A281889 -id:A281889 - cp's OEIS frontend

A027746 Irregular triangle in which first row is 1, n-th row (n>1) gives prime factors of n with repetition.

1, 2, 3, 2, 2, 5, 2, 3, 7, 2, 2, 2, 3, 3, 2, 5, 11, 2, 2, 3, 13, 2, 7, 3, 5, 2, 2, 2, 2, 17, 2, 3, 3, 19, 2, 2, 5, 3, 7, 2, 11, 23, 2, 2, 2, 3, 5, 5, 2, 13, 3, 3, 3, 2, 2, 7, 29, 2, 3, 5, 31, 2, 2, 2, 2, 2, 3, 11, 2, 17, 5, 7, 2, 2, 3, 3, 37, 2, 19, 3, 13, 2, 2, 2, 5, 41, 2, 3, 7, 43, 2, 2, 11, 3, 3, 5

Offset: 1

Maghraoui Abdelkader

n-th row has length A001222(n) (n>1).

			Triangle begins
  1;
  2;
  3;
  2, 2;
  5;
  2, 3;
  7;
  2, 2, 2;
  3, 3;
  2, 5;
  11;
  2, 2, 3;
  ...

N. J. A. Sloane, First 2048 rows of triangle, flattened
S. von Worley (?), Animated Factorization Diagrams, Oct. 2012.
Brent Yorgey, Factorization diagrams, The Math Less Traveled, Oct 05 2012.

Cf. A000027, A001222, A027748.
a(A022559(A000040(n))+1) = A000040(n).
Column 1 is A020639, columns 2 and 3 correspond to A014673 and A115561.
A281890 measures frequency of each prime in each column, with A281889 giving median values.
Cf. A175943 (partial products), A265110 (partial row products), A265111.

import Data.List (unfoldr)
a027746 n k = a027746_tabl !! (n-1) !! (k-1)
a027746_tabl = map a027746_row [1..]
a027746_row 1 = [1]
a027746_row n = unfoldr fact n where
   fact 1 = Nothing
   fact x = Just (p, x `div` p) where p = a020639 x
-- Reinhard Zumkeller, Aug 27 2011

P:=proc(n) local FM: FM:=ifactors(n)[2]: seq(seq(FM[j][1],k=1..FM[j][2]),j=1..nops(FM)) end: 1; for n from 2 to 45 do P(n) od; # yields sequence in triangular form; Emeric Deutsch, Feb 13 2005

row[n_] := Flatten[ Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]]; Flatten[ Table[ row[n], {n, 1, 45}]] (* Jean-François Alcover, Dec 01 2011 *)

A027746_row(n,o=[1])=if(n>1,concat(apply(t->vector(t[2],i,t[1]), Vec(factor(n)~))),o) \\ Use %(n,[]) if you want the more natural [] for the first row. - M. F. Hasler, Jul 29 2015

def factors(n: int) -> list[int]:
    p = n
    L:list[int] = []
    for f in range(2, p + 1):
        if f * f > p: break
        while True:
            q, r = divmod(p, f)
            if r != 0: break
            L.append(f)
            p = q
            if p == 1: return L
    L.append(p)
    return L  # Peter Luschny, Jul 18 2024

v=[1]
for k in [2..45]: v.extend(p for (p, m) in factor(k) for _ in range(m))
print(v) # Giuseppe Coppoletta, Dec 29 2017

Product_{k=1..A001222(n)} T(n,k) = n.
From Reinhard Zumkeller, Aug 27 2011: (Start)
A001414(n) = Sum_{k=1..A001222(n)} T(n,k), n>1;
A006530(n) = T(n,A001222(n)) = Max_{k=1..A001222(n)} T(n,k);
A020639(n) = T(n,1) = Min_{k=1..A001222(n)} T(n,k). (End)

More terms from James Sellers

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

Original entry on oeis.org

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

A016088 a(n) = smallest prime p such that Sum_{primes q = 2, ..., p} 1/q exceeds n.

Original entry on oeis.org

2, 5, 277, 5195977, 1801241230056600523

Offset: 0

Robert G. Wilson v

The indices of these primes are in A046024: a(n) = A000040(A046024(n)).
Comment from Eric Bach, Jun 03 2005: Schoenfeld (Math. Comp. 1976) has explicit estimates that imply, assuming the Riemann hypothesis, that the sum first exceeds 4 for some x in the range (1.80124093... * 10^18, 1.80124152... * 10^18).
Eric Bach (Sep 14 2005) comments that the next element in the sequence is about 4.2 * 10^49, so it may remain unknown for all eternity.
Comment from Richard C. Schroeppel, Nov 09 2006: The Bach-Sorenson algorithm takes around x^(1/3) space and x^(2/3) time. When x = 4 * 10^49, these are roughly 10^16 and 10^33. We have facilities today that handle 10^16 storage. Current world computing capability is ~10^25 instructions/year (10^8.5 machines, 10^9.5 inst/sec, 10^7.5 sec/year). The algorithm seems very parallelizable. So with current resources, we could have the value for a(5) in a mere 10^8 years. This might seem like a long time, but it's far short of eternity.
The sequence is less than 2^3^n for all n >= 1. Moreover, the limit a of a_n^e^(-n) seems to exist and is approximately 2.16 and thus a^e^n is an estimate for the sequence which is not completely wrong. - Wolfgang Burmeister (Wolfgang-Burmeister(AT)t-online.de), May 05 2007
Sequence can be approximated by the simple expression a(n) = exp(exp(n-0.2615)) due to the behavior of the sum of reciprocals of primes. This gives: a(4)=1.801..*10^18; a(5)=4.2..*10^49 and a(6)=7.7..*10^134. - Carmine Suriano, Mar 25 2014

Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Plenum Press, NY and London, 1996, page 64.

E. Bach, D. Klyve, and J. P. Sorenson, Computing prime harmonic sums, Math. Comp. 78 (268) (2009) 2283-2305.
Eric Bach and Jonathan Sorenson, Computing Prime Harmonic Sums [Wayback Machine cached version]
J. Barkley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions theta(x) and psi(x), Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. Math. Comp. 29 (1975), 243-269.
Lowell Schoenfeld, Sharper bounds for the Chebyshev functions theta (x) and psi (x). II. Math. Comp. 30 (1976), number 134, 337-360.
Lowell Schoenfeld, Corrigendum: "Sharper bounds for the Chebyshev functions theta (x) and psi (x). II" (Math. Comput. 30 (1976), number 134, 337-360), Math. Comp. 30 (1976), number 136, 900.
Index entries for sequences related to decimal expansion of 1/n

Cf. A046024, A223037, A281889.

s = 0; k = 1; Do[ While[ s = N[ s + 1/Prime[ k ], 36 ]; s <= n, k++ ]; Print[ Prime[ k ] ]; k++, {n, 1, 3} ]
s = 0; n = 0; For[k = 1, k > 0, k++, If[(s = N[s + 1/(p = Prime[k]), 40]) > n, Print[p|s]; n++ ]] (* Wolfgang Burmeister (Wolfgang-Burmeister(AT)t-online.de), May 05 2007 *)

a(n)=my(t); forprime(p=2,, t+=1./p; if(t>n, return(p))) \\ Charles R Greathouse IV, Apr 29 2015

From Jonathan Sondow, Apr 17 2013: (Start)
a(n) = A000040(A000720(A223037(n))+1).
a(n) ~ prime(floor(e^e^n)) = A000040(A096232(n)) as n -> oo (see comment in A223037). (End)

a(0) from Wolfgang Burmeister (Wolfgang-Burmeister(AT)t-online.de), May 05 2007
a(3) corrected by Ulrich Schimke (UlrSchimke(AT)aol.com)
a(4) computed by Eric Bach and Jon Sorenson, Sep 14 2005. They used a variant of the Lagarias-Miller-Odlyzko algorithm for pi(x) and found that sum_{p <= 1801241230056600467} 1/p = 3.99999999999999999966 and sum_{p <= 1801241230056600523} 1/p = 4.00000000000000000021. There are no primes between 1801241230056600467 and 1801241230056600523. Total computing time was about two weeks, divided between two workstations (i.e., about a week on each).

A014673 Smallest prime factor of greatest proper divisor of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 2, 3, 5, 1, 2, 1, 7, 5, 2, 1, 3, 1, 2, 7, 11, 1, 2, 5, 13, 3, 2, 1, 3, 1, 2, 11, 17, 7, 2, 1, 19, 13, 2, 1, 3, 1, 2, 3, 23, 1, 2, 7, 5, 17, 2, 1, 3, 11, 2, 19, 29, 1, 2, 1, 31, 3, 2, 13, 3, 1, 2, 23, 5, 1, 2, 1, 37, 5, 2, 11, 3, 1, 2, 3, 41, 1, 2, 17, 43, 29, 2, 1, 3, 13, 2, 31, 47

Offset: 1

Reinhard Zumkeller, Jun 24 2003

nonn

For n > 1: a(n) = 1 iff n is prime; a(A001358(n)) = A084127(n); a(A025475(n)) = A020639(A025475(n)). [corrected by Peter Munn, Feb 19 2017]
When n is composite, this is the 2nd factor when n is written as a product of primes in nondecreasing order. For example, 12 = 2*2*3, so a(12) = 2. - Peter Munn, Feb 19 2017
For all sufficiently large n the median value of a(1), a(2), ... a(n) is A281889(2) = 7. - Peter Munn, Jul 12 2019

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

Cf. A001222, A001358, A020639, A025475, A027746, A032742, A054576, A084127, A085392, A085393, A115561, A117357, A117358, A281889.

PrimeFactors[ n_ ] := Flatten[ Table[ # [ [ 1 ] ], {1} ] & /@ FactorInteger[ n ] ]; f[ n_ ] := Block[ {gpd = Divisors[ n ][ [ -2 ] ]}, If[ gpd == 1, 1, PrimeFactors[ gpd ][ [ 1 ] ] ] ]; Table[ If[ n == 1, 1, f[ n ] ], {n, 1, 95} ]
(* Second program: *)
Table[If[Or[PrimeQ@ n, n == 1], 1, FactorInteger[n/SelectFirst[Prime@ Range@ PrimePi[Sqrt@ n], Divisible[n, #] &]][[1, 1]] ], {n, 94}] (* Michael De Vlieger, Aug 14 2017 *)

lpf(n)=if(n>1,factor(n)[1,1],1)
a(n)=lpf(n/lpf(n)) \\ Charles R Greathouse IV, May 09 2013

a(n)=if(n<4||isprime(n),return(1)); my(f=factor(n)); if(f[1,2]>1, f[1,1], f[2,1]) \\ Charles R Greathouse IV, May 09 2013

(define (A014673 n) (A020639 (/ n (A020639 n)))) ;; Code for A020639 given under that entry - Antti Karttunen, Aug 12 2017

a(n) = A020639(A032742(n)).
A117357(n) = A020639(A054576(n)); A117358(n) = A032742(A054576(n)) = A054576(n)/A117357(n). - Reinhard Zumkeller, Mar 10 2006
If A001222(n) >= 2, a(n) = A027746(n,2), otherwise a(n) = 1. - Peter Munn, Jul 13 2019

A284411 Least prime p such that more than half of all integers are divisible by n distinct primes not greater than p.

Original entry on oeis.org

3, 37, 42719, 5737850066077

Offset: 1

Peter Munn, Mar 26 2017

The proportion of all integers that satisfy the divisibility criterion for p=prime(m) is determined using the proportion that satisfy it over any interval of primorial(m)=A002110(m) integers.
a(4) is from De Koninck, 2009; calculation credited to David Grégoire.
a(5) is about 7.887*10^34 assuming the Riemann Hypothesis, and about 7*10^34 unconditionally (De Koninck and Tenenbaum, 2002). - Amiram Eldar, Dec 05 2024

			Exactly half of the integers are divisible by 2, so a(1)>2. Two-thirds of all integers are divisible by 2 or 3, so a(1) = 3.

Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, pp. 13, 216 and 368.

Jean-Marie De Koninck and Gérald Tenenbaum, Sur la loi de répartition du k-ième facteur premier d'un entier, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 133, No. 2 (2002), pp. 191-204.
Gérald Tenenbaum, Some of Erdős' unconventional problems in number theory, thirty-four years later, Erdős Centennial, Janos Bolyai Math. Soc., 2013, 651-681. HAL Id: hal-01281530.

Cf. A002110, A077761, A096294, A194156, A281889.

Cf. A038110, A038111, A342479, A342480, A378720, A378721.

a(n) is least p=prime(m) such that 2*Sum_{k=0..n-1} A096294(m,k) < A002110(m).

log(log(a(n))) = n - b + O(1/sqrt(n)), where b = 1/3 + A077761 (De Koninck and Tenenbaum, 2002). - Amiram Eldar, Dec 05 2024

Definition edited by N. J. A. Sloane, Apr 01 2017

A115561 a(n) = lpf((n/lpf(n))/lpf(n/lpf(n))), where lpf=A020639, least prime factor.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 5, 1, 1, 1, 2, 1, 1, 3, 7, 1, 5, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 7, 1, 11, 5, 1, 1, 2, 1, 5, 1, 13, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1, 7, 2, 1, 11, 1, 17, 1, 7, 1, 2, 1, 1, 5, 19, 1, 13, 1, 2, 3, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 23, 1, 1, 1, 2, 1, 7, 11, 5, 1

Offset: 1

Reinhard Zumkeller, Mar 10 2006

nonn

From Peter Munn, Jul 14 2019: (Start)
a(n) = 1 if and only if n is 1 or a prime or semiprime. Otherwise a(n) is the 3rd factor when n is written as a product of primes in nondecreasing order. For example, 60 = 2*2*3*5, so a(60) = 3.
Although values equal to 1 are predominant at low indices, their asymptotic density is 0, whereas for values equal to prime(k) for k > 0 the asymptotic density is positive, namely A281890(k,3)/A002110(k)^3. For all sufficiently large n the median value of a(1), a(2), ... a(n) is A281889(3) = 433.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Least Prime Factor

Cf. A001222, A002110, A014673, A020639, A027746, A032742, A054576, A117358, A281889, A281890.

f[n_] := FactorInteger[n][[1, 1]]; Table[f[#/f@ #] &[n/f@ n], {n, 101}] (* Michael De Vlieger, Aug 14 2017 *)

a020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1) \\ after M. F. Hasler in A020639
a(n) = a020639((n/a020639(n))/a020639(n/a020639(n))) \\ Felix Fröhlich, Jul 15 2019

from sympy import divisors, primefactors
def a032752(n): return 1 if n==1 else divisors(n)[-2]
def a020639(n): return 1 if n==1 else primefactors(n)[0]
def a(n): return a020639(a032752(a032752(n)))
print([a(n) for n in range(1, 102)]) # Indranil Ghosh, Aug 12 2017

a(n) = A020639(A054576(n)).

If A001222(n) >= 3, a(n) = A027746(n,3), otherwise a(n) = 1. - Peter Munn, Jul 13 2019

A126283 Largest number k for which the n-th prime is the median of the largest prime dividing the first k integers.

Original entry on oeis.org

4, 18, 40, 76, 116, 182, 246, 330, 426, 532, 652, 770, 904, 1058, 1210, 1386, 1560, 1752, 1956, 2162, 2394, 2640, 2894, 3150, 3422, 3680, 3984, 4302, 4628, 4974, 5294, 5650, 5914, 6006, 6372, 6746, 7146, 7536, 7938, 8386, 8794, 9222, 9702, 10156

Offset: 1

Mark Thornquist (mthornqu(AT)fhcrc.org) & Robert G. Wilson v, Dec 15 2006

nonn

a(14) = 1058 is the first term where a(n) exceeds A290154(n). - Peter Munn, Aug 02 2019

			a(1)=4 because the median of {2,3,2} = {2, *2*,3} is 2 (the * surrounds the median) and for any number greater than 4 the median is greater than 2.
a(1)=18 because the median of {2,3,2,5,3,7,2,3,5,11,3,13,7,5,2,17,3} = {2,2,2,2,3,3,3,3, *3*,5,5,5,7,7,11,13,17}.

Other sequences about medians of prime factors: A124202, A126282, A281889, A284411, A290154, A308904.

Cf. A000040, A309366.

t = Table[0, {100}]; lst = {}; Do[lpf = FactorInteger[n][[ -1, 1]]; AppendTo[lst, lpf]; mdn = Median@lst; If[PrimeQ@ mdn, t[[PrimePi@mdn]] = n], {n, 2, 10^4}]; t

A126282 Median of the largest prime dividing the first 10^n numbers greater than 1.

Original entry on oeis.org

3, 11, 43, 191, 797, 3259, 13267, 54049, 219277, 887707

Offset: 1

Mark Thornquist (mthornqu(AT)fhcrc.org) and Robert G. Wilson v, Dec 15 2006

A randomly selected number <= 10^n (uniform distribution from 2 to 10^n) has a 50% probability of having a prime factor at least as large as a(n).

			The largest prime divisors of the nonunit 1-digit numbers are 2, 3, 2, 5, 3, 7, 2 and 3 respectively, with median 3.

Cf. A124202, A126283, A281889, A284411.

f[n_Integer(* n must be even so as to find a true median, not an average, and n must be greater than *)] := Block[{cnt, lmt = n/2, p = PrimePi[n/2], q = PrimePi[n]}, cnt = q - p; p--; While[cnt < lmt, cnt = cnt + Floor[n/Prime@ p]; p-- ]; p++; Prime@ p]; MapAt[# + 1 &, Reap[Do[Sow@ f[10^n], {n, 6}]][[-1, -1]], 1]

A309366 When the positive integers are written as products of primes in nondecreasing order, a(n) is the least prime to occur more frequently in n-th position than in any other position.

Original entry on oeis.org

2, 5, 71, 43103

Offset: 1

Peter Munn, Jul 25 2019

In such products of primes, prime(m) occurs in n-th position A281890(m,n) times in every interval of A002110(m)^n positive integers, as explained in A281890. A002110(m) = primorial(m), product of first m primes.
For n >= 2, a(n) is the least prime to occur more frequently in n-th position than (n-1)-th position.
Primes p satisfying a(n) <= p < a(n+1) appear to occur more frequently in n-th position than in any other position.
The next term, a(5), is estimated to be ~ 6*10^11.

			a(1) = prime(1) = 2, since 2 occurs in n-th position when an integer divisible by 2^n is written as a product of primes in nondecreasing order, thus more frequently in 1st position than in other positions.
Prime(2) = 3 occurs more often in 1st position than 2nd position, specifically once for every 6 consecutive integers (since A281890(2,1) = 1 and primorial(2) = 6) compared with 5 times for every 36 consecutive integers (since A281890(2,2) = 5 and primorial(2)^2 = 36). As 2 and 3 each occur more frequently in 1st position than 2nd position, a(2) > 3.
Prime(3) = 5 occurs in 1st position A281890(3,1) = 2 times in primorial(3) = 30, in 2nd position A281890(3,2) = 62 times in 30^2, in 3rd position A281890(3,3) = 1322 times in 30^3, and decreasingly frequently in subsequent positions. 2/30 < 62/30^2 and 62/30^2 > 1322/30^3. Thus 5 occurs most frequently in 2nd position and is the first prime to do so, so a(2) = 5.

Cf. A002110, A027746, A126283, A281889, A281890.

a(1) = prime(1) = 2.

For n >= 2, a(n) = min{ k : k = prime(m), A281890(m,n) > A002110(m) * A281890(m,n-1) }.

A180126 a(n) is the least k such that for numbers x >= k, PrimePi(n,x) > PrimePi(n-1,x), where PrimePi(n,x) is the number of n-almost-primes <= x.

Original entry on oeis.org

3, 34, 15530, 151165607042

Offset: 1

T. D. Noe, Aug 11 2010

Note that a(n) is an n-almost-prime. An n-almost-prime is a number having exactly n prime factors (counted with multiplicity). So 1 is the only 0-almost-prime; 1-almost-primes are the usual prime numbers; 2-almost-primes are also called semiprimes. The first three terms are mentioned in A125149.

For 2 <= n <= 4, the values for a(n)/a(n-1) (11.3, 456.8, 9733780.2) are each a little larger than A281889(n), "the median n-th least prime factor of the integers". - Peter Munn, Jan 04 2023

Cf. A078840, A125149, A281889.

Name edited by Peter Munn, Jan 04 2023

cp's OEIS Frontend

A027746 Irregular triangle in which first row is 1, n-th row (n>1) gives prime factors of n with repetition.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Sage

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Python

Formula

A016088 a(n) = smallest prime p such that Sum_{primes q = 2, ..., p} 1/q exceeds n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A014673 Smallest prime factor of greatest proper divisor of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Scheme

Formula

A284411 Least prime p such that more than half of all integers are divisible by n distinct primes not greater than p.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

Extensions

A115561 a(n) = lpf((n/lpf(n))/lpf(n/lpf(n))), where lpf=A020639, least prime factor.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI