A254926 -id:A254926 - cp's OEIS frontend

A349309 Numbers k such that A254926(k) = A254926(k+1).

7, 26, 124, 342, 1330, 2196, 12166, 24388, 29790, 79506, 103822, 148876, 205378, 226980, 300762, 357910, 493038, 571786, 1030300, 1092726, 1225042, 2248090, 2685618, 3307948, 3442950, 3869892, 4657462, 5177716, 5735338, 6967870, 7645372, 9393930, 11089566, 11697082

Offset: 1

Amiram Eldar, Nov 14 2021

nonn

			7 is a term since A254926(7) = A254926(8) = 7.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A254926.

Similar sequences: A001274, A287055, A293184, A326403, A336673, A349307, A349308.

f[p_, e_] := If[e < 3, p^e, p^e - p^(e - 3)]; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[10^6], s[#] == s[# + 1] &]

from math import prod
from itertools import count, islice
from sympy import factorint
def A349309_gen(startvalue=1): # generator of terms >= startvalue
    a = prod(p**e - (p**(e-3) if e >= 3 else 0) for p, e in factorint(max(startvalue,1)).items())
    for k in count(max(startvalue,1)):
        b = prod(p**e - (p**(e-3) if e >= 3 else 0) for p, e in factorint(k+1).items())
        if a == b:
            yield k
        a = b
A349309_list = list(islice(A349309_gen(),10)) # Chai Wah Wu, Jan 24 2022

A384049 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is cubefree.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 7, 9, 10, 11, 12, 13, 14, 15, 15, 17, 18, 19, 20, 21, 22, 23, 21, 25, 26, 26, 28, 29, 30, 31, 31, 33, 34, 35, 36, 37, 38, 39, 35, 41, 42, 43, 44, 45, 46, 47, 45, 49, 50, 51, 52, 53, 52, 55, 49, 57, 58, 59, 60, 61, 62, 63, 63, 65, 66, 67, 68

Offset: 1

Amiram Eldar, May 18 2025

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

Unitary analog of A254926.
Cf. A004709, A065468, A360539, A360540, A384046, A384047.
The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), this sequence (cubefree), A384050 (powerful), A384051 (cubefull), A384052 (square), A384053 (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).

f[p_, e_] := p^e - If[e < 3, 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - if(f[i,2] < 3, 0, 1));}

Multiplicative with a(p^e) = p^e if e <= 2, and p^e - 1 if e >= 3.
a(n) = n * A047994(n) / A384051(n).
a(n) = A047994(A360540(n)) * A360539(n).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^s - 1/p^(3*s) + 1/p^(4*s-1)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/(p^5*(p+1))) = 0.988504... (A065468).

A254981 a(n) is the sum of the divisors d of n such that n/d is cubefree.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 14, 13, 18, 12, 28, 14, 24, 24, 28, 18, 39, 20, 42, 32, 36, 24, 56, 31, 42, 39, 56, 30, 72, 32, 56, 48, 54, 48, 91, 38, 60, 56, 84, 42, 96, 44, 84, 78, 72, 48, 112, 57, 93, 72, 98, 54, 117, 72, 112, 80, 90, 60, 168, 62, 96, 104, 112, 84, 144

Offset: 1

Álvar Ibeas, Feb 11 2015

Inverse Möbius transform of A254926.

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

Cf. A000203, A001615, A001694, A212793, A254926.

nn = 66; f[list_, i_] := list[[i]]; a = Table[If[Max[FactorInteger[n][[All, 2]]] < 3, 1, 0], {n, 1, nn}]; b =Table[n, {n, 1, nn}]; Table[
DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Feb 22 2015 *)
f[p_, e_] := p^(e-2) * (1 + p + p^2); f[p_, 1] := 1 + p; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 27 2023 *)

a212793(n) = {my(f = factor(n)); for (i=1, #f~, if ((f[i, 2]) >=3, return(0)); ); return (1); }
a(n) = sumdiv(n, d, d*a212793(n/d)); \\ Michel Marcus, Feb 11 2015

a(n) = sumdiv(n, d, if (ispower(d, 3), moebius(sqrtnint(d, 3))*sigma(n/d), 0)); \\ Michel Marcus, Mar 04 2015

a(n) = Sum_{d | n} d * A212793(n/d) = n * Sum_{d | n} A212793(d) / d.
a(n) = Sum_{d^3 | n} mu(d) * A000203(n/d^3).
Multiplicative with a(p) = 1 + p; a(p^e) = p^(e-2) * (1 + p + p^2), for e>1.
Dirichlet g.f.: zeta(s) * zeta(s-1) / zeta(3s).
If n is powerful, a(n^k) = n^(k-1) * a(n).
For k>1, a(n^k) = n^(k-1) * a(n) * Product_{p prime, ord(n,p)=1} (p^3-1) / (p^3-p).
Sum_{k=1..n} a(k) ~ 315*n^2 / (4*Pi^4). - Vaclav Kotesovec, Feb 03 2019

A384655 a(n) = Sum_{k=1..n} A051903(gcd(n,k)).

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 1, 7, 4, 6, 1, 11, 1, 8, 7, 15, 1, 14, 1, 17, 9, 12, 1, 25, 6, 14, 13, 23, 1, 22, 1, 31, 13, 18, 11, 36, 1, 20, 15, 39, 1, 30, 1, 35, 26, 24, 1, 53, 8, 32, 19, 41, 1, 44, 15, 53, 21, 30, 1, 59, 1, 32, 34, 63, 17, 46, 1, 53, 25, 46, 1, 81, 1, 38

Offset: 1

Amiram Eldar, Jun 06 2025

nonn

The terms of this sequence can be calculated efficiently using the 1st formula. The value the of function f(n, k) is equal to the number of integers i from 1 to n such that gcd(i, n) is 1 if k = 1, or k-free if k >= 2 (k-free numbers are numbers that are not divisible by a k-th power other than 1). E.g., f(n, 1) = A000010(n), f(n, 2) = A063659(n), and f(n, 3) = A254926(n).

			a(4) = A051903(gcd(4,1)) + A051903(gcd(4,2)) + A051903(gcd(4,3)) + A051903(gcd(4,4)) = A051903(1) + A051903(2) + A051903(1) + A051903(4) = 0 + 1 + 0 + 2 = 3.

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

Cf. A000010, A033150, A005117, A050873, A051903, A051953, A063659, A067259, A254926, A357310, A383156, A384656 (unitary analog).

e[n_] := If[n == 1, 0, Max[FactorInteger[n][[;;, 2]]]]; a[n_] := Sum[e[GCD[n, k]], {k, 1, n}]; Array[a, 100]
(* or *)
f[p_, e_, k_] := p^e - If[e < k, 0, p^(e - k)]; a[n_] := Module[{fct = FactorInteger[n], emax, s}, emax = Max[fct[[;; , 2]]]; s = emax * n; Do[s -= Times @@ (f[#1, #2, k] & @@@ fct), {k, 1, emax}]; s]; a[1] = 0; Array[a, 100]

e(n) = if(n == 1, 0, vecmax(factor(n)[,2]));
a(n) = sum(k = 1, n, e(gcd(n, k)));

a(n) = if(n == 1, 0, my(f = factor(n), p = f[,1], e = f[,2], emax = vecmax(e), s = emax*n); for(k = 1, emax, s -= prod(i = 1, #p, p[i]^e[i] - if(e[i] < k, 0, p[i]^(e[i]-k)))); s);

a(n) = Sum_{k=1..A051903(n)} (n - f(n, k)) = A051903(n) * n  - Sum_{k=1..A051903(n)} f(n, k), where f(n, k) is multiplicative for a given k, with f(p^e, k) = p^e - p^(e-k) if e >= k and f(p^e, k) = p^e if e < k.
a(n) = 1 if and only if n is prime.
a(n) >= 2 if and only if n is composite.
a(n) >= A051953(n) with equality if and only if n is squarefree.
a(n) >= 2*n - A000010(n) - A063659(n) with equality if and only if n is cubefree that is not squarefree (i.e., n in A067259, or equivalently, A051903(n) = 2).
a(p^e) = (p^e-1)/(p-1) for a prime p and e >= 1.
a(n) < c*n and lim sun_{n->oo} a(n)/n = c, where c is Niven's constant (A033150).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Sum{k>=1} (1-1/zeta(2*k)) = 0.49056393035179738598... .

A384039 The number of integers k from 1 to n such that gcd(n,k) is a powerful number.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 6, 6, 7, 4, 10, 6, 12, 6, 8, 12, 16, 7, 18, 12, 12, 10, 22, 12, 21, 12, 21, 18, 28, 8, 30, 24, 20, 16, 24, 21, 36, 18, 24, 24, 40, 12, 42, 30, 28, 22, 46, 24, 43, 21, 32, 36, 52, 21, 40, 36, 36, 28, 58, 24, 60, 30, 42, 48, 48, 20, 66, 48, 44

Offset: 1

Amiram Eldar, May 18 2025

The number of integers k from 1 to n such that the powerfree part (A055231) of gcd(n,k) is 1.

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

Cf. A000010, A001694, A005117, A050873, A055231, A063658, A206369.

The number of integers k from 1 to n such that gcd(n,k) is: A026741 (odd), A062570 (power of 2), A063659 (squarefree), A078429 (cube), A116512 (power of a prime), A117494 (prime), A126246 (1 or 2), A206369 (square), A254926 (cubefree), A372671 (3-smooth), this sequence (powerful), A384040 (cubefull), A384041 (exponentially odd), A384042 (5-rough).

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

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

Multiplicative with a(p^e) = (p^2-p+1)*p^(e-2) if e >= 2, and p-1 otherwise.
a(n) >= A000010(n), with equality if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^s + 1/p^(2*s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 1/p^4) = 0.66922021803510257394... .

A384040 The number of integers k from 1 to n such that gcd(n,k) is a cubefull number.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 5, 6, 4, 10, 4, 12, 6, 8, 10, 16, 6, 18, 8, 12, 10, 22, 10, 20, 12, 19, 12, 28, 8, 30, 20, 20, 16, 24, 12, 36, 18, 24, 20, 40, 12, 42, 20, 24, 22, 46, 20, 42, 20, 32, 24, 52, 19, 40, 30, 36, 28, 58, 16, 60, 30, 36, 40, 48, 20, 66, 32, 44, 24

Offset: 1

Amiram Eldar, May 18 2025

The number of integers k from 1 to n such that the cubefree part (A360539) of gcd(n,k) is 1.

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

Cf. A005117, A036966, A078429, A360539, A254926.

The number of integers k from 1 to n such that gcd(n,k) is: A026741 (odd), A062570 (power of 2), A063659 (squarefree), A078429 (cube), A116512 (power of a prime), A117494 (prime), A126246 (1 or 2), A206369 (square), A254926 (cubefree), A372671 (3-smooth), A384039 (powerful), this sequence (cubefull), A384041 (exponentially odd), A384042 (5-rough).

f[p_, e_] := Switch[e, 1, p-1, 2, p^2-p, , (p^3-p^2+1)*p^(e-3)]; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, f[i,1]-1, if(f[i,2] == 2, f[i,1]*(f[i,1]-1), (f[i,1]^3-f[i,1]^2+1)*f[i,1]^(f[i,2]-3))));}

Multiplicative with a(p^e) = (p^3-p^2+1)*p^(e-3) if e >= 3, p*(p-1) if e = 2, and p-1 otherwise.
a(n) >= A384039(n), with equality if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^s + 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 1/p^6) = 0.62159731307414305346... .

A384041 The number of integers k from 1 to n such that gcd(n,k) is an exponentially odd number.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 7, 8, 10, 11, 9, 13, 14, 15, 13, 17, 16, 19, 15, 21, 22, 23, 21, 24, 26, 25, 21, 29, 30, 31, 27, 33, 34, 35, 24, 37, 38, 39, 35, 41, 42, 43, 33, 40, 46, 47, 39, 48, 48, 51, 39, 53, 50, 55, 49, 57, 58, 59, 45, 61, 62, 56, 53, 65, 66, 67, 51

Offset: 1

Amiram Eldar, May 18 2025

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

Cf. A000010, A268335.

The number of integers k from 1 to n such that gcd(n,k) is: A026741 (odd), A062570 (power of 2), A063659 (squarefree), A078429 (cube), A116512 (power of a prime), A117494 (prime), A126246 (1 or 2), A206369 (square), A254926 (cubefree), A372671 (3-smooth), A384039 (powerful), A384040 (cubefull), this sequence (exponentially odd), A384042 (5-rough).

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

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

Multiplicative with a(p^e) = ((p^2+p-1)*p^(e-1) - (-1)^e)/(p+1).
a(n) >= A000010(n), with equality if and only if n = 1.
Dirichlet g.f.: (zeta(s-1)*zeta(2*s)/zeta(s)) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 1/(p^2+1)) = 0.93749428273130025078... .

A384042 The number of integers k from 1 to n such that gcd(n,k) is a 5-rough number (A007310).

Original entry on oeis.org

1, 1, 2, 2, 5, 2, 7, 4, 6, 5, 11, 4, 13, 7, 10, 8, 17, 6, 19, 10, 14, 11, 23, 8, 25, 13, 18, 14, 29, 10, 31, 16, 22, 17, 35, 12, 37, 19, 26, 20, 41, 14, 43, 22, 30, 23, 47, 16, 49, 25, 34, 26, 53, 18, 55, 28, 38, 29, 59, 20, 61, 31, 42, 32, 65, 22, 67, 34, 46

Offset: 1

Amiram Eldar, May 18 2025

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

Cf. A000010, A003586, A007310, A065330, A065331, A372671.

The number of integers k from 1 to n such that gcd(n,k) is: A026741 (odd), A062570 (power of 2), A063659 (squarefree), A078429 (cube), A116512 (power of a prime), A117494 (prime), A126246 (1 or 2), A206369 (square), A254926 (cubefree), A372671 (3-smooth), A384039 (powerful), A384040 (cubefull), A384041 (exponentially odd), this sequence (5-rough).

f[p_, e_] := If[p < 5, (p-1)*p^(e-1), p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] < 5, (f[i,1]-1)*f[i,1]^(f[i,2]-1), f[i,1]^f[i,2]));}

Multiplicative with a(p^e) = (p-1)*p^(e-1) if p <= 3 and p^e if p >= 5.
a(n) >= A000010(n), with equality if and only if n is 3-smooth (A003586).
a(n) = A000010(A065331(n)) * A065330(n).
a(n) = 2 * n * phi(n)/phi(6*n) = n * A000010(n) / A372671(n).
Dirichlet g.f.: zeta(s-1) * (1-1/2^s) * (1-1/3^s).
Sum_{k=1..n} a(k) ~ n^2 / 3.

A309287 Square array T(v, m), read by antidiagonals, for the Rogel-Klee arithmetic function: number of positive integers h in the set [m] for which gcd(h, m) is v-th-power-free, i.e., gcd(h, m) is not divisible by any v-th power of an integer > 1 (with v, m >= 1).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 3, 2, 1, 4, 3, 3, 2, 1, 2, 5, 4, 3, 2, 1, 6, 6, 5, 4, 3, 2, 1, 4, 7, 6, 5, 4, 3, 2, 1, 6, 6, 7, 6, 5, 4, 3, 2, 1, 4, 8, 7, 7, 6, 5, 4, 3, 2, 1, 10, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 4, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 9, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 6, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Petros Hadjicostas, Jul 21 2019

For fixed v >= 1, T(v, .) is a multiplicative arithmetic function with T(v, 1) = 1; T(v, p^e) = p^e, if e < v; and T(v, p^e) = p^e - p^(e-v) if e >= v (where p is a prime >= 2).
Here, T(v=1, m) = phi(m) is the number of arithmetic progressions (s*m + k: s >= 0), k = 1, ..., m, that contain infinitely many primes (by Dirichlet's theorem). For v >= 2, T(v, m) is the number of these arithmetic progressions that contain infinitely many v-th-power-free numbers.
In Section 6 of his paper, Cohen (1959) mentions that this function was introduced by Rogel (1900) in an article published in a Bohemian journal. Roger's (1900) paper is a continuation of Rogel (1897) and the two should be read together.
McCarthy (1958) uses the asymptotic result given in the FORMULA section below to prove that the probability that the GCD of two positive integers is v-th-power-free is 1/zeta(2*v).

			Table for T(v, m) (with rows v >= 1 and columns m >= 1) begins as follows:
  v=1: 1, 1, 2, 2, 4, 2, 6, 4, 6,  4, 10,  4, 12,  6,  8,  8, ...
  v=2: 1, 2, 3, 3, 5, 6, 7, 6, 8, 10, 11,  9, 13, 14, 15, 12, ...
  v=3: 1, 2, 3, 4, 5, 6, 7, 7, 9, 10, 11, 12, 13, 14, 15, 14, ...
  v=4: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 15, ...
  v=5: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...
  v=6: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...
  v=7: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...
  ...
Clearly, lim_{v -> infinity} T(v, m) = m.

Paul J. McCarthy, Introduction to Arithmetical Functions, Springer-Verlag, 1986; see pp. 38-40 and 69.

Eckford Cohen, A class of residue systems (mod r) and related arithmetical functions. I. A generalization of the Moebius function, Pacific J. Math. 9(1) (1959), 13-24; see Section 6 where T(v, m) = Phi_v(m).
Eckford Cohen, A generalized Euler phi-function, Math. Mag. 41 (1968), 276-279; here T(v, m) = phi_v(m).
E. K. Haviland, An analogue of Euler's phi-function, Duke Math. J. 11 (1944), 869-872; here T(v=2, m) = rho(m).
V. L. Klee, Jr., A generalization of Euler's phi function, Amer. Math. Monthly, 55(6) (1948), 358-359; here T(v, m) = Phi_v(m).
Paul J. McCarthy, On a certain family of arithmetic functions, Amer. Math. Monthly 65 (1958), 586-590; here, T(v, m) = T_v(m).
Franz Rogel, Entwicklung einiger zahlentheoreticher Funktionen in unendliche Reihen, S.-B. Kgl. Bohmischen Ges. Wiss. Article XLVI/XLIV (1897), Prague (26 pages). [This paper deals with arithmetic functions, especially the Euler phi function. It was continued three years later with the next paper, which contains his function phi_k(n). As stated at the end of the volume, in the table of contents, there is a mistake in numbering the article, so two Roman numerals appear in the literature for labeling this article!]
Franz Rogel, Entwicklung einiger zahlentheoreticher Funktionen in unendliche Reihen, S.-B. Kgl. Bohmischen Ges. Wiss. Article XXX (1900), Prague (9 pages). [This is a continuation of the previous article, which was written three years earlier and has the same title. The numbering of the equations continues from the previous paper, but this paper is the one that introduces the function phi_k(n). In our notation, T(v, m) = phi_v(m). Cohen (1959) refers to this paper and correctly attributes this function to F. Rogel.]

A000010 (row v = 1 is Euler's phi function), A063659 (row v = 2 is Haviland's function), A254926 (row v = 3).

/* Modification of Michel Marcus's program from sequence A254926: */
T(v, m) = {f = factor(m); for (i=1, #f~, if ((e=f[i, 2])>=v, f[i, 1] = f[i, 1]^e - f[i, 1]^(e-v); f[i, 2]=1); ); factorback(f); }
/* Print the first 40 terms of each of the first 10 rows: */
{ for (v=1, 10, for (m=1, 40, print1(T(v, m), ", "); ); print(); ); }

T(v, m) = m * Product_{p prime and p^v|m} (1 - p^(-v)) for v, m >= 1.
T(v, m) = Sum_{n >= 1} mu(n) * [m, n^v] * (m/n^v), where [m, n^v] = 1 when m is a multiple of n^v, and = 0 otherwise. [This is Eq. (53) in Rogel (1900) and Eq. (6.1) in Cohen (1959).]
Dirichlet g.f. for row v: Sum_{m >= 1} T(v, m)/m^s = zeta(s-1)/zeta(v*s) for Re(s) > 1.
Asymptotics: Sum_{m = 1..n} T(v, m) = n^2/(2*zeta(2*v)) + O(n) for v >= 2 and = n^2/(2*zeta(2)) + O(n*log(n)) for v = 1 (for Euler's phi-function).
Analog of Fermat's theorem: if gcd(a, m) = 1 with a >= 1, then m/gcd(a^T(v, m) - 1, m) is v-th-power-free. (For v = 1, this means m/gcd(a^T(v=1, m) - 1, m) = 1.)
T(v, m^v)/m^v = Sum_{d|m} mu(d)/d^v for m, v >= 1. (It generalizes the formula phi(m)/m = Sum_{d|m} mu(d)/d since phi(m) = T(v=1, m).)

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A349309 Numbers k such that A254926(k) = A254926(k+1).

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A384049 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is cubefree.

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A254981 a(n) is the sum of the divisors d of n such that n/d is cubefree.

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A384655 a(n) = Sum_{k=1..n} A051903(gcd(n,k)).

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A384039 The number of integers k from 1 to n such that gcd(n,k) is a powerful number.

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A384040 The number of integers k from 1 to n such that gcd(n,k) is a cubefull number.

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A384041 The number of integers k from 1 to n such that gcd(n,k) is an exponentially odd number.

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A384042 The number of integers k from 1 to n such that gcd(n,k) is a 5-rough number (A007310).

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