A082897 -id:A082897 - cp's OEIS frontend

A333873 Numbers that equal to the sum of their iterated absolute Möbius divisor function (A173557).

3, 5, 17, 257, 413, 611, 1391, 1589, 1903, 2327, 5599, 27959, 29623, 36647, 36983, 38863, 42851, 43919, 46463, 49513, 65537, 76759, 82969, 86567, 88759, 96839, 111179, 116479, 129307, 171191, 184979, 213041, 277619, 301157, 310519, 346151, 362263, 372227, 375167

Offset: 1

Amiram Eldar, Apr 08 2020

nonn

A variant of A082897 (perfect totient numbers) in which the absolute Möbius divisor function (A173557) replaces the Euler totient function (A000010).

			5 is a term since A173557(5) = 4, A173557(4) = 1, and 4 + 1 = 5.

Amiram Eldar, Table of n, a(n) for n = 1..350 (terms below 4*10^9)
Daeyeoul Kim, Umit Sarp, and Sebahattin Ikikardes, Iterating the Sum of Möbius Divisor Function and Euler Totient Function, Mathematics, Vol. 7, No. 11 (2019), pp. 1083-1094.

A019434 is a subsequence.

Cf. A082897, A173557, A286067, A330273, A333871.

f[p_, e_] := p - 1; u[1] = 1; u[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^4], Plus @@ FixedPointList[u, #] == 2*# + 1 &]

A335121 Admirable totient numbers: numbers that are equal to the sum of their iterated phi, with one of them taken with a minus sign.

Original entry on oeis.org

5, 7, 33, 35, 55, 87, 95, 175, 201, 215, 219, 245, 531, 747, 927, 939, 1047, 1295, 1463, 1473, 1551, 1855, 2015, 2103, 2421, 2431, 2547, 2619, 2631, 2765, 3535, 4833, 5067, 5215, 7655, 7743, 7851, 10503, 11127, 11307, 13055, 13707, 16247, 16593, 17805, 18471

Offset: 1

Amiram Eldar, May 24 2020

nonn

Analogous to A111592 (admirable numbers) as A082897 (perfect totient numbers) is analogous to A000396 (perfect numbers).

			5 is a term since the values of the iterated phi of 5 are 4, 2 and 1 and 5 = 4 + 2 - 1.

Amiram Eldar, Table of n, a(n) for n = 1..171 (terms below 10^9)

Subsequence of A286265.

Cf. A000010, A053478, A082897, A111592, A288452.

admTotQ[n_] := Module[{s = Most @ Rest @ FixedPointList[EulerPhi, n]}, (ab = Plus @@ s - n) > 0 && EvenQ[ab] && ab/2 < n && MemberQ[s, ab/2]]; Select[Range[8000], admTotQ]

A348216 Numbers k such that A348215(k) = k.

Original entry on oeis.org

120, 1320, 2760, 3480, 3720, 4920, 5160, 5640, 6360, 7080, 7320, 8040, 8520, 8760, 9480, 9960, 10680, 11640, 12120, 12360, 12840, 13080, 13560, 14520, 15240, 15720, 16440, 16680, 17880, 18120, 18840, 19560, 20040, 20760, 21480, 21720, 22920, 23160, 23640, 23880

Offset: 1

Amiram Eldar, Oct 07 2021

nonn

Are there odd terms in this sequence? There are none below 10^8.

			120 is a term since the iterations of the map x -> A348158(x) starting from 120 are 120 -> 63 -> 57 and A348215(120) = 57 + 63 = 120.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A082897, A348158, A348213, A348214, A348215.

f[n_] := Plus @@ DeleteDuplicates @ Map[EulerPhi, Divisors[n]]; s[n_] := Plus @@ Most @ FixedPointList[f, n] - n; Select[Range[24000], s[#] == # &]

A181627 Number of iterations of phi(n) if n is a perfect totient number.

Original entry on oeis.org

2, 3, 4, 4, 5, 5, 6, 7, 6, 8, 7, 8, 8, 7, 8, 10, 11, 11, 11, 11, 9, 10, 12, 10, 14, 13, 11, 16, 14, 12, 16, 17, 13, 14, 19, 15, 20, 16, 17, 18, 18, 19, 24, 19, 20, 21, 22, 29, 32, 28, 30, 22, 29, 23, 30, 32, 24, 25, 31, 35, 26, 34, 35, 27

Offset: 1

Peter Luschny, Nov 02 2010

Let phi^{i} denote the i-th iteration of phi. a(n) is the smallest integer k such that phi^{k}(n) = 1 and Sum_{1<=i<=a(n)} phi^{i}(n) = n.

Paul Loomis, Michael Plytage and John Polhill, Summing up the Euler phi function, The College Mathematics Journal, Vol. 39, No. 1, Jan. 2008, pp. 34-42.
Peter Luschny, Sequences related to Euler's totient function.

Cf. A000010, A082897, A053478, A049108.

lst = (* get list from A082897 *); f[n_] := Length@ FixedPointList[ EulerPhi@ # &, n] - 2; f@# & /@ lst (* Robert G. Wilson v, Nov 06 2010 *)

a(n) = A049108(A082897(n)) - 1. - Amiram Eldar, Apr 14 2023

More terms from Robert G. Wilson v, Nov 06 2010

More terms from Amiram Eldar, Apr 14 2023

A286266 Number of totient abundant numbers <= 10^n.

Original entry on oeis.org

2, 36, 383, 3708, 35731, 347505, 3407290, 33579303, 332026623, 3290205509

Offset: 1

Amiram Eldar, May 05 2017

Totient abundant numbers are defined in A286265.

a(3)-a(8) were calculated by Loomis & Luca (2008).

			There are 2 totient abundant numbers <= 10^1 (5 and 7), thus a(1)=2.

Paul Loomis and Florian Luca, On totient abundant numbers, Electronic Journal of Combinatorial Number Theory, Vol. 8, #A06 (2008).

Cf. A000010, A082897, A092693, A286265.

Accumulate@ Table[Count[Select[Range[10^(n - 1) + 1, 10^n], (Total@ FixedPointList[EulerPhi, #] - (# + 1)) > # &], k_ /; k <= 10^n], {n, 6}] (* Michael De Vlieger, May 06 2017, after Alonso del Arte at A092693 *)

s(n) = {n=eulerphi(n); if(n==1, 1, n+s(n));}
lista(nmax) = {my(c = 0, r = 10); for(k = 1, 10^nmax, if(s(k) > k, c++); if(k == r, print1(c, ", "); r *= 10));} \\ Amiram Eldar, Mar 26 2023

a(9)-a(10) from Amiram Eldar, Mar 26 2023

A333104 Unitary quasiperfect cototient numbers: numbers k such that the sum of the iterated unitary cototient function of k is equal to k+1.

Original entry on oeis.org

10, 22, 98, 118, 230, 266, 1452, 88894, 114214, 1274198, 51675986, 61177358, 82986118

Offset: 1

Amiram Eldar, Mar 07 2020

a(14) > 10^9.

			10 is a term since A323410(10) = 6, A323410(6) = 4, A323410(4) = 1 and 6 + 4 + 1 = 11 = 10 + 1.

Cf. A082897, A286067, A323410, A330273, A333103.

uphi[0] = 0; uphi[1] = 1; uphi[n_] := (Times @@ (Table[#[[1]]^#[[2]] - 1, {1}] & /@ FactorInteger[n]))[[1]]; ucot[n_] := n - uphi[n]; Select[Range[10^4], Plus @@ FixedPointList[ucot, #] == 2*# + 1 &]

A343243 Sociable totient numbers of order 3: numbers k such that s(s(s(k))) = k, but s(k) != k, where s(k) = A092693(k) is the sum of iterated phi function.

Original entry on oeis.org

20339, 21159, 23883, 35503, 43255, 45375, 365599, 476343, 493047, 746383, 979839, 1097367, 3331135, 3816831, 3972543, 57720703, 68705247, 78376959, 3031407415, 3742563231, 3866214695

Offset: 1

Amiram Eldar, Apr 08 2021

The numbers k such that s(k) = k are the perfect totient numbers (A082897).

a(22) > 2*10^10, if it exists.

			20339 is a term since s(20339) = 23883, s(23883) = 21159 and s(21159) = 20339.

Cf. A000010, A082897, A092693, A286233.

totSum[n_] := Plus @@ FixedPointList[EulerPhi, n] - n - 1; soc3TotQ[n_] := Nest[totSum, n, 3] == n && totSum[n] != n; Select[Range[2, 10^6], soc3TotQ]

cp's OEIS Frontend

A333873 Numbers that equal to the sum of their iterated absolute Möbius divisor function (A173557).

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A335121 Admirable totient numbers: numbers that are equal to the sum of their iterated phi, with one of them taken with a minus sign.

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A348216 Numbers k such that A348215(k) = k.

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A181627 Number of iterations of phi(n) if n is a perfect totient number.

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A286266 Number of totient abundant numbers <= 10^n.

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A333104 Unitary quasiperfect cototient numbers: numbers k such that the sum of the iterated unitary cototient function of k is equal to k+1.

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A343243 Sociable totient numbers of order 3: numbers k such that s(s(s(k))) = k, but s(k) != k, where s(k) = A092693(k) is the sum of iterated phi function.

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