A361967 -id:A361967 - cp's OEIS frontend

A361971 Record values in A361967.

2, 3, 4, 5, 8, 11, 12, 14, 17, 23, 30, 31, 40, 64, 85, 95, 119, 147, 152, 207, 232, 257, 283, 344, 421, 469, 645, 956, 1034, 1306, 1578, 1797, 1943, 2304, 2334, 2877, 3217, 3396, 3536, 3973, 4378, 5171, 5457, 5464, 5659, 7586, 8317, 8430, 10609, 12566, 14469

Offset: 1

Amiram Eldar, Apr 01 2023

nonn

The unitary version of A131934.

Cf. A361966, A361967, A361968.

solnum[n_] :=  Length[invUPhi[n]]; seq[kmax_] := Module[{s = {}, solmax=0}, Do[sol = solnum[k]; If[sol > solmax, solmax = sol; AppendTo[s, sol]], {k, 1, kmax}]; s]; seq[10^5] (* using the function invUPhi from A361966 *)

a(n) = A361967(A361968(n)).

a(43)-a(51) from Amiram Eldar, Apr 10 2023

A327837 Decimal expansion of the asymptotic mean of the number of exponential divisors function (A049419).

Original entry on oeis.org

1, 6, 0, 2, 3, 1, 7, 1, 0, 2, 3, 0, 5, 4, 1, 8, 0, 5, 2, 3, 4, 9, 6, 2, 6, 3, 1, 5, 6, 2, 1, 1, 6, 1, 0, 0, 3, 7, 7, 6, 9, 3, 9, 4, 9, 5, 7, 8, 5, 5, 7, 2, 7, 3, 7, 7, 4, 6, 5, 3, 5, 2, 8, 5, 9, 8, 7, 8, 8, 8, 8, 6, 0, 2, 1, 6, 3, 3, 5, 4, 7, 2, 7, 5, 6, 6, 7, 3, 3, 9, 0, 4, 9, 4, 8, 8, 0, 6, 4, 1, 8, 0, 7, 5, 7

Offset: 1

Amiram Eldar, Sep 27 2019

			1.602317102305418052349626315621161003776939495785572...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 52 (constant Z3).
V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions - II, Indian J. Pure Appl. Math., Vol. 11 (1980), pp. 1334-1355 (eq. 2.37 and 3.18, pp. 1346 and 1354).
Abdelhakim Smati and Jie Wu, On the exponential divisor function, Publications de l'Institut Mathématique, Vol. 61 (1997), pp. 21-32.
László Tóth, An order result for the exponential divisor function, Publ. Math. Debrecen, Vol. 71, No. 1-2 (2007), pp. 165-171, arXiv preprint,, arXiv:0708.3552 [math.NT], 2007.
László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1 (section 4.10, p. 30).
Jie Wu, Problème de diviseurs exponentiels et entiers exponentiellement sans facteur carré, Journal de théorie des nombres de Bordeaux, Vol. 7, No. 1, (1995), pp. 133-141.

Cf. A000005, A047994, A049419, A145353, A361013, A361967, A379517, A379518.

Cf. A059956 (constant for unitary divisors), A306071 (bi-unitary), A327576 (infinitary).

$MaxExtraPrecision = 1500; m = 1500; em = 500; f[x_] := 1 + Log[1 + Sum[x^e * (DivisorSigma[0, e] - DivisorSigma[0, e - 1]), {e, 2, em}]]; c = Rest[ CoefficientList[Series[f[x], {x, 0, m}], x] * Range[0, m] ]; RealDigits[ Exp[NSum[Indexed[c, k] * PrimeZetaP[k]/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Equals lim_{k->oo} A145353(k)/k.
Equals Product_{p prime} (1 + Sum_{e >= 2} p^(-e) * (d(e) - d(e-1))), where d(e) is the number of divisors of e (A000005).
Equals Product_{p prime} (1 - 1/p) * (2 - (log(p-1) + QPolyGamma(0, 1, 1/p)) / log(p)). - Vaclav Kotesovec, Feb 27 2023
From Amiram Eldar, Dec 24 2024: (Start)
Equals lim_{m->oo} (1/m) * Sum_{k=1..m} k/uphi(k) = lim_{m->oo} (1/m) * Sum_{k=1..m} A319677(k)/A319676(k), where uphi(k) is the unitary totient function (A047994).
Equals lim_{m->oo} (1/log(m)) * Sum_{k=1..m} 1/uphi(k) = lim_{m->oo} (1/log(m)) * A379517(m)/A379518(m).
Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A361967(k).
Equals Product_{p prime} ((1-1/p) * (1 + Sum_{k>=1} 1/(p^k-1))).
Equals Product_{p prime} (1 + (1-1/p) * Sum_{k>=1} 1/(p^k*(p^k-1))). (End)

More digits from Vaclav Kotesovec, Jun 13 2021

A361966 Irregular table read by rows in which the n-th row consists of all the numbers m such that uphi(m) = n, where uphi is the unitary totient function (A047994).

Original entry on oeis.org

1, 2, 3, 6, 4, 5, 10, 7, 12, 14, 8, 9, 15, 18, 30, 11, 22, 13, 20, 21, 26, 42, 24, 16, 17, 34, 19, 28, 38, 33, 66, 23, 46, 25, 35, 36, 39, 50, 60, 70, 78, 27, 54, 29, 40, 58, 31, 44, 48, 62, 32, 45, 51, 90, 102, 37, 52, 57, 74, 84, 114, 41, 55, 82, 110, 43, 56, 86

Offset: 1

Amiram Eldar, Apr 01 2023

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

Amiram Eldar, Table of n, a(n) for n = 1..16006 (first 10000 rows)

The unitary version of A032447.

Cf. A047994, A135347, A162247, A361967 (row lengths), A361968, A361969, A361970, A361971.

invUPhi[n_] := Module[{fct = f[n], sol}, sol = Times @@@ (1 + Select[fct, UnsameQ @@ # && (Length[#] == 1 || CoprimeQ @@ (# + 1)) && Times @@ PrimeNu[# + 1] == 1 &]); Sort@ Join[sol, 2*Select[sol, OddQ]]]; invUPhi[1] = {1, 2}; Table[invUPhi[n], {n, 1, 50}] // Flatten (* using the function f by T. D. Noe at A162247 *)

A361968 Unitary highly totient numbers: numbers k that have more solutions x to the equation uphi(x) = k than any smaller k, where uphi is the unitary totient function (A047994).

Original entry on oeis.org

1, 6, 8, 12, 24, 48, 96, 120, 144, 240, 480, 576, 720, 1440, 2880, 4320, 5760, 8640, 10080, 17280, 20160, 30240, 34560, 40320, 60480, 80640, 120960, 241920, 362880, 483840, 725760, 967680, 1209600, 1451520, 2177280, 2419200, 2903040, 3628800, 4354560, 4838400

Offset: 1

Amiram Eldar, Apr 01 2023

nonn

Indices of records of A361967.

The corresponding numbers of solutions are 2, 3, 4, 5, 8, 11, ... (A361971).

Amiram Eldar, Table of n, a(n) for n = 1..51

The unitary version of A097942.

Cf. A047994, A135347, A347771, A361966, A361967, A361969, A361970, A361971.

solnum[n_] :=  Length[invUPhi[n]]; seq[kmax_] := Module[{s = {}, solmax=0}, Do[sol = solnum[k]; If[sol > solmax, solmax = sol; AppendTo[s, k]], {k, 1, kmax}]; s]; seq[10^5] (* using the function invUPhi from A361966 *)

A361969 Numbers k with a single solution x to the equation uphi(x) = k, where uphi is the unitary totient function (A047994).

Original entry on oeis.org

3, 7, 14, 15, 31, 54, 62, 63, 127, 154, 174, 182, 186, 234, 246, 254, 255, 294, 308, 318, 322, 364, 406, 414, 496, 510, 511, 516, 534, 558, 574, 594, 644, 666, 678, 762, 804, 806, 812, 846, 870, 948, 1022, 1023, 1026, 1036, 1074, 1098, 1146, 1148, 1164, 1204, 1246

Offset: 1

Amiram Eldar, Apr 01 2023

nonn

Numbers k such that A361967(k) = 1.
According to Carmichael's totient function conjecture, there are no numbers with a single solution x to the corresponding equation phi(x) = k, with Euler's totient function (A000010).
A000225(m) = 2^m - 1 is a term for all m >= 2. These are the only odd terms.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Carmichael's totient function conjecture.

Cf. A000010, A000225, A047994, A135347, A361966, A361967, A361968, A361970, A361971.

Select[Range[1250], Length[invUPhi[#]] == 1 &] (* using the function invUPhi from A361966 *)

A362181 Number of numbers k such that A323410(k) = n.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 2, 1, 3, 1, 3, 2, 3, 3, 2, 2, 3, 3, 4, 3, 3, 3, 3, 3, 4, 4, 3, 3, 4, 5, 4, 5, 4, 5, 3, 4, 4, 5, 3, 5, 3, 5, 5, 5, 4, 6, 4, 6, 4, 6, 2, 7, 4, 6, 4, 6, 3, 7, 3, 5, 4, 6, 3, 8, 2, 6, 6, 7, 4, 8, 4, 6, 6, 7, 3, 9, 4, 7, 4, 5, 5, 9, 6, 9, 4, 7, 3

Offset: 2

Amiram Eldar, Apr 10 2023

nonn

The offset is 2 since A323410(p) = 1 for all prime powers p (A246655).

a(0) = 1, since there is only one solution, x = 1, to A323410(x) = 0.

Amiram Eldar, Table of n, a(n) for n = 2..10000

Row lengths of A362180.
The unitary version of A063740.
Cf. A246655, A323410, A362182 (positions of 0's), A362183 (indices of records), A362184, A362185 (positions of 1's), A362186.
Similar sequences: A014197, A361967.

ucototient[n_] := n - Times @@ (Power @@@ FactorInteger[n] - 1); ucototient[1] = 0; With[{max = 100}, ucot = Table[ucototient[n], {n, 1, max^2}]; Table[Length[Position[ucot, n]], {n, 2, max}] // Flatten]

a(A362182(n)) = 0.
a(A362185(n)) = 1.
a(A362186(n)) = n.

A347771 Unitary nontotient numbers: values not in range of unitary totient function uphi(n).

Original entry on oeis.org

5, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 33, 34, 35, 37, 38, 39, 41, 43, 45, 47, 49, 50, 51, 53, 55, 57, 59, 61, 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, 113, 114, 115, 117, 118, 119, 121, 122, 123, 125, 129, 131, 133, 134, 135

Offset: 1

Eric Chen, Sep 13 2021

nonn

Numbers not appearing in A047994.
Indices of -1 in A135347.
Unitary version of A007617.
This sequence to A047994 is A007617 to A000010.
This sequence to A135347 is A007617 to A049283 (for the case that no such numbers exist, A135347 uses -1 and A049283 uses 0).
All odd numbers not of the form 2^k-1 (i.e. not in A000225) are in this sequence, since uphi(n) = A047994(n) is an even number unless n is a power of 2 (A000079), in this case uphi(n) = n-1.
The intersection of this sequence and A049225 is empty, since for squarefree numbers, all divisors are unitary divisors, note that the intersection of this sequence and A002202 is not empty, the number 110 is in both sequences.

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

Cf. A000010, A002202, A005277, A007617, A047994, A049225, A049283, A135347, A361966, A361967.

Select[Range[135], Length[invUPhi[#]] == 0 &] (* Amiram Eldar, Apr 01 2023, using the function invUPhi from A361966 *)

A047994(n)=my(f=factor(n)~); prod(i=1, #f, f[1, i]^f[2, i]-1)
is(n)=for(k=1,n^2,if(A047994(k)==n,return(0)));1 \\ after A047994

A361967(a(n)) = 0. - Amiram Eldar, Apr 01 2023

A361970 a(n) is the least number k such that the equation uphi(x) = k has exactly n solutions, or -1 if no such k exists, where uphi is the unitary totient function (A047994).

Original entry on oeis.org

5, 1, 2, 6, 8, 12, 36, 156, 24, 552, 168, 48, 96, 420, 120, 192, 3264, 144, 384, 336, 1536, 288, 360, 240, 672, 1200, 3888, 1080, 4896, 1584, 480, 576, 7056, 4992, 864, 1872, 1152, 3120, 960, 2400, 720, 2520, 30960, 2688, 19968, 1680, 1728, 1920, 2016, 2304, 12000

Offset: 0

Amiram Eldar, Apr 01 2023

nonn

Is there any n for which a(n) = -1?

Amiram Eldar, Table of n, a(n) for n = 0..300

The unitary version of A007374.

Cf. A047994, A135347, A347771, A361966, A361967, A361968, A361969, A361971.

solnum[n_] :=  Length[invUPhi[n]]; seq[len_, kmax_] := Module[{s = Table[-1, {len}], c = 0, k = 1, ind}, While[k < kmax && c < len, ind = solnum[k] + 1; If[ind <= len && s[[ind]] < 0, c++; s[[ind]] = k]; k++]; s]; seq[50, 10^5] (* using the function invUPhi from A361966 *)

A361967(a(n)) = n.

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.

cp's OEIS Frontend

A361971 Record values in A361967.

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A327837 Decimal expansion of the asymptotic mean of the number of exponential divisors function (A049419).

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A361966 Irregular table read by rows in which the n-th row consists of all the numbers m such that uphi(m) = n, where uphi is the unitary totient function (A047994).

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A361968 Unitary highly totient numbers: numbers k that have more solutions x to the equation uphi(x) = k than any smaller k, where uphi is the unitary totient function (A047994).

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A361969 Numbers k with a single solution x to the equation uphi(x) = k, where uphi is the unitary totient function (A047994).

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A362181 Number of numbers k such that A323410(k) = n.

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A347771 Unitary nontotient numbers: values not in range of unitary totient function uphi(n).

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A361970 a(n) is the least number k such that the equation uphi(x) = k has exactly n solutions, or -1 if no such k exists, where uphi is the unitary totient function (A047994).

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