A080197 -id:A080197 - cp's OEIS frontend

A341749 Numbers k such that gcd(k, phi(k)) > log(log(k)).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 93, 94, 96

Offset: 1

Amiram Eldar, Feb 18 2021

nonn

First differs from A080197 at n = 28.
Erdős et al. (2008) proved that the asymptotic density of numbers k such that gcd(k, phi(k)) > (log(log(k)))^u for a real number u > 0 is equal to exp(-gamma) * Integral_{t=u..oo} rho(t) dt, where rho(t) is the Dickman-de Bruijn function and gamma is Euler's constant (A001620). For this sequence u = 1, and therefore its asymptotic density is 1 - exp(-gamma) = 0.43854... (A227242).
There are only 8 cyclic numbers (A003277) in this sequence: 1, 2, 3, 5, 7, 11, 13, 15. All the other terms are in A060679. The first term of A060679 which is not in this sequence is 1622.

			16 is a term since gcd(16, phi(16)) = gcd(16, 8) = 8 > log(log(16)) = 1.0197...
17 is not a term since gcd(17, phi(17)) = gcd(17, 16) = 1 < log(log(17)) = 1.0414...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Erdős, Florian Luca and Carl Pomerance, On the proportion of numbers coprime to a given integer, in: J.-M. De Koninck, A. Granville and F. Luca (eds.), Anatomy of Integers, AMS, 2008, pp. 47-64.
Wikipedia, Dickman function.

Cf. A000010, A001620, A003277, A009195, A060679, A080130, A227242, A341750.

Select[Range[100], GCD[#, EulerPhi[#]] > Log[Log[#]] &]

isok(k) = (k==1) || (gcd(k, eulerphi(k)) > log(log(k))); \\ Michel Marcus, Feb 19 2021

A080785 Least p-smooth number not less n, where p is the smallest prime factor of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 8, 9, 16, 11, 16, 13, 16, 16, 16, 17, 32, 19, 32, 24, 32, 23, 32, 25, 32, 27, 32, 29, 32, 31, 32, 36, 64, 36, 64, 37, 64, 48, 64, 41, 64, 43, 64, 48, 64, 47, 64, 49, 64, 54, 64, 53, 64, 60, 64, 64, 64, 59, 64, 61, 64, 64, 64, 72, 128, 67, 128, 72, 128, 71

Offset: 1

Reinhard Zumkeller, Mar 12 2003

nonn

a(n)<=2^k for n<=2^k.

a(n)=n for n in A000961. - Ivan Neretin, Apr 30 2016

Ivan Neretin, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Smooth Number.

Cf. A020639, A006530.

Cf. A000079, A003586, A051037, A002473, A051038, A080197, A080681, A080682, A080683.

a[n_] := Module[{p, k}, p = FactorInteger[n][[1, 1]]; For[k = n, True, k++, If[FactorInteger[k][[-1, 1]] <= p, Return[k]]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 15 2021 *)

A144913 Integers which are the product of even powers of primes up to 13.

Original entry on oeis.org

4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 324, 400, 441, 484, 576, 625, 676, 729, 784, 900, 1024, 1089, 1225, 1296, 1521, 1600, 1764, 1936, 2025, 2304, 2401, 2500, 2704, 2916, 3025, 3136, 3600, 3969, 4096, 4225, 4356, 4900, 5184, 5625

Offset: 1

Reikku Kulon, Sep 25 2008, Sep 26 2008

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

Subsequence of A000290 and A080197.

Cf. A001694.

Select[ Range[75]^2, (fi = FactorInteger[#]; Max[ fi[[All, 1]] ] <= 13 && And @@ EvenQ /@ fi[[All, 2]]) &] (* Jean-François Alcover, Oct 10 2012 *)
eppQ[n_]:=Module[{fi=FactorInteger[n]},Max[fi[[All,1]]]<=13&&AllTrue[ fi[[All,2]],EvenQ]]; Select[Range[6000],eppQ] (* Harvey P. Dale, Dec 26 2021 *)

is(n)=if(issquare(n,&n), my(f=factor(n)[,1]); #f && f[#f]<14, 0) \\ Charles R Greathouse IV, Jun 17 2013

a(n) = A080197(n+1)^2. - Rémy Sigrist, Sep 26 2020

Sum_{n>=1} 1/a(n) = -1 + Product_{primes p <= 13} p^2/(p^2-1) = 273347/442368. - Amiram Eldar, Oct 14 2020

A373944 Lexicographically earliest sequence of distinct positive integers such that for A(k) <= n < A(k+1); rad(Product_{i = 1..n} a(i)) = A002110(k), where A = A002110, rad = A007947, k >= 0, n >= 1.

Original entry on oeis.org

1, 2, 4, 8, 16, 3, 6, 9, 12, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96, 108, 128, 144, 162, 192, 216, 243, 256, 288, 5, 10, 15, 20, 25, 30, 40, 45, 50, 60, 75, 80, 90, 100, 120, 125, 135, 150, 160, 180, 200, 225, 240, 250, 270, 300, 320, 324, 360, 375, 384, 400

Offset: 1

David James Sycamore, Jun 23 2024

Sequence is computed piecewise in blocks of A002110(k+1) - A002110(k) terms, for indices n in the range A002110(k) <= n < A002110(k+1), k = 0,1,2,... in which all terms are the ordered earliest prime(k)-smooth numbers not already recorded in earlier blocks. Since a(0) = 1, and for all k >= 1, all prime(k)-smooth numbers eventually appear in the sequence, this is a permutation of the positive integers, A000027.
From Michael De Vlieger, Jun 25 2024: (Start)
Let P(i) = A002110(i) be the product of i smallest primes.
Let rad = A007947 and let gpf = A006530.
Let S(i) = {k : rad(k) | P(i)}, the prime(i)-smooth numbers.
The notation S(i,j) denotes the j-th smallest term in i, i.e., the j-th term when S(i) is sorted.
This sequence can be seen as a table with row r = 0 {1}, r = 1 {2, 4, 8, 16}, etc.
Then row r contains k in S(r, 1..P(r+1)-1) such that terms k <= S(r-1, P(r)-1) such that gpf(k) < prime(r) are removed.
As a consequence, the sorted union of rows 0..r reconstructs S(r, 1..P(r+1)-1).
For example, A003586(1..29) is given by the sorted union of rows r = 0..2 of the sequence.
The sorted union of rows r = 0..3 gives A051037(1..209), etc.
For r > 1, P(r) is the P(r-1)-th term in row r. (End)

			k = 0 --> A(0) <= n < A(1) --> 1 <= n < 2 --> n = 1 --> a(1) = 1 since rad(1) = 1 = A(0).
k = 1 --> A(1) <= n < A(2) --> 2 <= n < 6 --> n = 2,3,4,5 --> a(2,3,4,5) = 2,4,8,16 (the first 4 terms of A000079, excluding 1).
k = 2 --> 6 <= n < 30 --> n = 6,7,8,9,...,29 --> a(6,7,8,9...,29) = 3,6,9,12,...,288 (the first 24 terms of A003586 excluding all above).
k = 3 --> 30 <= n < 210 --> n = 30,31,32,...,209 --> a(30,31,32,...,209) = 5,10,15,...,19200 (the first 180 terms of A051037 excluding all above).
Sequence can be presented as an irregular table T(n,k), in which the n-th row commences A008578(n); n >= 1, and T(n,k) is the k-th prime(n)-smooth number which has not appeared earlier.
Table starts:
  1;
  2,4,8,16;
  3,6,9,12,18,24,27,32,...,288;
  5,10,15,20,25,30,40,45,50,60,...,19200;
  7,14,21,28,...,13829760;

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n) n = 1..510509.

Cf. A000040, A002110, A002473, A003586, A007947, A051037, A051038, A080197, A080681, A080682, A080683.

(* First, load function f from A162306 *)
P = m = 1; Flatten@ Join[{{1}}, Reap[Do[P *= Prime[i]; (Sow@ Select[#, Nand[# <= m, FactorInteger[#][[-1, 1]] < Prime[i]] &]; m = #[[-1]]) &@ f[P, P^4][[;; P*Prime[i + 1] - 1]], {i, 3}] ][[-1, 1]]] (* Michael De Vlieger, Jun 24 2024 *)

a(A002110(n)) = A000040(n), n >= 1.

A277620 Positive integers that are composed of prime factors 2,3,5 and 11.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 16, 18, 20, 22, 24, 25, 27, 30, 32, 33, 36, 40, 44, 45, 48, 50, 54, 55, 60, 64, 66, 72, 75, 80, 81, 88, 90, 96, 99, 100, 108, 110, 120, 121, 125, 128, 132, 135, 144, 150, 160, 162, 165, 176, 180, 192, 198, 200, 216, 220

Offset: 1

Jason Gottfried, Oct 24 2016

nonn

A003592 is a subsequence, A080197 is a subsequence of this sequence.

A363794 a(n) = smallest prime(n)-smooth number k such that r(k) >= r(P(n+1)), where r(n) = A010846(n) and P(n) = A002110(n).

Original entry on oeis.org

16, 72, 540, 6300, 92400, 1681680, 36756720, 921470550, 27886608750, 970453984500, 37905932634570

Offset: 1

Michael De Vlieger, Jun 22 2023

Let R = r(P(n)) = A010846(A002110(n)) = A363061(n).
Let S(n) be the sorted tensor product of prime power ranges {p(i)^e : i<=n, e>=0}, e.g., S(1) = A000079, S(2) = A003586, S(3) = A051037, etc.
Let T(n) = A002110(n)*S(n). Note that S(1) = T(1) since omega(A002110(1)) = 1.
Let S(n,i) be the i-th term in S(n).
Then a(n) is the smallest S(n,i), i >= R, such that S(n,i) is also in T. Equivalently, a(n) is the smallest S(n,i), i >= R, such that rad(S(n,i)) = A002110(n), where rad(n) = A007947(n).

			a(1) = 16 since r(2^4) = 5 and r(6) = 5; numbers in row 16 of A162306 are its divisors {1, 2, 4, 8, 16}, while row 6 of A162306 is {1, 2, 3, 4, 6}.
a(2) = 72 = A003586(18) since r(72) = r(30) = 18. 72 is the 8th term in A003586 that is not in A000961.
a(3) = 540 since r(540) = 69 which exceeds r(210) = 68.
a(4) = 6300 since r(6300) = 290 which exceeds r(2310) = 283, etc.
Table showing the relationship of a(n) to r(P(n)) = A363061(n), with p(n) = prime(n), P(n+1) = A002110(n+1), r(a(n)) = A010846(a(n)), and j the index such that S(r(a(n))) = T(j) = a(n). a(n) = m*P(n).
   n p(n)        P(n+1)          a(n)  r(P(n))  r(a(n))   j    m
  --------------------------------------------------------------
   1   2             6            16        5        5    4    8
   2   3            30            72       18       18    8   12
   3   5           210           540       68       69   13   18
   4   7          2310          6300      283      290   22   30
   5  11         30030         92400     1161     1165   29   40
   6  13        510510       1681680     4843     4848   42   56
   7  17       9699690      36756720    19985    19994   53   72
   8  19     223092870     921470550    83074    83435   68   95
   9  23    6469693230   27886608750   349670   351047   89  125
  10  29  200560490130  970453984500  1456458  1457926  107  150

Cf. A000079, A000961, A002110, A002473, A003586, A007947, A010846, A051037, A051038, A080197, A080681, A080682, A080683, A162306, A363061.

nn = 6; rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; f[x_] := FactorInteger[x][[-1, 1]]; S = Array[Product[Prime[i], {i, #}] &, nn + 1]; Table[Set[{p, q}, Prime[n + {0, 1}]]; r = Count[Range[S[[n + 1]]], _?(f[#] <= q &)]; c = k = 1; While[Or[c < r, rad[k] != S[[n]]], If[f[k] <= p, c++]; k++]; k, {n, nn}]

a(n) >= A363061(n).

cp's OEIS Frontend

A341749 Numbers k such that gcd(k, phi(k)) > log(log(k)).

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A080785 Least p-smooth number not less n, where p is the smallest prime factor of n.

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A144913 Integers which are the product of even powers of primes up to 13.

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A373944 Lexicographically earliest sequence of distinct positive integers such that for A(k) <= n < A(k+1); rad(Product_{i = 1..n} a(i)) = A002110(k), where A = A002110, rad = A007947, k >= 0, n >= 1.

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A277620 Positive integers that are composed of prime factors 2,3,5 and 11.

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A363794 a(n) = smallest prime(n)-smooth number k such that r(k) >= r(P(n+1)), where r(n) = A010846(n) and P(n) = A002110(n).

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