A324707 -id:A324707 - cp's OEIS frontend

A324708 Lesser of tri-unitary amicable numbers pair: numbers (m, n) such that tsigma(m) = tsigma(n) = m + n, where tsigma(n) is the sum of the tri-unitary divisors of n (A324706).

114, 594, 1140, 5940, 8640, 10744, 12285, 13500, 44772, 60858, 62100, 67095, 67158, 79296, 79650, 79750, 118500, 142310, 143808, 177750, 185368, 298188, 308220, 356408, 377784, 462330, 545238, 600392, 608580, 609928, 624184, 635624, 643336, 643776, 669900

Offset: 1

Amiram Eldar, Mar 11 2019

nonn

The larger counterparts are in A324709.

			114 is in the sequence since it is the lesser of the amicable pair (114, 126): tsigma(114) = tsigma(126) = 114 + 126.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A002025, A002952, A126169, A292980, A322541, A324706, A324707, A324709.

f[p_, e_] := If[e == 3, (p^4-1)/(p-1), If[e==6, (p^8-1)/(p^2-1), p^e+1]]; tsigma[1]=1; tsigma[n_]:= Times @@ f @@@ FactorInteger[n]; s[n_] := tsigma[n] - n; seq={}; Do[m=s[n]; If[m>n && s[m]==n, AppendTo[seq, n]] ,{n,1,700000}]; seq

A324709 Larger of tri-unitary amicable numbers pair: numbers (m, n) such that tsigma(m) = tsigma(n) = m + n, where tsigma(n) is the sum of the tri-unitary divisors of n (A324706).

Original entry on oeis.org

126, 846, 1260, 8460, 11760, 10856, 14595, 17700, 49308, 83142, 62700, 71145, 73962, 83904, 107550, 88730, 131100, 168730, 149952, 196650, 203432, 306612, 365700, 399592, 419256, 548550, 721962, 669688, 831420, 686072, 691256, 712216, 652664, 661824, 827700

Offset: 1

Amiram Eldar, Mar 11 2019

nonn

The terms are ordered according to their lesser counterparts (A324708).

			126 is in the sequence since it is the larger of the amicable pair (114, 126): tsigma(114) = tsigma(126) = 114 + 126.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A002046, A002953, A126170, A292981, A322542, A324706, A324707, A324708.

f[p_, e_] := If[e == 3, (p^4-1)/(p-1), If[e==6, (p^8-1)/(p^2-1), p^e+1]]; tsigma[1]=1; tsigma[n_]:= Times @@ f @@@ FactorInteger[n]; s[n_] := tsigma[n] - n; seq={}; Do[m=s[n]; If[m>n && s[m]==n, AppendTo[seq, m]] ,{n,1,700000}]; seq

A322609 Numbers k such that s(k) = 2*k, where s(k) is the sum of divisors of k that have a square factor (A162296).

Original entry on oeis.org

24, 54, 112, 150, 294, 726, 1014, 1734, 1984, 2166, 3174, 5046, 5766, 8214, 10086, 11094, 13254, 16854, 19900, 20886, 22326, 26934, 30246, 31974, 32512, 37446, 41334, 47526, 56454, 61206, 63654, 68694, 71286, 76614, 96774, 102966, 112614, 115926, 133206

Offset: 1

Amiram Eldar, Dec 20 2018

nonn

This sequence is infinite since 6*p^2 is included for all primes p. Terms that are not of the form 6*p^2: 112, 1984, 19900, 32512, 134201344, ...
Includes 4*k if k is an even perfect number: see A000396. - Robert Israel, Jan 06 2019
From Amiram Eldar, Oct 01 2022: (Start)
24 = 6*prime(1)^2 = 4*A000396(1) is the only term that is common to the two forms that are mentioned above.
19900 is the only term below 10^11 which is not of any of these two forms. Are there any other such terms?
All the known nonunitary perfect numbers (A064591) are also of the form 4*k, where k is an even perfect number.
Equivalently, numbers k such that A325314(k) = -k. (End)

			24 is a term since A162296(24) = 48 = 2*24.

Amiram Eldar, Table of n, a(n) for n = 1..12091 (terms below 10^11; terms 1..300 from Robert Israel)

Cf. A162296, A325314.
Subsequence of A005101 and A013929.
Numbers k such that A162296(k) = m*k: A005117 (m=0), A001248 (m=1), this sequence (m=2), A357493 (m=3), A357494 (m=4).
Similar sequences: A000396, A002827, A007357, A054979, A064591, A322486, A323757, A324707, A327633.

filter:= proc(n) convert(remove(numtheory:-issqrfree,numtheory:-divisors(n)),`+`)=2*n end proc:
select(filter, [$1..200000]); # Robert Israel, Jan 06 2019

s[1]=0; s[n_] := DivisorSigma[1,n] - Times@@(1+FactorInteger[n][[;;,1]]); Select[Range[10000], s[#] == 2# &]

s(n) = sumdiv(n, d, d*(1-moebius(d)^2)); \\ A162296
isok(n) = s(n) == 2*n; \\ Michel Marcus, Dec 20 2018

from sympy import divisors, factorint
A322609_list = [k for k in range(1,10**3) if sum(d for d in divisors(k,generator=True) if max(factorint(d).values(),default=1) >= 2) == 2*k] # Chai Wah Wu, Sep 19 2021

A324706 The sum of the tri-unitary divisors of n.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 15, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 60, 26, 42, 40, 40, 30, 72, 32, 33, 48, 54, 48, 50, 38, 60, 56, 90, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 120, 72, 120, 80, 90, 60, 120, 62, 96, 80, 85, 84, 144

Offset: 1

Amiram Eldar, Mar 11 2019

A divisor d of n is tri-unitary if the greatest common bi-unitary divisor of d and n/d is 1.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Graeme L. Cohen, On an integer's infinitary divisors, Mathematics of Computation, Vol. 54, No. 189 (1990), pp. 395-411.
Pentti Haukkanen, On the k-ary convolution of arithmetical functions, The Fibonacci Quarterly, Vol. 38, No. 5 (2000) pp. 440-445.

Cf. A000203, A034448, A049417, A072691, A188999, A322485, A324707, A324708, A324709.

f[p_, e_] := If[e == 3, (p^4-1)/(p-1), If[e==6, (p^8-1)/(p^2-1), p^e+1]]; a[1]=1; a[n_]:= Times @@ f @@@ FactorInteger[n]; Array[a, 100]

A324706(n) = { my(f = factor(n)); prod(i=1, #f~, if(3==f[i,2], sigma(f[i,1]^f[i,2]), if(6==f[i,2], ((f[i,1]^8)-1)/((f[i,1]^2)-1), 1+(f[i,1]^f[i,2])))); }; \\ Antti Karttunen, Mar 12 2019

Multiplicative with a(p^3) = 1 + p + p^2 + p^3, a(p^6) = 1 + p^2 + p^4 + p^6, and a(p^e) = 1 + p^e otherwise.

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 - 1/p^3 + 1/p^4 - 2/p^6 + 2/p^8 - 1/p^9 - 1/p^12 + 1/p^13) = 0.72189237802... . - Amiram Eldar, Nov 24 2022

A335385 The number of tri-unitary divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 8, 2, 2, 4, 4, 4, 4, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 8, 4, 8, 4, 4, 2, 8, 2, 4, 4, 4, 4, 8, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4

Offset: 1

Amiram Eldar, Jun 04 2020

A divisor d of k is tri-unitary if the greatest common bi-unitary divisor of d and k/d is 1.

Differs from A037445 at n = 32, 96, 128, 160, 224, ...

			a(4) = 2 since 4 has 2 tri-unitary divisors, 1 and 4. 2 is not a tri-unitary divisor of 4 since the greatest common bi-unitary divisor of 2 and 4/2 = 2 is 2 and not 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Graeme L. Cohen, On an integer's infinitary divisors, Mathematics of Computation, Vol. 54, No. 189 (1990), pp. 395-411.
Pentti Haukkanen, On the k-ary convolution of arithmetical functions, The Fibonacci Quarterly, Vol. 38, No. 5 (2000), pp. 440-445.

Cf. A222266, A324706, A324707, A324708, A324709.

Cf. A000005, A034444, A037445, A048105, A049419, A286324, A322483.

f[p_, e_] := If[e == 3 || e == 6, 4, 2]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100]

a(n) = vecprod(apply(x -> if(x == 3 || x == 6, 4, 2), factor(n)[, 2])); \\ Amiram Eldar, Dec 18 2023

Multiplicative with a(p^e) = 4 if e = 3 or 6, and a(p^e) = 2 otherwise.

A335387 Tri-unitary harmonic numbers: numbers k such that the harmonic mean of the tri-unitary divisors of k is an integer.

Original entry on oeis.org

1, 6, 45, 60, 90, 270, 420, 630, 2970, 5460, 8190, 9100, 15925, 27300, 36720, 40950, 46494, 47520, 54600, 81900, 95550, 136500, 163800, 172900, 204750, 232470, 245700, 257040, 332640, 409500, 464940, 491400, 646425, 716625, 790398, 791700, 819000, 900900, 929880

Offset: 1

Amiram Eldar, Jun 04 2020

nonn

Equivalently, numbers k such that A324706(k) | (k * A335385(k)).

Differs from A063947 from n >= 18.

			45 is a term since its tri-unitary divisors are {1, 5, 9, 45} and their harmonic mean, 3, in an integer.

Amiram Eldar, Table of n, a(n) for n = 1..200

A324707 is a subsequence.
Analogous sequences: A001599 (harmonic numbers), A006086 (unitary), A063947 (infinitary), A286325 (bi-unitary), A319745 (nonunitary).
Cf. A324706, A335385.

f1[p_, e_] := If[e == 3 || e == 6, 4, 2]; f2[p_, e_] := If[e == 3, (p^4 - 1)/(p - 1), If[e == 6, (p^8 - 1)/(p^2 - 1), p^e + 1]]; f[p_, e_] := p^e * f1[p, e]/f2[p, e]; tuhQ[1] = True; tuhQ[n_] := IntegerQ[Times @@ (f @@@ FactorInteger[n])]; Select[Range[10^4], tuhQ]

A360524 Numbers k such that A360522(k) = 2*k.

Original entry on oeis.org

6, 12, 198, 240, 264, 270, 396, 540, 6720, 7920, 11880, 13770, 27540, 221760, 337440, 605880, 2500344, 6072570, 11135520, 12145140, 267193080, 441692160, 1112629770, 2225259540, 14575841280, 48955709880

Offset: 1

Amiram Eldar, Feb 10 2023

Analogous to perfect numbers (A000396) with A360522 instead of A000203.

a(27) > 10^11, if it exists.

			6 is a term since A360522(6) = 12 = 2 * 6.

Cf. A000203, A360522.

Similar sequences: A000396, A002827, A007357, A054979, A322486, A324707.

f[p_, e_] := p^e + e; q[n_] := Times @@ f @@@ FactorInteger[n] == 2*n; Select[Range[10^6], q]

is(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] + f[i,2]) == 2*n;}

cp's OEIS Frontend

A324708 Lesser of tri-unitary amicable numbers pair: numbers (m, n) such that tsigma(m) = tsigma(n) = m + n, where tsigma(n) is the sum of the tri-unitary divisors of n (A324706).

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A324709 Larger of tri-unitary amicable numbers pair: numbers (m, n) such that tsigma(m) = tsigma(n) = m + n, where tsigma(n) is the sum of the tri-unitary divisors of n (A324706).

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A322609 Numbers k such that s(k) = 2*k, where s(k) is the sum of divisors of k that have a square factor (A162296).

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A324706 The sum of the tri-unitary divisors of n.

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A335385 The number of tri-unitary divisors of n.

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A335387 Tri-unitary harmonic numbers: numbers k such that the harmonic mean of the tri-unitary divisors of k is an integer.

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A360524 Numbers k such that A360522(k) = 2*k.

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