A080683 -id:A080683 - cp's OEIS frontend

A184677 Number of numbers <= p^2 with largest prime factor <= p, where p is the n-th prime; a(0) = 1.

1, 3, 7, 16, 30, 61, 88, 138, 177, 248, 361, 423, 569, 690, 777, 924, 1137, 1370, 1495, 1765, 1979, 2129, 2452, 2711, 3075, 3563, 3871, 4078, 4412, 4639, 4996, 6027, 6427, 6988, 7272, 8181, 8494, 9135, 9803, 10320, 11031, 11768, 12140, 13315, 13713, 14330

Offset: 0

Reinhard Zumkeller, Jun 27 2011

nonn

a(n) = #{m: m<=A001248(n) and A006530(m)<=A000040(n)} for n > 0.

			a(1) = #{1,2,4} = 3 = number of binary powers <= 4;
a(2) = #{1,2,3,4,6,8,9} = 7 = number of 3-smooth numbers <= 9;
a(3) = #{1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25} = 16 = number of 5-smooth numbers <= 25.

Charles R Greathouse IV, Table of n, a(n) for n = 0..1000

Smooth numbers: A000079, A003586, A051037, A002473, A051038, A080197, A080681, A080682, A080683.

Cf. A027424.

Block[{nn = 45, w}, w = Array[FactorInteger[#][[All, 1]] &, Prime[nn]^2]; {1}~Join~Table[Count[w[[1 ;; p^2]], ?(AllTrue[#, # <= p &] &)], {p, Prime@ Range@ nn}]] (* _Michael De Vlieger, Mar 13 2021 *)

a(n)=if(n==0, return(1)); my(p=prime(n),s=p); forfactored(k=p+1,p^2, if(vecmax(k[2][,1])<=p, s++)); s \\ Charles R Greathouse IV, Nov 27 2017

A318298 Numbers whose set of decimal digits coincides with the set of the indices of their prime factors.

Original entry on oeis.org

12, 14, 154, 1196, 14112, 21888, 53625, 226512, 279174, 358435, 821142, 1222452, 1665664, 2228814, 2454375, 2614248, 2872116, 4425729, 5751746, 8653645, 9551256, 15261246, 19427226, 19644898, 19775998, 21271488, 27676935, 29591892, 29956212, 41878242, 45574144

Offset: 1

Michel Lagneau, Aug 24 2018

It is impossible to find a number with 9 distinct decimal digits because the prime factors 2 and 5 generate d_k = 0.

The finite subsequence containing the smallest numbers having at least j distinct digits for j = 2, 3, ..., 8, is 12, 154, 53625, 279174, 19427226, 82447365 and 41762985264.

			1196 is in the sequence because the prime factors are {2, 13, 23} = {prime(1), prime(6), prime(9)}, and 1196 contains the decimal digits 1, 6, 9.

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

Cf. A001221, A080683, A097227, A290675.

with(numtheory):nn:=10^8:
for n from 1 to nn do:
lst:={}:d:=factorset(n):n0:=nops(d):
q:=convert(n,base,10):n1:=nops(q):
p:=product(‘q[i]’, ‘i’=1..n1):
if p<>0
  then
  for i from 1 to n1 do :
   lst:=lst union {ithprime(q[i])}:
  od:
   if lst = d
    then
     print(n):
     else
     fi:fi:
od:

ok[n_] := Block[{f = First /@ FactorInteger[n], d}, Last@f < 24 && Min[d = Union@ IntegerDigits@ n] > 0 && Prime[d] == f]; Select[Range[10^6], ok] (* Giovanni Resta, Aug 24 2018 *)

A330583 The orders, with repetition, of the non-cyclic finite simple groups whose orders are 23-smooth.

Original entry on oeis.org

60, 168, 360, 504, 660, 1092, 2448, 2520, 3420, 4080, 5616, 6048, 6072, 7800, 7920, 9828, 20160, 20160, 25920, 29120, 58800, 62400, 95040, 126000, 175560, 181440, 262080, 443520, 604800, 979200, 1451520, 1814400

Offset: 1

Hal M. Switkay, Dec 18 2019

nonn

This is the intersection of A109379 and A080683. It should be a finite set; a proof thereof would be welcome.

			This list contains the orders of all non-cyclic finite simple groups < 12180. However, 29|12180, which is the order of L2(29).

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].

Hal M. Switkay, Table of n, a(n) for n = 1..102
David A. Madore, Orders of non-abelian simple groups
R. A. Wilson et al., ATLAS of Finite Group Representations - Version 3

Cf. A109379, A080683.

A330584 The orders, with repetition, of the non-cyclic finite simple groups that are subquotients of the automorphism groups of sublattices of the Leech lattice.

Original entry on oeis.org

60, 168, 360, 504, 660, 1092, 2448, 2520, 3420, 4080, 5616, 6048, 6072, 7800, 7920, 20160, 20160, 25920, 62400, 95040, 126000, 181440, 443520, 604800, 979200, 1451520, 1814400, 3265920, 4245696, 10200960

Offset: 1

Hal M. Switkay, Dec 18 2019

Note: not every sublattice of the Leech lattice is necessarily a section of the Leech lattice. For example, every Niemeyer lattice is commensurable with the Leech lattice; thus the orders of the simple components of their automorphism groups are in this list, even when those groups are not sections of Co0.
By a theorem of Conway and Sloane, any simple group with a cover that has a crystallographic representation in <= 21 dimensions is in this list.
This is a subsequence of A330583.

			All simple groups of order less than 9828 have crystallographic representations within sublattices of the Leech lattice. The smallest nontrivial crystallographic representation of L2(27), of order 9828, is 26-dimensional.

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].
J. H. Conway, N. J. A. Sloane, Sphere Packings, Lattices, and Groups. Springer, 3rd ed., 1999.

Hal M. Switkay, Table of n, a(n) for n = 1..56
J. H. Conway, N. J. A. Sloane, Low-dimensional lattices V: Integral coordinates for integral lattices, Proc. Royal Soc. A 426 (1989), 211-232.
David A. Madore, Orders of non-abelian simple groups
R. A. Wilson et al., ATLAS of Finite Group Representations - Version 3

Cf. A109379, A080683, A330583.

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 *)

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.

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

A184677 Number of numbers <= p^2 with largest prime factor <= p, where p is the n-th prime; a(0) = 1.

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A318298 Numbers whose set of decimal digits coincides with the set of the indices of their prime factors.

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A330583 The orders, with repetition, of the non-cyclic finite simple groups whose orders are 23-smooth.

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A330584 The orders, with repetition, of the non-cyclic finite simple groups that are subquotients of the automorphism groups of sublattices of the Leech lattice.

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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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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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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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