A051038 -id:A051038 - cp's OEIS frontend

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

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 *)

A133581 (k^2)-th k-smooth number for k = prime(n).

Original entry on oeis.org

8, 16, 54, 112, 396, 512, 1008, 1155, 1794, 3312, 3520, 5488, 6776, 7020, 8405, 11180, 14384, 14720, 18241, 20339, 20709, 24769, 27094, 31648, 38994, 41890, 42336, 45318, 45825, 48852, 66234, 69874, 76857, 77441, 91719, 92323, 100215, 108376, 112896, 121539

Offset: 1

Jonathan Vos Post, Dec 26 2007

An integer is k-smooth if it has no prime factors > k.

			a(1) = 8 = A000079(4).
a(2) = 16 = A003586(9).
a(3) = 54 = A051037(25).

Eric Weisstein's World of Mathematics, Smooth Number

Cf. A000040, A000079, A001248, A003586, A051037, A002473, A051038.

from sympy import integer_log, prime, prevprime
def A133581(n):
    if n==1: return 8
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x,m): return sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1)) if m==3 else sum(g(x//(m**i),prevprime(m))for i in range(integer_log(x,m)[0]+1))
    k = prime(n)
    def f(x): return k**2+x-g(x,k)
    return bisection(f,k**2,k**2) # Chai Wah Wu, Sep 17 2024

a(n) = A001248(n)-th integer which has no prime factors > A000040(n).

Corrected and extended by D. S. McNeil, Dec 08 2010

a(33)-a(40) from Chai Wah Wu, Sep 17 2024

A219785 Primes not neighboring an 11-smooth number.

Original entry on oeis.org

103, 137, 157, 173, 227, 229, 233, 277, 283, 311, 313, 317, 347, 367, 373, 389, 409, 443, 457, 467, 509, 521, 523, 547, 557, 563, 569, 571, 607, 613, 619, 643, 653, 677, 683, 691, 709, 733, 739, 743, 761, 773, 787, 797, 821, 823, 827, 829, 853, 857, 859, 877

Offset: 1

Jonathan Vos Post, Nov 27 2012

			103 is in the sequence because it is prime and the closest 11-smooth numbers are 100 and 105, which differ from 103 by 3 and -2 respectively, neither being -1 or +1.
137 is in the sequence because it is prime and neither 137 - 1 = 136 = 2^3 * 17 nor 137 + 1 = 138 = 2 * 3 * 23 are 11-smooth.

Cf. A000040, A002473, A051037, A051038, A080194, A219528, A219697.

mx = 2^10; t11 = Select[Sort[Flatten[Table[2^i 3^j 5^k 7^l 11^m, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx]}, {k, 0, Log[5, mx]}, {l, 0, Log[7, mx]}, {m, 0, Log[11, mx]}]]], # <= mx &]; Complement[Prime[Range[PrimePi[mx]]], Union[Select[t11 + 1, PrimeQ], Select[t11 - 1, PrimeQ]]] (* T. D. Noe, Nov 27 2012 *)

Numbers k such that k is prime and k is neither (2^i * 3^j * 5^k * 7^l * 11^m) - 1 nor (2^i * 3^j * 5^k * 7^l * 11^m) + 1 for any i, j, k, l, m >= 0.

A219790 Smallest prime not neighboring a prime(n)-smooth number.

Original entry on oeis.org

11, 29, 43, 67, 103, 137, 173, 173, 173, 283, 283, 283, 283, 283, 317, 653, 653, 653, 653, 653, 653, 653, 653, 653, 653, 653, 653, 653, 787, 787, 787, 907, 907, 907, 907, 1433, 1433, 1433, 1433, 1433, 1447, 1543, 1543, 1867, 1867, 1867, 1867, 1867, 1867

Offset: 1

Jonathan Vos Post, Nov 27 2012

			a(2) = 29, the smallest prime not neighboring a 3-smooth number, since 3 is the 2nd prime; i.e., not of the form 2^j*3^k +/- 1. 43-1 = 2*3*7, 43+1 = 2*2*11, so neither are 5-smooth.
a(3) = 43, the smallest prime not neighboring a 5-smooth number, since 5 is the 3rd prime, and 43-1 = 42 = 2 * 3 * 7 is not 5 smooth, and 43+1 = 44 = 2^2 * 11 is not 5 smooth. - corrected by _Jason Kimberley_, Nov 29 2012
a(4) = 67, the smallest prime not neighboring a 7-smooth number, since 7 is the 4th prime, and 67-1 = 66 = 2 * 3 * 11 is not 7 smooth, and 67+1 = 68 = 2^2 * 17 is not 7 smooth. - corrected by _Jason Kimberley_, Nov 29 2012
a(5) = 103, the smallest prime not neighboring a 11-smooth number, since 11 is the 5th prime, and 103-1 = 102 = 2 * 3 * 17 is not 11 smooth, and 103+1 = 104 = 2^3 * 13 is not 11 smooth.
a(6) = 137, the smallest prime not neighboring a 13-smooth number, since 13 is the 6th prime, and 137-1 = 136 = 2^3 * 17 is not 13 smooth, and 137+1 = 138 = 2 * 3 * 23 is not 13 smooth.

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

Cf. A000040, A002473, A051037, A051038, A080194, A219528, A219697, A219785.

a(n)=my(p=prime(n));forprime(q=6*p-1,,if(vecmax(factor(q-1)[,1])>p && vecmax(factor(q+1)[,1])>p,return(q))) \\ Charles R Greathouse IV, Nov 28 2012

a(n) > 6p for n > 1, where p is the n-th prime. - Charles R Greathouse IV, Nov 28 2012

a(3) and a(4) corrected by Charles R Greathouse IV, Nov 28 2012

a(1) and a(7)-a(53) from Charles R Greathouse IV, Nov 28 2012

A298882 a(1) = 1, and for any n > 1, if n is the k-th number with least prime factor p, then a(n) is the k-th number with greatest prime factor p.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 16, 6, 32, 11, 64, 13, 128, 9, 256, 17, 512, 19, 1024, 12, 2048, 23, 4096, 10, 8192, 18, 16384, 29, 32768, 31, 65536, 24, 131072, 15, 262144, 37, 524288, 27, 1048576, 41, 2097152, 43, 4194304, 36, 8388608, 47, 16777216, 14, 33554432, 48

Offset: 1

Rémy Sigrist, Jan 28 2018

nonn

This sequence is a permutation of the natural numbers, with inverse A298268.
For any prime p and k > 0:
- if s_p(k) is the k-th p-smooth number and r_p(k) is the k-th p-rough number,
- then a(p * r_p(k)) = p * s_p(k),
- for example: a(11 * A008364(k)) = 11 * A051038(k).

			The first terms, alongside A020639(n), are:
  n     a(n)    lpf(n)
  --    ----    ------
   1       1      1
   2       2      2
   3       3      3
   4       4      2
   5       5      5
   6       8      2
   7       7      7
   8      16      2
   9       6      3
  10      32      2
  11      11     11
  12      64      2
  13      13     13
  14     128      2
  15       9      3
  16     256      2
  17      17     17
  18     512      2
  19      19     19
  20    1024      2

Index entries for sequences that are permutations of the natural numbers

Cf. A008364, A020639, A046022, A051038, A055396, A078898, A083140, A125624, A298268 (inverse).

a(1) = 1.
a(A083140(n, k)) = A125624(n, k) for any n > 0 and k > 0.
a(n) = A125624(A055396(n), A078898(n)) for any n > 1.
Empirically:
- a(n) = n iff n belongs to A046022,
- a(2 * k) = 2^k for any k > 0,
- a(p^2) = 2 * p for any prime p,
- a(p * q) = 3 * p for any pair of consecutive odd primes (p, q).

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.

A232668 Natural numbers that are not (primes, 11-smooth, perfect powers or base-10 palindromes).

Original entry on oeis.org

26, 34, 38, 39, 46, 51, 52, 57, 58, 62, 65, 68, 69, 74, 76, 78, 82, 85, 86, 87, 91, 92, 93, 94, 95, 102, 104, 106, 114, 115, 116, 117, 118, 119, 122, 123, 124, 129, 130, 133, 134, 136, 138, 142, 143, 145, 146, 148, 152, 153, 155, 156, 158, 159, 164

Offset: 1

John R Phelan, Nov 27 2013

The intention was to generate a sequence of uninteresting numbers. - John R Phelan, Dec 01 2014

			16 is not in the sequence since it's a perfect power, 2^4.
19 is not in the sequence since it's prime.
18 is not in the sequence since it's 2*3*3, so it's 11-smooth.
22 is not in the sequence since it's a base 10 palindrome.
26 is in the sequence since it's 2*13, so it's not prime, not 11-smooth, not a base-10 palindrome, and not a perfect power.

Wikipedia, Complement (set theory)
Wikipedia, Interesting number paradox

This sequence is A000027 \ A000040 \ A051038 \ A002113 \ A001597.

public class Nnn {public static void main(String[] args) {String str = ""; for (int i = 0; i < 1000000 && str.length() < 250; i++) {if (isPrime(i) || isSmooth(11,i) || isPerfectPower(i) || isPalindrome(i)) {} else {str += i + ", ";}} System.out.println(str);} static boolean isPalindrome(int i) {return ((i+"").equals(new StringBuilder(i+"").reverse().toString()));} static boolean isSmooth(int s, int n) {if (n<2) return true; for (int i=2;i<=s;i++) {while (n%i==0) n=n/i;} return n==1;} static boolean isPerfectPower(int n) {for (int i=2;i<=Math.sqrt(n);i++) {int j=i*i; while (j

A000027 \ A000040 \ A051038 \ A002113 \ A001597.
A \ B represents set "subtraction", all the elements in A that are not in B.
In other words, start with the Natural numbers (A000027).
Remove the prime numbers (A000040).
Remove the 11-smooth numbers, numbers whose prime divisors are all <= 11 (A051038).
Remove the base-10 palindromes (A002113).
Remove the perfect powers, m^k where m > 0 and k >= 2 (A001597).
And what's left is this sequence.
a(n) ~ n; in particular, a(n) = n + n/log n + o(n/log n). - Charles R Greathouse IV, Nov 27 2013

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

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

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Mathematica

A133581 (k^2)-th k-smooth number for k = prime(n).

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Python

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A219785 Primes not neighboring an 11-smooth number.

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A219790 Smallest prime not neighboring a prime(n)-smooth number.

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PARI

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A298882 a(1) = 1, and for any n > 1, if n is the k-th number with least prime factor p, then a(n) is the k-th number with greatest prime factor p.

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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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Mathematica

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A232668 Natural numbers that are not (primes, 11-smooth, perfect powers or base-10 palindromes).

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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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