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

%I A322691 #43 Mar 26 2019 18:36:03
%S A322691 15132960,15870624,15966240,15975036,16854684,15175160,15572856,
%T A322691 16579134,16629354,17492046,17671392,18346968,18644448,20598318,
%U A322691 20608038,26382240,27668256,27843360,27850284,28026540,28020384,29474016,29563296,29667924,31301556,30743000,31130008,31356440,34531750
%N A322691 Five-column table read by rows: Primitive distinct quintuples that have the same value of phi, sigma, and tau.
%C A322691 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.
%C A322691 From _David A. Corneth_, Feb 15 2019: (Start)
%C A322691 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].
%C A322691 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)
%C A322691 From _David A. Corneth_, Feb 17 2019: (Start)
%C A322691 A quintuple (e1, e2, e3, e4, e5) is valid and primitive if and only if
%C A322691 1. The elements are in increasing order.
%C A322691 2. Every element e of the quintuple has the same value for phi(e), sigma(e) and tau(e).
%C A322691 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).
%C A322691 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)
%H A322691 Jud McCranie, <a href="/A322691/b322691.txt">Table of n, a(n) for n = 1..5000</a>
%H A322691 David A. Corneth, <a href="/A322691/a322691.gp.txt">first 1000 quintuplets (according to Jud McCranies terms) along with the values for phi, sigma and tau. </a>
%e A322691 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.
%Y A322691 Cf. A134922, A322681, A322688, A322689, A322690, A322692, A322693, A322694, A322695, A322696, A322697, A306430.
%K A322691 nonn,tabf
%O A322691 1,1
%A A322691 _Jud McCranie_, Dec 30 2018