A182987 -id:A182987 - cp's OEIS frontend

A033676 Largest divisor of n <= sqrt(n).

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 4, 1, 3, 1, 4, 3, 2, 1, 4, 5, 2, 3, 4, 1, 5, 1, 4, 3, 2, 5, 6, 1, 2, 3, 5, 1, 6, 1, 4, 5, 2, 1, 6, 7, 5, 3, 4, 1, 6, 5, 7, 3, 2, 1, 6, 1, 2, 7, 8, 5, 6, 1, 4, 3, 7, 1, 8, 1, 2, 5, 4, 7, 6, 1, 8, 9, 2, 1, 7, 5, 2, 3

Offset: 1

N. J. A. Sloane

a(n) = sqrt(n) is a new record if and only if n is a square. - Zak Seidov, Jul 17 2009
a(n) = A060775(n) unless n is a square, when a(n) = A033677(n) = sqrt(n) is strictly larger than A060775(n). It would be nice to have an efficient algorithm to calculate these terms when n has a large number of divisors, as for example in A060776, A060777 and related problems such as A182987. - M. F. Hasler, Sep 20 2011
a(n) = 1 when n = 1 or n is prime. - Alonso del Arte, Nov 25 2012
a(n) is the smallest central divisor of n. Column 1 of A207375. - Omar E. Pol, Feb 26 2019
a(n^4+n^2+1) = n^2-n+1: suppose that n^2-n+k divides n^4+n^2+1 = (n^2-n+k)*(n^2+n-k+2) - (k-1)*(2*n+1-k) for 2 <= k <= 2*n, then (k-1)*(2*n+1-k) >= n^2-n+k, or n^2 - (2*k-1)*n + (k^2-k+1) = (n-k+1/2)^2 + 3/4 < 0, which is impossible. Hence the next smallest divisor of n^4+n^2+1 than n^2-n+1 is at least n^2-n+(2*n+1) = n^2+n+1 > sqrt(n^4+n^2+1). - Jianing Song, Oct 23 2022

G. Tenenbaum, pp. 268 ff, in: R. L. Graham et al., eds., Mathematics of Paul Erdős I.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A033677 (n/a(n)), A000196 (sqrt), A027750 (list of divisors), A056737 (n/a(n) - a(n)), A219695 (half of this for odd numbers), A207375 (list the central divisor(s)).
The strictly inferior case is A060775. Cf. also A140271.
Indices of given values: A008578 (1 and the prime numbers: a(n) = 1), A161344 (a(n) = 2), A161345 (a(n) = 3), A161424 (4), A161835 (5), A162526 (6), A162527 (7), A162528 (8), A162529 (9), A162530 (10), A162531 (11), A162532 (12), A282668 (indices of primes).

a033676 n = last $ takeWhile (<= a000196 n) $ a027750_row n
-- Reinhard Zumkeller, Jun 04 2012

A033676 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do if d^2 <= n then a := max(a,d) ; end if; end do: a; end proc: # R. J. Mathar, Aug 09 2009

largestDivisorLEQR[n_Integer] := Module[{dvs = Divisors[n]}, dvs[[Ceiling[Length@dvs/2]]]]; largestDivisorLEQR /@ Range[100] (* Borislav Stanimirov, Mar 28 2010 *)
Table[Last[Select[Divisors[n],#<=Sqrt[n]&]],{n,100}] (* Harvey P. Dale, Mar 17 2017 *)

A033676(n) = {local(d);if(n<2,1,d=divisors(n);d[(length(d)+1)\2])} \\ Michael B. Porter, Jan 30 2010

from sympy import divisors
def A033676(n):
    d = divisors(n)
    return d[(len(d)-1)//2]  # Chai Wah Wu, Apr 05 2021

a(n) = n / A033677(n).
a(n) = A161906(n,A038548(n)). - Reinhard Zumkeller, Mar 08 2013
a(n) = A162348(2n-1). - Daniel Forgues, Sep 29 2014

A060796 Upper central divisor of n-th primorial.

Original entry on oeis.org

2, 3, 6, 15, 55, 182, 715, 3135, 15015, 81345, 448630, 2733549, 17490603, 114388729, 785147363, 5708795638, 43850489690, 342503171205, 2803419704514, 23622001517543, 201817933409378, 1793779635410490, 16342166369958702, 154171363634898185, 1518410187442699518, 15259831781575946565

Offset: 1

Labos Elemer, Apr 27 2001

nonn

Also: Write product of first n primes as x*y with x < y and x maximal; sequence gives value of y. Indeed, p(n)# = primorial(n) = A002110(n) is never a square for n >= 1; all exponents in the prime factorization are 1. Therefore primorial(n) has N = 2^n distinct divisors. Since this is an even number, the N divisors can be grouped in N/2 pairs {d(k), d(N+1-k)} with product equal to p(n)#. One of the two is always smaller and one is larger than sqrt(p(n)#). This sequence gives the (2^(n-1)+1)-th divisor, which is the smallest one larger than sqrt(p(n)#). - M. F. Hasler, Sep 20 2011

			n = 8, q(8) = 2*3*5*7*11*13*17*19 = 9699690. Its 128th and 129th divisors are {3094, 3135}: a(8) = 3135, and 3094 < A000196(9699690) = 3114 < 3135. [Corrected by _M. F. Hasler_, Sep 20 2011]

Max Alekseyev, Table of n, a(n) for n = 1..70

Cf. A060755, A000196, A002110, A033677, A060776, A060777, A061057, A060795 (x), A061060 (y-x), A182987 (x+y), A061030, A061031, A061032, A061033, A200744.

k = 1; Do[k *= Prime[n]; l = Divisors[k]; x = Length[l]; Print[l[[x/2 + 1]]], {n, 1, 24}] (* Ryan Propper, Jul 25 2005 *)

A060796(n) = divisors(prod(k=1,n,prime(k)))[2^(n-1)+1] \\ Requires stack size > 2^(n+5). - M. F. Hasler, Sep 20 2011

a(n) = A033677(A002110(n)).

a(n) = A002110(n)/A060795(n). - M. F. Hasler, Mar 21 2022

More terms from Ryan Propper, Jul 25 2005
a(24)-a(37) in b-file calculated from A182987 by M. F. Hasler, Sep 20 2011
a(38) from David A. Corneth, Mar 21 2022
a(39)-a(70) in b-file from Max Alekseyev, Apr 20 2022

A060795 Write product of first n primes as x*y with x

Original entry on oeis.org

1, 2, 5, 14, 42, 165, 714, 3094, 14858, 79534, 447051, 2714690, 17395070, 114371070, 783152070, 5708587335, 43848093003, 342444658094, 2803119896185, 23619540863730, 201813981102615, 1793779293633437, 16342050964565645, 154170926013430326, 1518409177581024365

Offset: 1

Labos Elemer, Apr 27 2001

nonn

Or, lower central divisor of n-th primorial.

Subsequence of A005117 (squarefree numbers). - Michel Marcus, Feb 22 2016

			n = 8: q(8) = 2*3*5*7*11*13*17*19 = 9699690. Its 128th and 129th divisors are {3094, 3135}: a(8) = 3094 and 3094 < A000196(9699690) = 3114 < 3135. [Corrected by _Colin Barker_, Oct 22 2010]
2*3*5*7 = 210 = 14*15 with difference of 1, so a(4) = 14.

Max Alekseyev, Table of n, a(n) for n = 1..70

Cf. A000196, A060776, A060777, A061057, A060796 (y), A061060 (y-x), A182987 (x+y), A061030, A061031, A061032, A061033, A060755, A033677, A200743.

F:= proc(n) local P,N,M;
     P:= {seq(ithprime(i),i=1..n)};
     N:= floor(sqrt(convert(P,`*`)));
     M:= map(convert, combinat:-powerset(P),`*`);
     max(select(`<=`,M,N))
end proc:
map(F, [$1..20]); # Robert Israel, Feb 22 2016

a[n_] := (m = Times @@ Prime[Range[n]] ; dd = Divisors[m]; dd[[Length[dd]/2 // Floor]]); Table[Print[an = a[n]]; an, {n, 1, 25}] (* Jean-François Alcover, Oct 15 2016 *)

a(n) = my(m=prod(i=1, n, prime(i))); divisors(m)[numdiv(m)\2]; \\ Michel Marcus, Feb 22 2016

a(n) = A060775(A002110(n)). - Labos Elemer, Apr 27 2001

a(n) = A002110(n)/A060796(n). - M. F. Hasler, Mar 21 2022

More terms from Ed Pegg Jr, May 28 2001
a(16)-a(23) computed by Jud McCranie, Apr 15 2000
a(24) and a(25) from Robert Israel, Feb 22 2016
a(25) corrected by Jean-François Alcover, Oct 15 2016
a(26)-a(33) in b-file from Amiram Eldar, Apr 09 2020
Up to a(38) using b-file of A060796, by M. F. Hasler, Mar 21 2022
a(39)-a(70) in b-file from Max Alekseyev, Apr 20 2022

A061060 Write product of first n primes as x*y with x

Original entry on oeis.org

1, 1, 1, 1, 13, 17, 1, 41, 157, 1811, 1579, 18859, 95533, 17659, 1995293, 208303, 2396687, 58513111, 299808329, 2460653813, 3952306763, 341777053, 115405393057, 437621467859, 1009861675153, 6660853109087, 29075165225531

Offset: 1

Ed Pegg Jr, May 28 2001

nonn

			a(4)=1: 2*3*5*7 = 210 = 14*15, so we can take x=14, y=15, with difference of 1.
Also: n=3: 2*3-5=1; n=4: 3*5-2*7=1; n=5: 5*11-2*3*7=13; n=6: 2*7*13-3*5*11=17; n=7: 5*11*13-2*3*7*17=1; n=8: 3*5*11*19-2*7*13*17=41

Max Alekseyev, Table of n, a(n) for n = 1..70
C. Aebi and G. Cairns, Partitions of primes, Parabola 45, Issue 1 (2009). - _Jonathan Sondow_, Jun 21 2012
Carlos Rivera, Conjecture 18. Minimal Primorial Partitions, The Prime Puzzles & Problems Connection.

Cf. A038667, A060776, A060777, A061057, A060795, A060796, A061030-A061033, A182987, A263292, A352813.

A061060aux := proc(l1,l2) local resul ; resul := product(l1[i],i=1..nops(l1)) ; resul := resul-product(l2[i],i=1..nops(l2)) ; RETURN(abs(resul)) ; end:
A061060 := proc(n) local plist,i,subl,resul,j,l1,l2,k,d ; plist := [] ; resul := 1 ; for i from 1 to n do resul := resul*ithprime(i) ; plist := [op(plist), ithprime(i)] ; od; for i from 1 to n/2 do subl := combinat[choose](plist,i) ; for j from 1 to nops(subl) do l1 := op(j,subl) ; l2 := convert(plist,set) minus convert(l1,set) ; d := A061060aux(l1,l2) ; if d < resul then resul := d ; fi ; od; od ; RETURN(resul) ; end:
for n from 3 to 19 do printf("%d,",A061060(n)) ; od ; # R. J. Mathar, Aug 26 2006 [This Maple program was attached to A121315. However I think it belongs here, so I renamed the variables and moved it to this entry. - N. J. A. Sloane, Sep 16 2005]

(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) f[n_] := Block[{arrayofnprimes = Array[Prime, n], primorial = Times @@ Array[Prime, n], diffmin = Infinity, adiff, sub}, If[n == 1, 1, Do[sub = Times @@ NthSubset[i, arrayofnprimes]; adiff = Abs[primorial/sub - sub]; If[adiff < diffmin, diffmin = adiff], {i, 2, 2^n/2}]; diffmin]]; Do[ Print@f@n, {n, 30}] (* Robert G. Wilson v, Sep 14 2006 *)

Conjecture: Limit_{N->oo} (Sum_{n=1..N} log(a(n))) / (Sum_{n=1..N} prime(n)) = 1/e (A068985). - Alain Rocchelli, Nov 13 2023

Terms a(16)-a(45) in b-file computed by Jud McCranie, Apr 15 2000; Jan 12 2016
a(46)-a(60) in b-file from Don Reble, Jul 11 2020
a(61)-a(70) in b-file from Max Alekseyev, Apr 20 2022

A057230 Numbers k such that k = p+q = r+s with pq = rs = primorial number(A002110) (pq) < (rs).

Original entry on oeis.org

31, 107, 391, 467, 34049, 67973, 176413

Offset: 1

Naohiro Nomoto, Sep 19 2000

The corresponding pairs of primorials are (3#, 4#), (4#, 5#), (5#, 6#), (5#, 6#), (7#, 9#), (8#, 9#), (8#, 10#). No other terms found up to 23#. - Michel Marcus, Feb 21 2016

a(8) > 3203982595205562774973. - Sean A. Irvine, May 26 2022

			31 = 30+1 = 21+10, where 30=30*1 and 210=21*10 are primorial numbers.

Sean A. Irvine, Java program (github)

Cf. A002110, A006862, A182987.

isprimo(n) = {if (n==1, return (1)); if (!issquarefree(n), return(0)); f = factor(n); #f~ == primepi(vecmax(f[,1]));}
isok(n) = {c = 0; for (na=1, n\2, if (isprimo(na*(n - na)), c++); if (c == 2, return(1)););} \\ Michel Marcus, Feb 20 2016

A237110 Maximum number of distinct prime factors of pairs of coprime g and h (g < h) adding to n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 4, 2, 4, 3, 3, 3, 3, 2, 4, 3, 3, 3, 4, 2, 4, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 3, 4, 3, 4, 3, 4, 2, 4, 3, 3, 3, 4, 3, 4, 4, 4, 3, 4, 3, 4, 4, 3, 4, 4, 3, 4, 3, 4, 3, 4, 3, 4, 4, 4, 3, 4

Offset: 3

Lei Zhou, Feb 06 2014

nonn

This sequence is defined for n >= 3.
The difference between this sequence and A237354 is that A237354 allows g and h have common factors while in this sequence g and h must be coprime.
The smallest n that makes a(n)=k gives the sequence A182987, Least a+b such that ab=A002110(n).
The largest n that makes a(n)=k forms a sequence starting with 6, 60, 420, 6930, 30030, which are Prime(2)#, 2*Prime(3)#, 2*Prime(4)#, 3*Prime(5)#, where p# denotes the product of prime numbers up to p.
The largest n that makes a(n)=5 is not found yet; it is greater than Prime(6)#.

			n=3, 3=1+2. 1 has no prime factors. 2 has one.  So a(3)=0+1=1;
n=5, 5=1+4=1+2^2, gives number of prime factors 0+1=1, and 5=2+3, gives 1+1=2.  So a(5)=2;
...
n=97, 97=1+96=1+2^5*3, gives number of distinct prime factors of g=1 and h=96 0+2=2.  Checking all pairs of g, h from 1, 96 through 47, 49 with GCD[g, h]=1, we find that for 97=42+55=2*3*7+5*11 we get 3+2=5 prime factors from g and h.  So a(97)=5.

Lei Zhou, Table of n, a(n) for n = 3..10000

Cf. A237354, A182987, A002110

Table[ct = 0; Do[h = n - g; If[GCD[g,h]==1,c=Length[FactorInteger[g]]+Length[FactorInteger[h]]; If[g == 1, c--]; If[h == 1, c--]; If[c > ct, ct = c]], {g, 1, Floor[n/2]}]; ct, {n, 3, 89}]

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A033676 Largest divisor of n <= sqrt(n).

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A060796 Upper central divisor of n-th primorial.

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A060795 Write product of first n primes as x*y with x

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A061060 Write product of first n primes as x*y with x

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A057230 Numbers k such that k = p+q = r+s with p*q = r*s = primorial number(A002110) (p*q) < (r*s).

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A237110 Maximum number of distinct prime factors of pairs of coprime g and h (g < h) adding to n.

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A057230 Numbers k such that k = p+q = r+s with pq = rs = primorial number(A002110) (pq) < (rs).