A333953 -id:A333953 - cp's OEIS frontend

A333963 Intersection of A307866 and A333953, together with the number 2.

1, 2, 4, 6, 8, 12, 24, 36, 48, 72, 96, 120, 144, 240, 288, 360, 480, 576, 720, 1152, 1440, 2160, 2880, 4320, 5760, 8640, 11520, 17280, 25920, 30240, 34560, 51840, 60480, 69120, 103680, 120960, 172800, 207360, 241920, 345600, 362880, 414720, 483840, 725760

Offset: 1

Amiram Eldar and David A. Corneth, Apr 15 2020

nonn

Numbers that are both recursively superabundant numbers and K-champion numbers.

Cf. A307866, A333952, A333953.

A283052 Numbers k such that uphi(k)/phi(k) > uphi(m)/phi(m) for all m < k, where phi(k) is the Euler totient function (A000010) and uphi(k) is the unitary totient function (A047994).

Original entry on oeis.org

1, 4, 8, 16, 32, 36, 72, 144, 216, 288, 432, 864, 1728, 2592, 3600, 5400, 7200, 10800, 21600, 43200, 64800, 108000, 129600, 216000, 259200, 324000, 529200, 1058400, 2116800, 3175200, 5292000, 6350400, 10584000, 12700800, 15876000, 31752000, 63504000, 95256000

Offset: 1

Amiram Eldar, May 19 2017

nonn

This sequence is infinite.
a(1) = 1, a(6) = 36, a(15) = 3600 and a(32) = 6350400 are the smallest numbers n such that uphi(n)/phi(n) = 1, 2, 3 and 4. They are squares of 1, 6, 60, and 2520.
Also, coreful superabundant numbers: numbers k with a record value of the coreful abundancy index, A057723(k)/k > A057723(m)/m for all m < k. The two sequences are equivalent since A057723(k)/k = A047994(k)/A000010(k) for all k. - Amiram Eldar, Dec 28 2020

			uphi(k)/phi(k) = 1, 1, 1, 3/2 for k = 1, 2, 3, 4, thus a(1) = 1 and a(2) = 4 since a(4) > a(m) for m < 4.

Amiram Eldar, Table of n, a(n) for n = 1..46

Cf. A000010, A047994, A057723, A285906.

Similar sequences: A002110, A004394, A037992, A061742, A292984, A329882, A333953.

uphi[n_] := If[n == 1, 1, (Times @@ (Table[#[[1]]^#[[2]] - 1, {1}] & /@
FactorInteger[n]))[[1]]]; a = {}; rmax = 0; For[k = 0, k < 10^9, k++; r = uphi[k]/EulerPhi[k]; If[r > rmax, rmax = r; a = AppendTo[a, k]]]; a

uphi(n) = my(f = factor(n)); prod(i=1, #f~, f[i,1]^f[i,2]-1);
lista(nn) = {my(rmax = 0); for (n=1, nn, if ((newr=uphi(n)/eulerphi(n)) > rmax, print1(n, ", "); rmax = newr););} \\ Michel Marcus, May 20 2017

A333952 Recursively highly composite numbers: numbers m such that A067824(m) > A067824(k) for all k < m.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 24, 36, 48, 72, 96, 120, 144, 192, 240, 288, 360, 432, 480, 576, 720, 864, 960, 1152, 1440, 1728, 1920, 2160, 2304, 2880, 3456, 4320, 5760, 6912, 8640, 11520, 17280, 23040, 25920, 30240, 34560, 46080, 51840, 60480, 69120, 86400, 103680, 120960

Offset: 1

Amiram Eldar, Apr 11 2020

nonn

This sequence is not to be confused with A333931.
The corresponding record values are 1, 2, 4, 6, 8, 16, 40, 52, 96, ...
Fink (2019) defined this sequence. He asked whether 720 is the largest term that is also highly composite number (A002182).
This is, except the terms 2, the sequence records of indices of A074206 for positive n as a(n) = 2*A074206(n), n>1, i.e. A307866. (formula from - Vladeta Jovovic, Jul 03 2005) - David A. Corneth, Apr 13 2020

			The first 6 terms of A067824 are 1, 2, 2, 4, 2, 6. The record values occur at 1, 2, 4, 6, the first 4 terms of this sequence.

David A. Corneth, Table of n, a(n) for n = 1..291 (first 120 terms from Amiram Eldar)
Thomas Fink, Recursively divisible numbers, arXiv:1912.07979 [math.NT], 2019. See section 5.
T. M. A. Fink, Number of ordered factorizations and recursive divisors, arXiv:2307.16691 [math.NT], 2023.

Cf. A002182, A067824, A307866, A333953.

d[1] = 1; d[n_] := d[n] = 1 + DivisorSum[n, d[#] &, # < n &]; seq={}; dm = 0; Do[d1 = d[n]; If[d1 > dm, dm = d1; AppendTo[seq, n]], {n, 1, 10^4}]; seq

A333962 a(n) = A330575(A025487(n)).

Original entry on oeis.org

1, 3, 8, 14, 20, 42, 48, 116, 104, 112, 176, 304, 346, 256, 524, 768, 1044, 576, 1472, 1584, 1888, 1056, 2088, 2960, 1280, 3968, 5208, 4544, 3858, 6216, 8032, 2816, 10368, 15960, 10752, 12612, 17712, 10532, 21088, 6144, 22416, 26432, 19128, 24096, 46512, 25088, 38400

Offset: 1

David A. Corneth, Apr 15 2020

nonn

These numbers provide candidate record values of A330575(k)/k as k must be a product of primorials for A330575(k)/k to be a record.

Amiram Eldar, Table of n, a(n) for n = 1..750

Cf. A025487, A330575, A333953.

cp's OEIS Frontend

A333963 Intersection of A307866 and A333953, together with the number 2.

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A283052 Numbers k such that uphi(k)/phi(k) > uphi(m)/phi(m) for all m < k, where phi(k) is the Euler totient function (A000010) and uphi(k) is the unitary totient function (A047994).

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A333952 Recursively highly composite numbers: numbers m such that A067824(m) > A067824(k) for all k < m.

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A333962 a(n) = A330575(A025487(n)).

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