A331273 -id:A331273 - cp's OEIS frontend

A331407 Numbers at which the sum of the iterated exponential totient function (A331273) attains a record.

1, 2, 8, 32, 128, 864, 3456, 7776, 31104, 279936, 497664, 1990656, 4478976, 17915904, 62208000, 97200000, 559872000, 874800000, 1555200000, 6220800000, 13996800000, 55987200000

Offset: 1

Amiram Eldar, Feb 25 2020

Analogous to A181659 with the exponential totient function (A072911) instead of the Euler totient function phi (A000010).

The corresponding record values are 0, 1, 3, 5, 7, 11, 13, 19, 27, 37, 43, 51, 61, 75, 83, 101, 123, 147, 165, 195, 243, 293, ...

Cf. A072911, A181659, A331273.

ephi[n_] := Times @@ EulerPhi[FactorInteger[n][[;; , 2]]]; s[n_] := Plus @@ FixedPointList[ephi, n] - n - 1; seq = {}; smax = -1; Do[s1 = s[n]; If[s1 > smax, smax = s1; AppendTo[seq, n]], {n, 1, 5000}]; seq

A372601 The maximal exponent in the prime factorization of the largest exponentially odd divisor of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, May 07 2024

First differs from A331273 at n = 64.

Differs from A363332 at n = 1, 216, 432, 648, 864, 1000, ... .

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

Cf. A051903, A109613, A350390, A368711.
Cf. A331273, A363332.
Similar sequences: A007424, A368781, A372602, A372603, A372604.

f[n_] := n - If[EvenQ[n], 1, 0]; a[n_] := f[Max[FactorInteger[n][[;; , 2]]]]; a[1] = 0; Array[a, 100]

s(n) = (n+1) \ 2 * 2 - 1;
a(n) = if(n>1, s(vecmax(factor(n)[,2])), 0);

a(n) = A051903(A350390(n)).
a(n) = A109613(A051903(n)-1) for n >= 2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + 2 * Sum_{i>=1} (1 - (1/zeta(2*i+1))) = 1.42929441950714075659... .

A372604 The maximal exponent in the prime factorization of the largest divisor of n whose number of divisors is a power of 2.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, May 07 2024

First differs from A331273 at n = 32.
Differs from A368247 at n = 1, 128, 216, 256, 384, 432, 512, ... .
All the terms are of the form 2^k-1 (A000225).

			4 has 3 divisors, 1, 2 and 4. The number of divisors of 4 is 3, which is not a power of 2. The number of divisors of 2 is 2, which is a power of 2. Therefore, A372379(4) = 2 and a(4) = A051903(2) = 1.

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

Cf. A000225, A051903, A092323, A372379, A372466.
Cf. A331273, A368247.
Similar sequences: A007424, A368781, A372601, A372602, A372603.

f[n_] := 2^Floor[Log2[n + 1]] - 1; a[n_] := f[Max[FactorInteger[n][[;; , 2]]]]; a[1] = 0; Array[a, 100]

s(n) = 2^exponent(n+1) - 1;
a(n) = if(n>1, s(vecmax(factor(n)[,2])), 0);

a(n) = A051903(A372379(n)).
a(n) = A092323(A051903(n)+1).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{i>=1} 2^i * (1 - 1/zeta(2^(i+1)-1)) = 1.36955053734097783559... .

A333611 Sum of the iterated infinitary totient function iphi (A091732).

Original entry on oeis.org

0, 1, 3, 6, 10, 3, 9, 6, 14, 10, 20, 9, 21, 9, 14, 29, 45, 14, 32, 21, 21, 20, 42, 9, 33, 21, 45, 32, 60, 14, 44, 29, 41, 45, 33, 33, 69, 32, 33, 21, 61, 21, 63, 44, 61, 42, 88, 44, 92, 33, 61, 69, 121, 45, 61, 32, 69, 60, 118, 33, 93, 44, 92, 106, 92, 41, 107

Offset: 1

Amiram Eldar, Mar 28 2020

nonn

			a(3) = iphi(3) + iphi(iphi(3)) = 2 + 1 = 3.

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

Cf. A091732, A092693, A329153, A330273, A331273, A333609.

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_] := Plus @@ NestWhileList[iphi, n, # != 1 &] - n; Array[a, 100]

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A331407 Numbers at which the sum of the iterated exponential totient function (A331273) attains a record.

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A372601 The maximal exponent in the prime factorization of the largest exponentially odd divisor of n.

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A372604 The maximal exponent in the prime factorization of the largest divisor of n whose number of divisors is a power of 2.

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A333611 Sum of the iterated infinitary totient function iphi (A091732).

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Mathematica