A006530 -id:A006530 - cp's OEIS frontend

A063539 Numbers n that are sqrt(n-1)-smooth: largest prime factor of n (=A006530(n)) < sqrt(n).

1, 8, 12, 16, 18, 24, 27, 30, 32, 36, 40, 45, 48, 50, 54, 56, 60, 63, 64, 70, 72, 75, 80, 81, 84, 90, 96, 98, 100, 105, 108, 112, 120, 125, 126, 128, 132, 135, 140, 144, 147, 150, 154, 160, 162, 165, 168, 175, 176, 180, 182, 189, 192, 195, 196

Offset: 1

N. J. A. Sloane, Aug 14 2001

nonn

Sometimes (Weisstein) called the "usual numbers" as opposed to what Greene and Knuth define as "unusual numbers" (A063538), which turn out to not be so unusual after all (Greene and Knuth 1990, Finch 2001). - Jonathan Vos Post, Sep 11 2010
If we define a divisor d|n to be superior if d >= n/d, then superior divisors are counted by A038548 and listed by A161908. This sequence lists numbers without a superior prime divisor, which is unique (A341676) when it exists. For example, the set of superior prime divisors of each n starts: {},{2},{3},{2},{5},{3},{7},{},{3},{5},{11},{},{13},{7}. The positions of empty sets give the sequence. - Gus Wiseman, Feb 24 2021
As Jonathan Vos Post's comment suggests, the sqrt(n-1)-smooth numbers are asymptotically less dense than their "unusual" complement. This is part of a larger picture of "typical" relative sizes of a number's prime factors: see, for example, the medians of the n-th smallest prime factors of the positive integers in A281889. - Peter Munn, Mar 03 2021

			a(100) = 360; a(1000) = 3744; a(10000) = 37665; a(100000)=375084;
a(10^6) = 3697669; a(10^7) = 36519633; a(10^8) = 360856296;
a(10^9) = 3571942311; a(10^10) = 35410325861; a(10^11) = 351498917129. - _Giovanni Resta_, Apr 12 2020

Greene, D. H. and Knuth, D. E., Mathematics for the Analysis of Algorithms, 3rd ed. Boston, MA: Birkhäuser, pp. 95-98, 1990.

N. J. A. Sloane, Table of a(n) for n = 1..10622 [Extending and correcting earlier b-files from T. D. Noe and Marius A. Burtea]
M. Beeler, R. W. Gosper and R. Schroeppel, HAKMEM, ITEM 29
Steven Finch, "RE: Unusual Numbers.", Aug 27 2001.
Hugo Pfoertner, Illustration of deviation from linear fit 3.7642*n, (2020).
Hugo Pfoertner, Illustration of relative error of asymptotic approximation, (2020).
Project Euler, Problem 668: Square root smooth numbers
V. Ramaswami, On the number of positive integers less than x and free of prime divisors greater than x^c, Bull. Amer. Math. Soc. 55 (1949), 1122-1127.
Eric W. Weisstein, Rough Number. [From _Jonathan Vos Post_, Sep 11 2010]
Wikipedia, Dickman function

Set difference of A048098 and A001248.
Complement of A063538.
Cf. A006530.
The following are all different versions of sqrt(n)-smooth numbers: A048098, A063539, A064775, A295084, A333535, A333536.
Positions of zeros in A341591.
A001221 counts prime divisors, with sum A001414.
A001222 counts prime-power divisors.
A033677 selects the smallest superior divisor.
A038548 counts superior (or inferior) divisors.
A051283 lists numbers without a superior prime-power divisor.
A056924 counts strictly superior (or strictly inferior) divisors.
A059172 lists numbers without a superior squarefree divisor.
A063962 counts inferior prime divisors.
A116882/A116883 list numbers with/without a superior odd divisor.
A161908 lists superior divisors.
A207375 lists central divisors.
A217581 selects the greatest inferior prime divisor.
A341642 counts strictly superior prime divisors.
A341676 gives unique superior prime divisors, with strict case A341643.
- Inferior: A033676, A066839, A069288, A161906, A333749, A333750.
- Superior: A070038, A072500, A341592, A341593, A341675.
- Strictly inferior: A060775, A070039, A333805, A333806, A341596, A341674.
- Strictly superior: A064052, A140271, A238535, A341594, A341595, A341644, A341645, A341646, A341673.
Cf. A000005, A000203, A020639, A112798, A281889.

[1] cat [m:m in [2..200]| Max(PrimeFactors(m)) lt Sqrt(m) ]; // Marius A. Burtea, May 08 2019

N:= 1000: # to get all terms <= N
Primes:= select(isprime, [2, seq(2*i+1, i=1..floor((N-1)/2))]):
S:= {$1..N} minus {seq(seq(m*p, m = 1 .. min(p, N/p)), p=Primes)}:
sort(convert(S, list)); # Robert Israel, Sep 02 2015

Prepend[Select[Range[192], FactorInteger[#][[-1, 1]] < Sqrt[#] &], 1] (* Ivan Neretin, Sep 02 2015 *)

from math import isqrt
from sympy import primepi
def A063539(n):
    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 f(x): return int(n+primepi(x//(y:=isqrt(x)))+sum(primepi(x//i)-primepi(i) for i in range(1,y)))
    return bisection(f,n,n) # Chai Wah Wu, Oct 05 2024

From Hugo Pfoertner, Apr 02 - Apr 12 2020: (Start)
For small n (e.g. n < 10000) a(n) can apparently be approximated by 3.7642*n.
Asymptotically, the number of sqrt(n)-smooth numbers < x is known to be (1-log(2))*x + O(x/log(x)), see Ramaswami (1949).
n = (1-log(2))*a(n) - 0.59436*a(n)/log(a(n)) is a fitted approximation. (End)
However, it is known that this fit only leads to an increase of accuracy in the range up to a(10^11). The improvement in accuracy suggested by the plot of the relative error for even larger n does not occur. For larger n the behavior of the error term O(x/log(x)) is not known. - Hugo Pfoertner, Nov 12 2023

A244991 Numbers whose greatest prime factor is a prime with an odd index; n such that A006530(n) is in A031368.

Original entry on oeis.org

2, 4, 5, 8, 10, 11, 15, 16, 17, 20, 22, 23, 25, 30, 31, 32, 33, 34, 40, 41, 44, 45, 46, 47, 50, 51, 55, 59, 60, 62, 64, 66, 67, 68, 69, 73, 75, 77, 80, 82, 83, 85, 88, 90, 92, 93, 94, 97, 99, 100, 102, 103, 109, 110, 115, 118, 119, 120, 121, 123, 124, 125, 127, 128

Offset: 1

Antti Karttunen, Jul 21 2014

nonn

Equally, numbers n for which A061395(n) is odd.
A122111 maps each one of these numbers to a unique term of A026424 and vice versa.
If the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), these are the Heinz numbers of partitions whose greatest part is odd, counted by A027193. - Gus Wiseman, Feb 08 2021

			From _Gus Wiseman_, Feb 08 2021: (Start)
The sequence of terms together with their prime indices begins:
      2: {1}           32: {1,1,1,1,1}     64: {1,1,1,1,1,1}
      4: {1,1}         33: {2,5}           66: {1,2,5}
      5: {3}           34: {1,7}           67: {19}
      8: {1,1,1}       40: {1,1,1,3}       68: {1,1,7}
     10: {1,3}         41: {13}            69: {2,9}
     11: {5}           44: {1,1,5}         73: {21}
     15: {2,3}         45: {2,2,3}         75: {2,3,3}
     16: {1,1,1,1}     46: {1,9}           77: {4,5}
     17: {7}           47: {15}            80: {1,1,1,1,3}
     20: {1,1,3}       50: {1,3,3}         82: {1,13}
     22: {1,5}         51: {2,7}           83: {23}
     23: {9}           55: {3,5}           85: {3,7}
     25: {3,3}         59: {17}            88: {1,1,1,5}
     30: {1,2,3}       60: {1,1,2,3}       90: {1,2,2,3}
     31: {11}          62: {1,11}          92: {1,1,9}
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10001

Complement: A244990.
Cf. A006530, A026424, A031368, A122111, A244321, A244322, A244989.
Looking at least instead of greatest prime index gives A026804.
The partitions with these Heinz numbers are counted by A027193.
The case where Omega is odd also is A340386.
A001222 counts prime factors.
A056239 adds up prime indices.
A300063 ranks partitions of odd numbers.
A061395 selects maximum prime index.
A066208 ranks partitions into odd parts.
A112798 lists the prime indices of each positive integer.
A340931 ranks odd-length partitions of odd numbers.
Cf. A000009, A058695, A072233, A160786, A300272, A340101, A340385, A340604.

Select[Range[100],OddQ[PrimePi[FactorInteger[#][[-1,1]]]]&] (* Gus Wiseman, Feb 08 2021 *)

For all n, A244989(a(n)) = n.

A063538 Numbers n that are not sqrt(n-1)-smooth: largest prime factor of n (=A006530(n)) >= sqrt(n).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91

Offset: 1

N. J. A. Sloane, Aug 14 2001

If we define a divisor d|n to be superior if d >= n/d, then superior divisors are counted by A038548 and listed by A161908. This sequence lists all numbers with a superior prime divisor, which is unique (A341676) when it exists. For example, 42 is in the sequence because it has a prime divisor 7 which is greater than the quotient 42/7 = 6. - Gus Wiseman, Feb 19 2021

D. H. Greene and D. E. Knuth, Mathematics for the Analysis of Algorithms; see pp. 95-98.

Robert Israel, Table of n, a(n) for n = 1..10000
Beeler, M., Gosper, R. W. and Schroeppel, R., HAKMEM, ITEM 29

Cf. A006530, A063762.
Complement of A063539. Supersequence of A001358 (semiprimes).
The strictly superior version is A064052 (complement: A048098), with associated unique prime divisor A341643.
The case of odd instead of prime divisors is A116883 (complement: A116882).
Also nonzeros of A341591 (number of superior prime divisors).
The unique superior prime divisors of the terms are A341676.
A001221 counts prime divisors, with sum A001414.
A033677 selects the smallest superior divisor.
A038548 counts superior (also inferior) divisors.
A161908 lists superior divisors.
- Inferior: A033676, A063962, A066839, A161906, A217581, A333749, A333750.
- Superior: A051283, A059172, A070038, A072500, A341592, A341593, A341675.
- Strictly Inferior: A056924, A060775, A070039, A333805, A333806, A341596, A341674, A341677.
- Strictly Superior: A140271, A238535, A341594, A341595, A341642, A341644, A341645/A341646, A341673.
Cf. A000005, A001055, A001222, A001248, A020639, A056239, A112798.

N:= 1000: # to get all terms <= N
Primes:= select(isprime, [2,seq(2*i+1, i=1..floor((N-1)/2))]):
S:= {seq(seq(m*p, m = 1 .. min(p, N/p)),p=Primes)}:
sort(convert(S,list)); # Robert Israel, Sep 01 2015

Select[Range[2, 91], FactorInteger[#][[-1, 1]] >= Sqrt[#] &] (* Ivan Neretin, Aug 30 2015 *)

from math import isqrt
from sympy import primepi
def A063538(n):
    def f(x): return int(n+x-primepi(x//(y:=isqrt(x)))-sum(primepi(x//i)-primepi(i) for i in range(1,y)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Oct 05 2024

Union of A001248 and A064052. - Gus Wiseman, Feb 24 2021

A241916 a(2^k) = 2^k, and for other numbers, if n = 2^e1 * 3^e2 * 5^e3 * ... p_k^e_k, then a(n) = 2^(e_k - 1) * 3^(e_{k-1}) * ... * p_{k-1}^e2 * p_k^(e1+1). Here p_k is the greatest prime factor of n (A006530), and e_k is its exponent (A071178), and the exponents e1, ..., e_{k-1} >= 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 7, 8, 6, 25, 11, 27, 13, 49, 15, 16, 17, 18, 19, 125, 35, 121, 23, 81, 10, 169, 12, 343, 29, 75, 31, 32, 77, 289, 21, 54, 37, 361, 143, 625, 41, 245, 43, 1331, 45, 529, 47, 243, 14, 50, 221, 2197, 53, 36, 55, 2401, 323, 841, 59, 375, 61, 961, 175, 64

Offset: 1

Antti Karttunen, May 03 2014

nonn

For other numbers than the powers of 2 (that are fixed), this permutation reverses the sequence of exponents in the prime factorization of n from the exponent of 2 to that of the largest prime factor, except that the exponents of 2 and the greatest prime factor present are adjusted by one. Note that some of the exponents might be zeros.
Self-inverse permutation of natural numbers, composition of A122111 & A241909 in either order: a(n) = A122111(A241909(n)) = A241909(A122111(n)).
This permutation preserves both bigomega and the (index of) largest prime factor: for all n it holds that A001222(a(n)) = A001222(n) and A006530(a(n)) = A006530(n) [equally: A061395(a(n)) = A061395(n)].
From the above it follows, that this fixes both primes (A000040) and powers of two (A000079), among other numbers.
Even positions from n=4 onward contain only terms of A070003, and the odd positions only the terms of A102750, apart from 1 which is at a(1), and 2 which is at a(2).

Antti Karttunen, Table of n, a(n) for n = 1..8192
A. Karttunen, A few notes on A122111, A241909 & A241916.
Index entries for sequences that are permutations of the natural numbers

A241912 gives the fixed points; A241913 their complement.
Cf. A006530, A137502, A070003, A102750, A278525.
{A000027, A122111, A241909, A241916} form a 4-group.
The sum of prime indices of a(n) is A243503(n).
Even bisection of A358195 = Heinz numbers of rows of A358172.
A112798 lists prime indices, length A001222, sum A056239.
Cf. A005940, A019565, A161511, A246277, A253565, A326844, A356958, A358137.

nn = 65; f[n_] := If[n == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ n]; g[w_List] := Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, w]; Table[If[IntegerQ@ #, n/4, g@ Reverse@(# - Join[{1}, ConstantArray[0, Length@ # - 2], {1}] &@ f@ n)] &@ Log2@ n, {n, 4, 4 nn, 4}] (* Michael De Vlieger, Aug 27 2016 *)

A209229(n) = (n && !bitand(n,n-1));
A241916(n) = if(1==A209229(n), n, my(f = factor(2*n), nbf = #f~, igp = primepi(f[nbf,1]), g = f); for(i=1,nbf,g[i,1] = prime(1+igp-primepi(f[i,1]))); factorback(g)/2); \\ Antti Karttunen, Jul 02 2018

(define (A241916 n) (A122111 (A241909 n)))

a(1)=1, and for n>1, a(n) = A006530(n) * A137502(n)/2.
a(n) = A122111(A241909(n)) = A241909(A122111(n)).
If 2n has prime factorization Product_{i=1..k} prime(x_i), then a(n) = Product_{i=1..k-1} prime(x_k-x_i+1). The opposite version is A000027, even bisection of A246277. - Gus Wiseman, Dec 28 2022

Description clarified by Antti Karttunen, Jul 02 2018

A071322 Alternating sum of all prime factors of n; primes nonincreasing, starting with the largest prime factor: A006530(n).

Original entry on oeis.org

0, 2, 3, 0, 5, 1, 7, 2, 0, 3, 11, 3, 13, 5, 2, 0, 17, 2, 19, 5, 4, 9, 23, 1, 0, 11, 3, 7, 29, 4, 31, 2, 8, 15, 2, 0, 37, 17, 10, 3, 41, 6, 43, 11, 5, 21, 47, 3, 0, 2, 14, 13, 53, 1, 6, 5, 16, 27, 59, 2, 61, 29, 7, 0, 8, 10, 67, 17, 20, 4, 71, 2, 73, 35

Offset: 1

Reinhard Zumkeller, May 18 2002

nonn

a(n) = 0 iff n square, a(A000290(n)) = 0;

a(n) = n iff n prime, a(A000040(n)) = A000040(n).

			72 = 2*2*2*3*3, therefore a(72) = 3 - 3 + 2 - 2 + 2 = 2;
90 = 2*3*3*5, therefore a(90) = 5 - 3 + 3 - 2 = 3.

Shreyansh Jaiswal, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale).

Cf. A001414, A071324, a(n) = abs(A071321(n)).

aspf[n_]:=Total[Times@@@Partition[Riffle[Reverse[Flatten[Table[#[[1]],{#[[2]]}]&/@FactorInteger[n]]],{1,-1},{2,-1,2}],2]]; Join[{0}, Array[ aspf,80,2]] (* Harvey P. Dale, Apr 19 2015 *)

from sympy import factorint
def c(n):
    fs = factorint(n, multiple=True)
    return sum(fs[::2])-sum(fs[1::2])
for n in range(1,75):
    print(abs(c(n)), end=", ") #Shreyansh Jaiswal, Apr 16 2025

A253560 Multiply n by its largest prime factor: a(n) = A006530(n) * n.

Original entry on oeis.org

1, 4, 9, 8, 25, 18, 49, 16, 27, 50, 121, 36, 169, 98, 75, 32, 289, 54, 361, 100, 147, 242, 529, 72, 125, 338, 81, 196, 841, 150, 961, 64, 363, 578, 245, 108, 1369, 722, 507, 200, 1681, 294, 1849, 484, 225, 1058, 2209, 144, 343, 250, 867, 676, 2809, 162, 605, 392, 1083, 1682, 3481, 300, 3721, 1922, 441

Offset: 1

Antti Karttunen, Jan 03 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Essentially the same as A129598, except that here we have a(1) = 1.
Cf. A070003 (same sequence without 1, sorted into ascending order).
Cf. A000040, A006530, A052126, A061395, A122111, A253550, A253561, A253563, A253565.
Differs from A072995 for the first time at n=15, where a(15) = 75, while A072995(15) = 225.

A253560 := proc(n)
    n*A006530(n) # code re-use
end proc:
seq( A253560(n),n=1..80) ; # R. J. Mathar, Jun 27 2024

a[n_] := n FactorInteger[n][[-1, 1]];
Array[a, 100] (* Jean-François Alcover, Feb 15 2021 *)

a(n) = if (n==1, 1, n*vecmax(factor(n)[,1])); \\ Michel Marcus, Feb 15 2021

Scheme
```
(define (A253560 n) (* (A006530 n) n))
```

a(1) = 1; for n > 1, a(n) = A006530(n) * n = A000040(A061395(n)) * n.
Other identities:
a(n) >= A253550(n) for all n >= 1.
a(n) = A129598(n) for all n >= 2.
A052126(a(n)) = n. [A052126 works as an inverse function for this injection.]

A251726 Numbers n > 1 for which gpf(n) < lpf(n)^2, where lpf and gpf (least and greatest prime factor of n) are given by A020639(n) and A006530(n).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 29, 31, 32, 35, 36, 37, 41, 43, 45, 47, 48, 49, 53, 54, 55, 59, 61, 63, 64, 65, 67, 71, 72, 73, 75, 77, 79, 81, 83, 85, 89, 91, 95, 96, 97, 101, 103, 105, 107, 108, 109, 113, 115, 119, 121, 125, 127, 128, 131, 133, 135, 137, 139, 143, 144

Offset: 1

Antti Karttunen, Dec 17 2014

nonn

Numbers n > 1 for which there exists r <= gpf(n) such that r^k <= lpf(n) and gpf(n) < r^(k+1) for some k >= 0, where lpf and gpf (least and greatest prime factor of n) are given by A020639(n) and A006530(n) (the original, equivalent definition of the sequence).
Numbers n > 1 such that A252375(n) < 1 + A006530(n). Equally, one can substitute A251725 for A252375.
These are numbers n all of whose prime factors "fit between" two consecutive powers of some positive integer which itself is <= the largest prime factor of n.
Conjecture: If any n is in the sequence, then so is A003961(n).
Note: if Legendre's or Brocard's conjecture is true, then the above conjecture is true as well. See my comments at A251728. - Antti Karttunen, Jan 01 2015

			For 35 = 5*7, 7 is less than 5^2, thus 35 is included.
For 90 = 2*3*3*5, 5 is not less than 2^2, thus 90 is NOT included.
For 105 = 3*5*7, 7 is less than 3^2, thus 105 is included.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Ratio A251726(n)/A251727(n) (plotted with OEIS-server's Plot2-utility)

Complement: A251727. Subsequences: A251728, A000961 (after 1).
Characteristic function: A252372. Inverse function: A252373.
Gives the positions of zeros in A252459 (after its initial zero), cf. also A284261.
Cf. A252370 (gives the difference between the prime indices of gpf and lpf for each a(n)).
Sequence gives all n > 1 for which A284252(n) (equally: A284254) is 1, and A284256(n) (equally A284258) is 0, and also n > 1 such that A284260(n) = A006530(n).
Cf. A003961, A006530, A020639, A252375, A251725, A253784.
Related permutations: A252757-A252758.

pfQ[n_]:=Module[{f=FactorInteger[n]},f[[-1,1]]Harvey P. Dale, May 01 2015 *)

for(n=2, 150, if(vecmax(factor(n)[,1]) < vecmin(factor(n)[,1])^2, print1(n,", "))) \\ Indranil Ghosh, Mar 24 2017

from sympy import primefactors
print([n for n in range(2, 150) if max(primefactors(n))Indranil Ghosh, Mar 24 2017

Other identities. For all n >= 1:

A252373(a(n)) = n. [A252373 works as an inverse or ranking function for this sequence.]

A new simpler definition found Jan 01 2015 and the original definition moved to the Comments section.

A070087 P(n) > P(n+1) where P(n) (A006530) is the largest prime factor of n.

Original entry on oeis.org

3, 5, 7, 11, 13, 14, 15, 17, 19, 23, 26, 29, 31, 34, 35, 37, 38, 39, 41, 43, 44, 47, 49, 51, 53, 55, 59, 61, 62, 63, 65, 67, 69, 71, 73, 74, 76, 79, 80, 83, 86, 87, 89, 94, 95, 97, 99, 101, 103, 104, 107, 109, 111, 113, 116, 118, 119, 122, 123, 124

Offset: 1

N. J. A. Sloane, May 13 2002

nonn

Erdos conjectured that this sequence has asymptotic density 1/2.

H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 210.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A006530, A070089.

f[n_] := FactorInteger[n][[ -1, 1]]; Select[ Range[125], f[ # ] > f[ # + 1] &]
With[{lpfn=Table[FactorInteger[n][[-1,1]],{n,200}]},Flatten[ Position[ Partition[ lpfn,2,1],?(#[[1]]>#[[2]]&),{1},Heads->False]]] (* _Harvey P. Dale, Sep 14 2014 *)

A070089 P(n) < P(n+1) where P(n) (A006530) is the largest prime factor of n.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 10, 12, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32, 33, 36, 40, 42, 45, 46, 48, 50, 52, 54, 56, 57, 58, 60, 64, 66, 68, 70, 72, 75, 77, 78, 81, 82, 84, 85, 88, 90, 91, 92, 93, 96, 98, 100, 102, 105, 106, 108, 110, 112, 114, 115, 117

Offset: 1

N. J. A. Sloane, May 13 2002

nonn

Erdős conjectured that this sequence has asymptotic density 1/2.
There are 500149 terms in this sequence up to 10^6, 4999951 up to 10^7, 49997566 up to 10^8, and 499992458 up to 10^9. With a binomial model with p = 1/2, these would be +0.3, -0.5, -0.0, and -0.5 standard deviations from their respective means. In other words, Erdős's conjecture seems solid. - Charles R Greathouse IV, Oct 27 2015
Erdős and Pomerance (1978) proved that the lower density of this sequence is at least 0.0099. This value was improved to 0.05544 (De La Bretèche et al., 2005), 0.1063 (Wang, 2017), 0.1356 (Wang, 2018), and 0.2017 (Lü and Wang, 2018). - Amiram Eldar, Aug 02 2020

H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 210.

T. D. Noe, Table of n, a(n) for n = 1..1000
Régis De La Bretèche, Carl Pomerance and Gérald Tenenbaum, Products of ratios of consecutive integers, The Ramanujan Journal, Vol. 9, No. 1-2 (2005), pp. 131-138, alternative link.
Paul Erdős and Carl Pomerance, On the largest prime factors of n and n + 1, Aequationes Math., Vol. 17, No. 1 (1978), pp. 311-321, alternative link.
Xiaodong Lü and Zhiwei Wang, On the largest prime factors of consecutive integers, 2018.
Zhiwei Wang, On the largest prime factors of consecutive integers in short intervals, Proceedings of the American Mathematical Society, Vol. 145, No. 8 (2017), pp. 3211-3220.
Zhiwei Wang, Sur les plus grands facteurs premiers d'entiers consécutifs, Mathematika, Vol. 64, No. 2 (2018), pp. 343-379, preprint, arXiv:1706.02980 [math.NT], 2017.

Cf. A006530, A070087.

f[n_] := FactorInteger[n][[ -1, 1]]; Select[ Range[125], f[ # ] < f[ # + 1] &]

gpf(n)=if(n<3,n,my(f=factor(n)[,1]); f[#f])
is(n)=gpf(n) < gpf(n+1) \\ Charles R Greathouse IV, Oct 27 2015

A070229 Next m>n such that m is divisible by lpf(n), lpf=A006530 largest prime factor.

Original entry on oeis.org

2, 4, 6, 6, 10, 9, 14, 10, 12, 15, 22, 15, 26, 21, 20, 18, 34, 21, 38, 25, 28, 33, 46, 27, 30, 39, 30, 35, 58, 35, 62, 34, 44, 51, 42, 39, 74, 57, 52, 45, 82, 49, 86, 55, 50, 69, 94, 51, 56, 55, 68, 65, 106, 57, 66, 63, 76, 87, 118, 65, 122

Offset: 1

Reinhard Zumkeller, May 07 2002

Or, for n>1, n + (largest prime divisor of n). [Anne Robinson, daughter of Herman P. Robinson, Oct 08 1981]

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
H. P. Robinson, Letter to N. J. A. Sloane, Oct 1981

Cf. A036441.
Cf. A006530, A061228, A256393.
Iterations: A076271 (start=1), A036441 (start=2), A123581 (start=3).

a070229 n = n + a006530 n
-- Reinhard Zumkeller, Nov 08 2015, Nov 14 2011

Table[n + Last@(First/@FactorInteger[n]), {n, 70}] (* Vladimir Joseph Stephan Orlovsky, Apr 08 2011 *)

vector(100, n, if(n>1, vecmax(factor(n)[, 1]), 1) + n) \\ Altug Alkan, Nov 13 2015

a(n) = (n/lpf(n)+1)*lpf(n).

a(n) = n+lpf(n) where lpf=A006530 largest prime factor. For example, a(14)=14+7=21 (instead of ((14/7)+1)*7). - Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Jun 14 2007

cp's OEIS Frontend

A063539 Numbers n that are sqrt(n-1)-smooth: largest prime factor of n (=A006530(n)) < sqrt(n).

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A244991 Numbers whose greatest prime factor is a prime with an odd index; n such that A006530(n) is in A031368.

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A063538 Numbers n that are not sqrt(n-1)-smooth: largest prime factor of n (=A006530(n)) >= sqrt(n).

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A241916 a(2^k) = 2^k, and for other numbers, if n = 2^e1 * 3^e2 * 5^e3 * ... p_k^e_k, then a(n) = 2^(e_k - 1) * 3^(e_{k-1}) * ... * p_{k-1}^e2 * p_k^(e1+1). Here p_k is the greatest prime factor of n (A006530), and e_k is its exponent (A071178), and the exponents e1, ..., e_{k-1} >= 0.

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A071322 Alternating sum of all prime factors of n; primes nonincreasing, starting with the largest prime factor: A006530(n).

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A253560 Multiply n by its largest prime factor: a(n) = A006530(n) * n.

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A251726 Numbers n > 1 for which gpf(n) < lpf(n)^2, where lpf and gpf (least and greatest prime factor of n) are given by A020639(n) and A006530(n).

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