A new idea for OEIS "triangle read by rows" sequence

[sweety439]

Triangle read by rows: a(m,n) = the gcd(m,n)-th number k such that sigma(k)/k = m/n (i.e. the abundancy of k is m/n, note that the abundancy is always >=1 and rational number), or 0 if no such k exists, for 1<=n<=m

All positive integers appear in this triangle exactly once (only 0 appears infinitely many times), however, computing this triangle is very hard, related topics: friendly number multiply perfect number hemiperfect number

This triangle begins with:

1 (1 is the first (and the only) number k such that sigma(k)/k = 1/1 = 1)
6, 0 (6 is the first number k such that sigma(k)/k = 2/1 = 2, and there is no second number k such that sigma(k)/k = 2/2 = 1, 1 is the only one such number)
120, 2, 0 (120 is the first number k such that sigma(k)/k = 3/1 = 3, 2 is the first number k such that sigma(k)/k = 3/2, and there is no third number k such that sigma(k)/k = 3/3 = 1, 1 is the only one such number)
30240, 28, 3, 0 (30240 is the first number k such that sigma(k)/k = 4/1 = 4, 28 is the second number k such that sigma(k)/k = 4/2 = 2, 3 is the first number k such that sigma(k)/k = 4/3, and there is no fourth number k such that sigma(k)/k = 4/4 = 1, 1 is the only one such number)
14182439040, 24, (unknown, there is no k <= 2^24 such that sigma(k)/k = 5/3), (unknown, there is no k <= 2^24 such that sigma(k)/k = 5/4), 0
154345556085770649600, 672 (the second k such that sigma(k)/k = 6/2 = 3), 496, 0 (2 is solitary number, thus there is no second number k such that sigma(k)/k = 3/2), 5, 0
141310897947438348259849402738485523264343544818565120000, 4320, 12, 4, (unknown), (unknown), 0
8268099687077761372899241948635962893501943883292455548843932421413884476391773708366277840568053624227289196057256213348352000000000, 32760, 84, 8128, 15, 0 (3 is solitary number, thus there is no second number k such that sigma(k)/k = 4/3), 7, 0
...

keywords: nonn, tabl, more, hard

[sweety439]

Problem: Prove or disprove such terms are 0:

a(5,3): Are there any k such that sigma(k)/k = 5/3?
a(5,4): Are there any k such that sigma(k)/k = 5/4?
a(7,5): Are there any k such that sigma(k)/k = 7/5?
a(7,6): Are there any k such that sigma(k)/k = 7/6?

I conjectured that no such k exists, but this can be hard to prove or disprove, like that it is hard to prove or disprove that 10 is solitary.

[sweety439]

b-file (or a-file):

(rows m = 1 to m = 12 of the triangle (indices i = 1 to i = 78), flattened, row m = 1 has index i = 1, row m = 2 has index i = {2,3}, row m = 3 has index i = {4,5,6}, row m = 4 has index i = {7,8,9,10}, ...)

1 1
2 6
3 0
4 120
5 2
6 0
7 30240
8 28
9 3
10 0
11 14182439040
12 24
13 0 (conjectured, b-file cannot contain conjectured terms)
14 0 (conjectured, b-file cannot contain conjectured terms)
15 0
16 154345556085770649600
17 672
18 496
19 0
20 5
21 0
22 141310897947438348259849402738485523264343544818565120000
23 4320
24 12
25 4
26 0 (conjectured, b-file cannot contain conjectured terms)
27 0 (conjectured, b-file cannot contain conjectured terms)
28 0
29 8268099687077761372899241948635962893501943883292455548843932421413884476391773708366277840568053624227289196057256213348352000000000
30 32760
31 84
32 8128
33 15
34 0
35 7
36 0
37 56130808183737158999998793684026231356147190822348283579122819870557664808030968216100782148452765644947099984854756332066651809002612793115408005967022213284272150201873375214629478176342119709234895003815657961417701371450048608475283004587476685222825422086715415685343739904000000000
38 8910720
39 523776
40 40
41 10
42 0
43 0 (conjectured, b-file cannot contain conjectured terms)
44 0 (conjectured, b-file cannot contain conjectured terms)
45 0
46 448565429898310924320164584477824539743733611787032214093531166213832480352545322596103983279901594851618303191976881540265700979715663129717227174064951388089181836554261404658300202164569689002163541973673164520563541918204730545126953234301917651168672330930792980648798941714800067628476201295868684534260358385663132152404556573415114464345129374402241250321888006130154014722016581041848460470899369707101530131054233224151410021638247842255328134212216267585925319281368941986654342379111273890042490862210239469135812761018153626037271326011687320220816667938885176742884376178910446080293797341901619200000000000000000000000000000
47 31998395520
48 1080
49 91963648
50 33550336
51 0 (conjectured, b-file cannot contain conjectured terms)
52 0 (conjectured, b-file cannot contain conjectured terms)
53 0 (conjectured, b-file cannot contain conjectured terms)
54 0 (conjectured, b-file cannot contain conjectured terms)
55 0
56 25185041348399291877483713498452839916966096879183629830423600245683852142136706259560800348986386756884928746427450335660601307023373038634588577582918892533431450402183724117455334595958960646443710559359553217238301822034060822235497467583505902376632937749821904538948595692372746818119489645548364913224810802818959192770727375458075135612292574997001997278799175655795461301046784886677005760822388567001630151101782815381184988668092886418826403557592465895134406328103108387886302588280426808773105723398511413080140942576327322081815502240446525869836049273257585109346805290065013209442964638084455345164451071503934096586345646621435252985421032613820965126563181089299972811779771827292133331127950543939413593680479692537947729759573681603882467419425304239282710389518990539657568601363717555733252090589049779121854544304430943422632475378859195505972332643589920186230935827402032392512358317904205719007276528974346996804088475093174195795846288979499984245887890932762691427735635608946229415529311685622909161472374878023577290544985026475478842990208730092690868098537241205864463576592480291550144216695662591474662609220161635272633460803908095561758934546539203777010617081602775055346295159952823126174561031213792909432811583524975624624923782045771007518368762761898257156254594714798330150065789591832417356833304083655085709749622056190377608624429624702581833657422942180419191308548660976118472588542171257328553869306196831264880724956685291206292203673425203402099172683799298900379846778295601867178657268395893138340588006559551079505099888372453843841958093814447626891996590684617850572626445740708184605705926892483089270269513227148864728809213020684836052682352877501319442767645251307037335411271471315001229908824788094350629237006350011614937855042347271887606679208087512719593728227162814693769216000000000000000000000000000000000000000000000000000000000000000000
57 17116004505600
58 35640
59 47616
60 0 (conjectured, b-file cannot contain conjectured terms)
61 0 (conjectured, b-file cannot contain conjectured terms)
62 0 (conjectured, b-file cannot contain conjectured terms)
63 0 (conjectured, b-file cannot contain conjectured terms)
64 0 (conjectured, b-file cannot contain conjectured terms)
65 0 (conjectured, b-file cannot contain conjectured terms)
66 0
67 (first 12-perfect number, should be exist)
68 9186050031556349952000
69 2178540
70 459818240
71 30
72 8589869056
73 14
74 0
75 0
76 0
77 11
78 0

[sweety439]

* Smallest base b such that n is a unique period (see unique prime) (cf. A085398):

a(n) for n = 1 to 100: 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 10, 2, 22, 2, 2, 4, 6, 2, 2, 2, 2, 2, 14, 3, 61, 2, 10, 2, 14, 2, 15, 25, 11, 2, 5, 5, 2, 6, 30, 11, 24, 2, 7, 2, 5, 7, 19, 3, 2, 2, 3, 3, 2, 9, 46, 47, 2, 3, 3, 3, 11, 16, 59, 7, 2, 2, 22, 2, 21, 61, 41, 7, 2, 2, 8, 5, 2, 2, 11, 4, 2, 6, 44, 4, 12, 2, 63, 20

* Smallest totient k > 1 such that n*k is not a totient, or 0 if no such k exists: (A301587: numbers n such that a(n) = 0) (A350085(n) = a(A007617(n)))

a(n) for n = 1 to 29: 0, 0, 30, 0, 10, 0, 2, 0, 10, 110, 22, 0, 2, 22, 6, 0, 2, 0, 2, 0, 54, 22, 10, 0, 2, 22, 22, <=28^2*29^110, 6

* Infinity-touchable numbers (assuming the strong version of Goldbach conjecture is true)

includes all numbers < 208 not in A005114, the first numbers which is neither in this sequence nor in A005114 are 208, 250, ...)

* Smallest starting value of exactly n-1 numbers with exactly n divisors, or 0 if no such number exists: (cf. A072507, A292580)

a(n) for n = 2 to 11: 5, 0, 33, 0, 10093613546512321, 0, 171893, 0, 0, 0

5 should be replaced by 2 if we use "at least n-1 numbers" instead of "exactly n-1 numbers"

a(12) <= 677667095479412562100444 (a(12) <= 247239052981730986799644 if we use "at least n-1 numbers" instead of "exactly n-1 numbers)

If k = floor(log_2(n-1)), there must be at least one term exactly divisible by 2^j for any j < k; hence the number of divisors must be divisible by j+1, or more generally by lcm_{i<=k} i. The only values of n divisible by this lcm are 1, 2, 3, 4, 6, 8, 12, 24, 60, 120 (e.g. for n = 30, there must be an element divisible by 8 but not by 16, so its number of divisors is divisible by 4, and for n = 36, there must be an element divisible by 16 but not by 32, so its number of divisors is divisible by 5), for n = 60, there must by two numbers 8k and 8(k+2) with k odd; then k and k+2 must each have 15 divisors, making them squares, thus a(n) = 0 for all n except 2, 3, 4, 6, 8, 12, 24, 120 (a(1) is not defined in this sequence), whether a(24) and a(120) exist is an open question, they exist if Schinzel's hypothesis H is true.

[sweety439]

store my text file here