A091732 -id:A091732 - cp's OEIS frontend

A362484 Irregular table read by rows in which the n-th row consists of all the numbers m such that iphi(m) = n, where iphi is the infinitary totient function A091732.

1, 2, 3, 6, 4, 8, 5, 10, 7, 12, 14, 24, 9, 15, 18, 30, 11, 22, 13, 20, 21, 26, 40, 42, 16, 32, 17, 27, 34, 54, 19, 28, 38, 56, 33, 66, 23, 46, 25, 35, 36, 39, 50, 60, 70, 72, 78, 120, 29, 58, 31, 44, 48, 62, 88, 96, 45, 51, 90, 102, 37, 52, 57, 74, 84, 104, 114, 168

Offset: 1

Amiram Eldar, Apr 22 2023

			The table begins:
  n   n-th row
  --  -----------------------
   1  1, 2;
   2  3, 6;
   3  4, 8;
   4  5, 10;
   5
   6  7, 12, 14, 24;
   7
   8  9, 15, 18, 30;
   9
  10  11, 22;
  11
  12  13, 20, 21, 26, 40, 42;

Amiram Eldar, Table of n, a(n) for n = 1..17858 (first 100000 rows)

Cf. A091732, A162247, A362485 (row lengths).

Similar sequences: A032447, A361966, A362213, A362180.

powQ[n_] := n == 2^IntegerExponent[n, 2]; powfQ[n_] := Length[fact = FactorInteger[n]] == 1 && powQ[fact[[1, 2]]];
invIPhi[n_] := Module[{fct = f[n], sol}, sol = Times @@@ (1 + Select[fct, UnsameQ @@ # && AllTrue[# + 1, powfQ] &]); Sort@ Join[sol, 2*sol]]; invIPhi[1] = {1, 2};
Table[invIPhi[n], {n, 1, 36}] // Flatten (* using the function f by T. D. Noe at A162247 *)

A326403 Numbers k such that iphi(k) = iphi(k+1), where iphi(k) is an infinitary analog to the Euler totient function (A091732).

Original entry on oeis.org

1, 20, 35, 143, 194, 208, 740, 1220, 1299, 1419, 1803, 1892, 3705, 3716, 3843, 5186, 5635, 7868, 10659, 13634, 13905, 17948, 18507, 18914, 18980, 21007, 25388, 25545, 30380, 31599, 32304, 34595, 37820, 47067, 70394, 73059, 78064, 87856, 94874, 105908, 116963

Offset: 1

Amiram Eldar, Sep 14 2019

nonn

			20 is in the sequence since iphi(20) = iphi(21) = 12.

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

Cf. A001274, A091732, A287055, A293184, A301866.

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); a[1] = 1; a[n] := Times @@ (Flatten @(f @@@ FactorInteger[n]) - 1); a1 = 1; s = {}; Do[a2 = a[n]; If[a1 == a2, AppendTo[s, n - 1]]; a1 = a2, {n, 2, 10^5}]; s

A362485 Number of numbers k such that iphi(k) = n, where iphi is the infinitary totient function A091732.

Original entry on oeis.org

2, 2, 2, 2, 0, 4, 0, 4, 0, 2, 0, 6, 0, 0, 2, 4, 0, 4, 0, 2, 0, 2, 0, 10, 0, 0, 0, 2, 0, 6, 0, 4, 0, 0, 0, 8, 0, 0, 0, 4, 0, 2, 0, 2, 2, 2, 0, 14, 0, 0, 0, 2, 0, 2, 0, 2, 0, 2, 0, 10, 0, 0, 0, 4, 0, 4, 0, 0, 0, 2, 0, 14, 0, 0, 0, 0, 0, 2, 0, 8, 0, 2, 0, 4, 0, 0

Offset: 1

Amiram Eldar, Apr 22 2023

nonn

a(n) is even for all n, because if k is a solution to iphi(k) = n, and A007814(k) is even, then 2*k is also a solution, i.e., iphi(2*k) = n.

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

Row lengths of A362484.
Cf. A007814, A091732, A362486 (positions of 0's), A362487 (indices of records).
Similar sequences: A014197, A063740, A361967, A362181.

a[n_] := Length[invIPhi[n]]; Array[a, 100] (* using the function invIPhi from A362484 *)

a(A362486(n)) = 0.

A330273 Infinitary perfect totient numbers: numbers that equal to the sum of their iterated infinitary totient function (A091732).

Original entry on oeis.org

3, 10, 21, 44, 93, 118, 170, 320, 548, 3596, 3620, 4772, 5564, 18260, 33051, 256425, 403700, 1071129, 1790160, 2318180, 3968852, 4027375, 10001319, 11270012, 12048740, 13358121, 31741593, 46271673, 56149161, 4344134553

Offset: 1

Amiram Eldar, Dec 13 2019

The infinitary version of A082897 (perfect totient numbers), in which the infinitary totient function iphi (A091732) replaces the Euler totient function (A000010).

			10 is an infinitary perfect totient number because iphi(10) + iphi(iphi(10)) + ... = 4 + 3 + 2 + 1 = 10.

Cf. A082897, A091732, A286067.

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); iphi[1] = 1; iphi[n] := iphi[n] = Times @@ (Flatten@(f @@@ FactorInteger[n]) - 1); infPerfTotQ[n_] := Plus @@ FixedPointList[iphi@# &, n] == 2 n + 1; Select[Range[1000], infPerfTotQ]

A333609 The number of iterations of the infinitary totient function iphi (A091732) required to reach from n to 1.

Original entry on oeis.org

0, 1, 2, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 3, 4, 5, 6, 4, 5, 4, 4, 5, 6, 3, 4, 4, 6, 5, 6, 4, 5, 5, 5, 6, 4, 4, 5, 5, 4, 4, 5, 4, 5, 5, 6, 6, 7, 5, 6, 4, 6, 5, 6, 6, 5, 5, 5, 6, 7, 4, 5, 5, 6, 7, 6, 5, 6, 6, 6, 4, 5, 4, 5, 5, 6, 7, 5, 4, 5, 5, 6, 5, 6, 5, 8, 5, 6

Offset: 1

Amiram Eldar, Mar 28 2020

nonn

			a(6) = 2 since there are 2 iterations from 6 to 1: iphi(6) = 2 and iphi(2) = 1.

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

Cf. A003434, A049865, A091732.

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], 1])); iphi[1] = 1; iphi[n_] := Times @@ (Flatten@(f @@@ FactorInteger[n]) - 1); a[n_] := Length @ NestWhileList[iphi, n, # != 1 &] - 1; Array[a, 100]

A340087 a(n) = gcd(n-1, A091732(n)), where A091732 is an infinitary analog of Euler's phi function.

Original entry on oeis.org

1, 1, 2, 3, 4, 1, 6, 1, 8, 1, 10, 1, 12, 1, 2, 15, 16, 1, 18, 1, 4, 1, 22, 1, 24, 1, 2, 9, 28, 1, 30, 1, 4, 1, 2, 1, 36, 1, 2, 3, 40, 1, 42, 1, 4, 1, 46, 1, 48, 1, 2, 3, 52, 1, 2, 1, 4, 1, 58, 1, 60, 1, 2, 9, 16, 5, 66, 1, 4, 3, 70, 1, 72, 1, 2, 3, 4, 1, 78, 1, 80, 1, 82, 1, 4, 1, 2, 3, 88, 1, 18, 1, 4, 1, 2, 5, 96

Offset: 1

Antti Karttunen, Dec 31 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A091732, A340088, A340089.

Cf. also A049559.

ispow2(n) = (n && !bitand(n,n-1));
A302777(n) = ispow2(isprimepower(n));
A091732(n) = { my(m=1); while(n > 1, fordiv(n, d, if((dA302777(n/d), m *= ((n/d)-1); n = d; break))); (m); };
A340087(n) = gcd(n-1, A091732(n));

a(n) = gcd(n-1, A091732(n)).
a(n) = A091732(n) / A340088(n).
For n > 1, a(n) = (n-1) / A340089(n).

A340088 a(n) = A091732(n) / gcd(n-1, A091732(n)), where A091732 is an infinitary analog of Euler's phi function.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, 6, 1, 6, 4, 1, 1, 8, 1, 12, 3, 10, 1, 6, 1, 12, 8, 2, 1, 8, 1, 15, 5, 16, 12, 24, 1, 18, 12, 4, 1, 12, 1, 30, 8, 22, 1, 30, 1, 24, 16, 12, 1, 16, 20, 18, 9, 28, 1, 24, 1, 30, 24, 5, 3, 4, 1, 48, 11, 8, 1, 24, 1, 36, 24, 18, 15, 24, 1, 60, 1, 40, 1, 36, 16, 42, 28, 10, 1, 32, 4, 66

Offset: 1

Antti Karttunen, Dec 31 2020

nonn

Conjecture: a(n) = 1 iff n = 1 or in A050376. This is an infinitary analog of Lehmer's totient conjecture from 1932.

For all i, j > 1: a(i) = a(j) => A302777(i) = A302777(j), if the above conjecture holds.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Wikipedia, Lehmer's totient problem

Cf. A050376, A084400, A091732, A302777, A340087, A340089.

Cf. also A160595.

ispow2(n) = (n && !bitand(n,n-1));
A302777(n) = ispow2(isprimepower(n));
A091732(n) = { my(m=1); while(n > 1, fordiv(n, d, if((dA302777(n/d), m *= ((n/d)-1); n = d; break))); (m); };
A340088(n) = { my(x=A091732(n)); (x/gcd(n-1, x)); };

a(n) = A091732(n) / A340087(n) = A091732(n) / gcd(n-1, A091732(n)).

For all n >= 1, a(A084400(n)) = 1.

A340089 a(n) = (n-1) / gcd(n-1, A091732(n)), where A091732 is an infinitary analog of Euler's phi function.

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 1, 7, 1, 9, 1, 11, 1, 13, 7, 1, 1, 17, 1, 19, 5, 21, 1, 23, 1, 25, 13, 3, 1, 29, 1, 31, 8, 33, 17, 35, 1, 37, 19, 13, 1, 41, 1, 43, 11, 45, 1, 47, 1, 49, 25, 17, 1, 53, 27, 55, 14, 57, 1, 59, 1, 61, 31, 7, 4, 13, 1, 67, 17, 23, 1, 71, 1, 73, 37, 25, 19, 77, 1, 79, 1, 81, 1, 83, 21, 85, 43, 29, 1, 89

Offset: 1

Antti Karttunen, Dec 31 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A091732, A340087, A340088.

Cf. also A160596.

ispow2(n) = (n && !bitand(n,n-1));
A302777(n) = ispow2(isprimepower(n));
A091732(n) = { my(m=1); while(n > 1, fordiv(n, d, if((dA302777(n/d), m *= ((n/d)-1); n = d; break))); (m); };
A340089(n) = ((n-1)/gcd(n-1, A091732(n)));

a(n) = (n-1) / A340087(n) = (n-1) / gcd(n-1, A091732(n)).

A362486 Infinitary nontotient numbers: values not in the range of the infinitary totient function iphi (A091732).

Original entry on oeis.org

5, 7, 9, 11, 13, 14, 17, 19, 21, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 47, 49, 50, 51, 53, 55, 57, 59, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 81, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98, 99, 101, 103, 105, 107, 109, 110, 111

Offset: 1

Amiram Eldar, Apr 22 2023

nonn

Numbers k such that A091732(x) = k has no solution, i.e., A362485(k) = 0.

Most of the odd numbers are in this sequence. Odd numbers that are not here are 1, 3, 15, 45, 255, 765, 3825, 11475, 65535, 196605, 983025, ..., which are the values of iphi at powers of 2.

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

Cf. A091732, A362484, A362485.

Similar sequences: A005277, A005278, A347771, A362182.

Select[Range[120], Length[invIPhi[#]] == 0 &] (* using the function invIPhi from A362484 *)

A362485(a(n)) = 0.

A323413 Infinitary analog of cototient function A051953: a(n) = n - A091732(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 4, 1, 5, 1, 6, 1, 6, 1, 8, 7, 1, 1, 10, 1, 8, 9, 12, 1, 18, 1, 14, 11, 10, 1, 22, 1, 17, 13, 18, 11, 12, 1, 20, 15, 28, 1, 30, 1, 14, 13, 24, 1, 18, 1, 26, 19, 16, 1, 38, 15, 38, 21, 30, 1, 36, 1, 32, 15, 19, 17, 46, 1, 20, 25, 46, 1, 48, 1, 38, 27, 22, 17, 54, 1, 20, 1, 42, 1, 48, 21, 44, 31, 58, 1, 58, 19

Offset: 1

Antti Karttunen, Jan 15 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..21000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A037445, A049417, A050376, A091732, A327575.

Cf. also A051953, A126168, A323410.

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], 1])); a[1] = 0; a[n_] := n - Times @@ (Flatten@(f @@@ FactorInteger[n]) - 1); Array[a, 100] (* Amiram Eldar, Jan 09 2021 *)

ispow2(n) = (n && !bitand(n,n-1));
A302777(n) = ispow2(isprimepower(n));
A091732(n) = { my(m=1); while(n > 1, fordiv(n, d, if((dA302777(n/d), m *= ((n/d)-1); n = d; break))); (m); };
A323413(n) = (n-A091732(n));

a(n) = n - A091732(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = 1/2 - A327575 = 0.171064... . - Amiram Eldar, Dec 15 2023

cp's OEIS Frontend

A362484 Irregular table read by rows in which the n-th row consists of all the numbers m such that iphi(m) = n, where iphi is the infinitary totient function A091732.

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A326403 Numbers k such that iphi(k) = iphi(k+1), where iphi(k) is an infinitary analog to the Euler totient function (A091732).

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A362485 Number of numbers k such that iphi(k) = n, where iphi is the infinitary totient function A091732.

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A330273 Infinitary perfect totient numbers: numbers that equal to the sum of their iterated infinitary totient function (A091732).

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A333609 The number of iterations of the infinitary totient function iphi (A091732) required to reach from n to 1.

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A340087 a(n) = gcd(n-1, A091732(n)), where A091732 is an infinitary analog of Euler's phi function.

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A340088 a(n) = A091732(n) / gcd(n-1, A091732(n)), where A091732 is an infinitary analog of Euler's phi function.

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A340089 a(n) = (n-1) / gcd(n-1, A091732(n)), where A091732 is an infinitary analog of Euler's phi function.

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A362486 Infinitary nontotient numbers: values not in the range of the infinitary totient function iphi (A091732).

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