4, e5) is valid and primitive if and only if

1. The elements are in increasing order.

2. Every element e of the quintuple has the same value for phi(e), sigma(e) and tau(e).

3. For every number k between e1 and e5 that's not in the quintuple, at least one of the following statements is false: phi(e1) = phi(k), sigma(e1) = sigma(k), tau(e1) = tau(k).

4. Let g be gcd(e1, e2, e3, e4, e5). Then for every d|g, (e1/d, e2/d, e3/d, e4/d, e5/d) is not a valid quintuple. Therefore, (e1, e2, e3, e4, e5) is primitive. (End)

Examples

			15132960, 15870624, 15966240, 15975036,and 16854684 have the same value of phi (3870720), sigma (55157760), and tau (192), so these five numbers are in the sequence.
		

Links

• Jud McCranie, Table of n, a(n) for n = 1..5000
• David A. Corneth, first 1000 quintuplets (according to Jud McCranies terms) along with the values for phi, sigma and tau.

Crossrefs

Cf. A134922, A322681, A322688, A322689, A322690, A322692, A322693, A322694, A322695, A322696, A322697, A306430.