A036450 -id:A036450 - cp's OEIS frontend

A053096 When the Euler phi function is iterated with initial value A002110(n) = primorial, a(n) = number of iterations required to reach the fixed number = 1.

1, 2, 4, 6, 9, 12, 16, 19, 23, 27, 31, 35, 40, 44, 49, 54, 59, 64, 69, 74, 79, 84, 90, 96, 102, 108, 114, 120, 125, 131, 136, 142, 149, 155, 161, 167, 173, 178, 185, 191, 198, 204, 210, 217, 223, 229, 235, 241, 248, 254, 261, 268, 275, 282, 290, 297, 304, 310

Offset: 1

Labos Elemer, Feb 28 2000

nonn

Analogous to A053025, A053034, A053044. For comparison: iteration of, e.g., A000005 to primorial i.v. is trivially computable: q(n)=A002110(n), d(q(n)) = 2^n, d(d(q(n))) = n+1 and so A036450(A002110(n)) = A000005(n+1).

			n=7, A002110(7)=510510; the corresponding iteration chain is {510510, 92160, 24576, 8192, 4096, 2048, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1}. Its length is 17, so the required number of iterations is a(7)=16.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000010, A002110, A003434.

Array[-2 + Length@ FixedPointList[EulerPhi, Product[Prime@ i, {i, #}]] &, 58] (* Michael De Vlieger, Nov 20 2017 *)

a(n)=my(t=prod(i=1,n,prime(i)-1),s=1); while(t>1, t=eulerphi(t); s++); s \\ Charles R Greathouse IV, Jan 06 2016

A003434(n)=my(s);while(n>1,n=eulerphi(n);s++);s
first(n)=my(s=1); vector(n,k,s+=A003434(prime(k))-1) \\ Charles R Greathouse IV, Jan 06 2016

a(n) is the smallest number such that Nest[EulerPhi, A002110, a(n)]=1

A174457 Infinitely refactorable numbers: numbers k such that each iteration under the map x -> A000005(x) produces a divisor of k.

Original entry on oeis.org

1, 2, 12, 24, 36, 60, 72, 84, 96, 108, 132, 156, 180, 204, 228, 240, 252, 276, 288, 348, 360, 372, 396, 444, 468, 480, 492, 504, 516, 564, 600, 612, 636, 640, 672, 684, 708, 720, 732, 792, 804, 828, 852, 864, 876, 936, 948, 972, 996, 1044, 1056, 1068, 1116, 1152

Offset: 1

Matthew Vandermast, Dec 04 2010

nonn

In other words, let d^1(n) = A000005(n) and, for all positive integers k, let d^(k+1)(n) = A000005(d^k(n)). Sequence lists numbers n with the property that every such value of d^k(n) divides n.
A141586 is a subsequence. Is A110821 a subsequence?
Not a subsequence of A141551: 504 is the smallest term in this sequence not member of A141551.
a(n) is even for all n, since for any n >= 2, d^k(n) = 2 for some k. Proof: {d^k(n)} is a nonincreasing sequence of k, so it must stablize at a fixed point of the map x -> A000005(x), namely x = 1 or 2. But d^k(n) = 1 for some k implies that n = 1. - Jianing Song, Apr 20 2022

			9 has 3 divisors, and 9 is a multiple of 3. But 3 has 2 divisors, and 9 is not a multiple of 2. Hence, 9 does not belong to this sequence.
36 has 9 divisors, 9 has 3 divisors, 3 has 2 divisors, and 9, 3, and 2 are all divisors of 36. (Since 2 has 2 divisors, all further steps produce a value of 2.) Hence, 36 belongs to this sequence.

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

Cf. A036459 (number of steps of the map), A000005 (d(n): number of divisors).
Cf. A010553 (d(d(n))), A036450 (d^3(n)), A036452 (d^4(n)), A036453 (d^5(n)).
Subsequence of A033950 (refactorable numbers: d(n) | n) and A141113 (d(d(n))| n).

is_A174457(n, d=n)=!until(d<3, n%(d=numdiv(d)) && return) \\ M. F. Hasler, Dec 05 2010, updated PARI syntax Apr 16 2022

Edited by M. F. Hasler, Apr 16 2022

A053023 Divisor function applied thrice to n!.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 6, 6, 6, 5, 8, 8, 8, 8, 8, 9, 8, 10, 8, 10, 10, 12, 10, 10, 10, 15, 16, 16, 18, 16, 8, 16, 10, 12, 16, 18, 14, 3, 14, 16, 24, 16, 16, 20, 24, 24, 28, 24, 24, 16, 24, 24, 32, 32, 28, 32, 18, 20, 24, 28, 36, 32, 36, 24, 21, 24, 20, 40, 40, 30, 30, 36, 40, 42, 24, 32

Offset: 1

Labos Elemer, Feb 24 2000

nonn

			A036450(x) = 8 if x = 11!, 12!, 13!, 14!, 15!. The reversed sequences of these d-iterates are {2, 3, 4, 8, 24, 540, 39916800}, {2, 3, 4, 8, 24, 792, 479001600}, {2, 3, 4, 8, 30, 1584, 6227020800}, {2, 3, 4, 8, 30, 2592, 87178291200}, {2, 3, 4, 8, 42, 4032, 1307674368000}.

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

Cf. A000005, A000142, A010553, A036450, A053021.

a[n_] := Nest[DivisorSigma[0, #]&, n!, 3]; Array[a, 100] (* Amiram Eldar, Aug 12 2024 *)

a(n) = numdiv(numdiv(numdiv(n!))); \\ Amiram Eldar, Aug 12 2024

a(n) = d(d(d(n!))) = A036450(A000142(n)) = A036450(n!).

A293167 a(n) = Sum_{k = 1..n} d(d(d(k))), where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 24, 26, 28, 30, 32, 34, 37, 39, 42, 44, 46, 48, 51, 53, 55, 57, 60, 62, 65, 67, 70, 72, 74, 76, 78, 80, 82, 84, 87, 89, 92, 94, 97, 100, 102, 104, 107, 109, 112, 114, 117, 119, 122, 124, 127, 129, 131, 133, 137, 139, 141, 144, 146, 148, 151, 153, 156, 158

Offset: 1

N. J. A. Sloane, Oct 17 2017

nonn

David A. Corneth, Table of n, a(n) for n = 1..10000
Richard Bellman and Harold N. Shapiro, On a problem in additive number theory, Annals Math., 49 (1948), 333-340.
Imre Kátai, On the iteration of the divisor-function, Publ. Math. Debrecen, Vol. 16 (1969), pp. 3-15.

Part of the sequence A000005, A006218, A010553, A036450, A139130.

Accumulate[Table[DivisorSigma[0, DivisorSigma[0, DivisorSigma[0, n]]], {n, 80}]] (* Alonso del Arte, Oct 17 2017 *)

a(n) = sum(k=1, n, numdiv(numdiv(numdiv(k)))); \\ Michel Marcus, Oct 17 2017

first(n) = {my(v = vector(n)); v[1] = 1; for(i=2,n,v[i] = v[i-1] + numdiv(numdiv(numdiv(i)))); v} \\ David A. Corneth, Oct 17 2017

a(1) = 1; a(n + 1) = a(n) + A036450(n + 1) for n > 0. - David A. Corneth, Oct 17 2017

a(n) = (1 + o(1)) * c * n * log(log(log(n))), where c > 0 is a constant (Kátai, 1969). - Amiram Eldar, Apr 17 2024

cp's OEIS Frontend

A053096 When the Euler phi function is iterated with initial value A002110(n) = primorial, a(n) = number of iterations required to reach the fixed number = 1.

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A174457 Infinitely refactorable numbers: numbers k such that each iteration under the map x -> A000005(x) produces a divisor of k.

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A053023 Divisor function applied thrice to n!.

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A293167 a(n) = Sum_{k = 1..n} d(d(d(k))), where d(k) is the number of divisors of k (A000005).

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