A300236 -id:A300236 - cp's OEIS frontend

A300239 Difference between A032742 (the largest proper divisor of n) and its Möbius transform (A300236).

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 5, 1, 6, 1, 1, 1, 8, 1, 1, 3, 8, 1, 11, 1, 8, 1, 1, 1, 12, 1, 1, 1, 12, 1, 15, 1, 12, 7, 1, 1, 16, 1, 9, 1, 14, 1, 15, 1, 16, 1, 1, 1, 22, 1, 1, 9, 16, 1, 23, 1, 18, 1, 17, 1, 24, 1, 1, 9, 20, 1, 27, 1, 24, 9, 1, 1, 30, 1, 1, 1, 24, 1, 29, 1, 24, 1, 1, 1, 32, 1, 13, 13, 30, 1, 35, 1, 28, 17

Offset: 1

Antti Karttunen, Mar 10 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A008683, A032742, A300236.

Table[n/FactorInteger[n][[1, 1]] - DivisorSum[n, # MoebiusMu[n/#]/FactorInteger[#][[1, 1]] &], {n, 105}] (* or *)
Fold[Function[{a, n}, Append[a, {Abs@ Total@ Map[MoebiusMu[n/#] a[[#, -1]] &, Most@ Divisors@ n], n/FactorInteger[n][[1, 1]]}]], {{0, 1}}, Range[2, 105]][[All, 1]] (* Michael De Vlieger, Mar 10 2018 *)

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A300239(n) = -sumdiv(n,d,(dA032742(d));

a(n) = A032742(n) - A300236(n).

a(n) = -Sum_{d|n, dA008683(n/d)*A032742(d).

A032742 a(1) = 1; for n > 1, a(n) = largest proper divisor of n (that is, for n>1, maximum divisor d of n in range 1 <= d < n).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 6, 1, 7, 5, 8, 1, 9, 1, 10, 7, 11, 1, 12, 5, 13, 9, 14, 1, 15, 1, 16, 11, 17, 7, 18, 1, 19, 13, 20, 1, 21, 1, 22, 15, 23, 1, 24, 7, 25, 17, 26, 1, 27, 11, 28, 19, 29, 1, 30, 1, 31, 21, 32, 13, 33, 1, 34, 23, 35, 1, 36, 1, 37, 25, 38, 11, 39, 1, 40

Offset: 1

Patrick De Geest, May 15 1998

It seems that a(n) = Max_{j=n+1..2n-1} gcd(n,j). - Labos Elemer, May 22 2002
This is correct: No integer in the range [n+1, 2n-1] has n as its divisor, but certainly at least one multiple of the largest proper divisor of n will occur there (e.g., if it is n/2, then at n + (n/2)). - Antti Karttunen, Dec 18 2014
The slopes of the visible lines made by the points in the scatter plot are 1/2, 1/3, 1/5, 1/7, ... (reciprocals of primes). - Moosa Nasir, Jun 19 2022

Rémi Eismann, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Moosa Nasir, Slopes.
Eric Weisstein's World of Mathematics, Proper Divisor.

Cf. A002110, A002808, A005867, A006530, A008578, A020639, A032741, A003961, A052126, A054576, A055396, A060681, A068319, A063928, A130064, A246277, A250245, A250246, A276085, A276086, A276151, A286477, A300236, A302025, A302026, A302032, A302042, A325563, A325567.

Maximal GCD of k positive integers with sum n for k = 2..10: this sequence (k=2,n>=2), A355249 (k=3), A355319 (k=4), A355366 (k=5), A355368 (k=6), A355402 (k=7), A354598 (k=8), A354599 (k=9), A354601 (k=10).

a032742 n = n `div` a020639 n  -- Reinhard Zumkeller, Oct 03 2012

A032742 :=proc(n) option remember; if n = 1 then 1; else numtheory[divisors](n) minus {n} ; max(op(%)) ; end if; end proc: # R. J. Mathar, Jun 13 2011
1, seq(n/min(numtheory:-factorset(n)), n=2..1000); # Robert Israel, Dec 18 2014

f[n_] := If[n == 1, 1, Divisors[n][[-2]]]; Table[f[n], {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Mar 03 2010 *)
Join[{1},Divisors[#][[-2]]&/@Range[2,80]] (* Harvey P. Dale, Nov 29 2011 *)
a[n_] := n/FactorInteger[n][[1, 1]]; Array[a, 100] (* Amiram Eldar, Nov 26 2020 *)
Table[Which[n==1,1,PrimeQ[n],1,True,Divisors[n][[-2]]],{n,80}] (* Harvey P. Dale, Feb 02 2022 *)

a(n)=if(n==1,1,n/factor(n)[1,1]) \\ Charles R Greathouse IV, Jun 15 2011

from sympy import factorint
def a(n): return 1 if n == 1 else n//min(factorint(n))
print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Jun 21 2022

(define (A032742 n) (/ n (A020639 n))) ;; Antti Karttunen, Dec 18 2014

a(n) = n / A020639(n).
Other identities and observations:
A054576(n) = a(a(n)); A117358(n) = a(a(a(n))) = a(A054576(n)); a(A008578(n)) = 1, a(A002808(n)) > 1. - Reinhard Zumkeller, Mar 10 2006
a(n) = A130064(n) / A006530(n). - Reinhard Zumkeller, May 05 2007
a(m)*a(n) < a(m*n) for m and n > 1. - Reinhard Zumkeller, Apr 11 2008
a(m*n) = max(m*a(n), n*a(m)). - Robert Israel, Dec 18 2014
From Antti Karttunen, Mar 31 2018: (Start)
a(n) = n - A060681(n).
For n > 1, a(n) = A003961^(r)(A246277(n)), where r = A055396(n)-1 and A003961^(r)(n) stands for shifting the prime factorization of n by r positions towards larger primes.
a(n) = A250246(A302042(A250245(n))) = A302026(A302032(A302025(n))).
For all n >= 1, A276085(a(A276086(n))) = A276151(n).
(End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Sum_{k>=1} A005867(k-1)/(prime(k)*A002110(k)) = 0.165049... . - Amiram Eldar, Nov 19 2022

Definition clarified by N. J. A. Sloane, Dec 26 2022

A305807 Dirichlet inverse of A032742 (the largest proper divisor of n).

Original entry on oeis.org

1, -1, -1, -1, -1, -1, -1, -1, -2, -3, -1, 1, -1, -5, -3, -1, -1, 0, -1, 1, -5, -9, -1, 3, -4, -11, -4, 1, -1, 5, -1, -1, -9, -15, -5, 6, -1, -17, -11, 5, -1, 7, -1, 1, -2, -21, -1, 5, -6, -8, -15, 1, -1, 4, -9, 7, -17, -27, -1, 19, -1, -29, -4, -1, -11, 11, -1, 1, -21, -3, -1, 8, -1, -35, -8, 1, -9, 13, -1, 9, -8, -39, -1, 29, -15, -41, -27, 11, -1

Offset: 1

Antti Karttunen, Jun 13 2018

sign

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A032742, A300236, A300239, A305808.

b[n_] := If[n == 1, 1, Divisors[n][[-2]]];
a[n_] := a[n] = If[n == 1, 1, -Sum[b[n/d] a[d], {d, Most@ Divisors[n]}]];
Array[a, 100] (* Jean-François Alcover, Feb 17 2020 *)

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A305807(n) = if(1==n,1,-sumdiv(n,d,if(dA032742(n/d)*A305807(d),0)));

a(1) = 1; for n > 1, a(n) = -Sum_{d|n, dA032742(n/d)*a(d).

A143112 A051731 * A032742 = sum of largest proper divisors of the divisors of n.

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 5, 8, 2, 14, 2, 10, 8, 16, 2, 18, 2, 20, 10, 14, 2, 30, 7, 16, 14, 26, 2, 32, 2, 32, 14, 20, 10, 44, 2, 22, 16, 44, 2, 42, 2, 38, 26, 26, 2, 62, 9, 38, 20, 44, 2, 54, 14, 58, 22, 32, 2, 80, 2, 34, 34, 64, 16, 62, 2, 56, 26, 58, 2, 96, 2, 40, 38, 62, 14, 72, 2, 92

Offset: 1

Gary W. Adamson and Mats Granvik, Jul 25 2008

nonn

Inverse Möbius transform of A032742. - Antti Karttunen, Sep 25 2018

			a(12) = 14. The divisors of 12 are shown in row 12 of triangle A127093: (1, 2, 3, 4, 0, 6, 0, 0, 0, 0, 0, 12). The largest proper divisors of these terms are (1, 1, 1, 2, 0, 3, 0, 0, 0, 0, 0, 6), sum = 14. Or, we can take row of triangle A051731: (1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1) dot (1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 6) = (1 + 1 + 1 + 2 + 0 + 3 + 0 + 0 + 0 + 0 + 0 + 6) = 14, where A032742 = (1, 1, 1, 2, 1, 3, 1, 4, 3, 5,...).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A051731, A127093, A032742, A300236, A305807, A305808.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A143112(n) = sumdiv(n,d,A032742(d)); \\ Antti Karttunen, Sep 25 2018

A051731 * A032742, where A051731 = the inverse Mobius transform and A032742 = the largest proper divisors of n: (1, 1, 1, 3, 1, 3, 1, 4, 3, 5, 1, 6, 1, 7,...).

a(n) = Sum_{d|n} A032742(d). - Antti Karttunen, Sep 25 2018

More terms from R. J. Mathar, Jan 19 2009

A300721 Möbius transform of A060681, the largest difference between consecutive divisors of n (ordered by size).

Original entry on oeis.org

0, 1, 2, 1, 4, 0, 6, 2, 4, 0, 10, 2, 12, 0, 4, 4, 16, 2, 18, 4, 6, 0, 22, 4, 16, 0, 12, 6, 28, 4, 30, 8, 10, 0, 18, 6, 36, 0, 12, 8, 40, 6, 42, 10, 16, 0, 46, 8, 36, 4, 16, 12, 52, 6, 30, 12, 18, 0, 58, 8, 60, 0, 24, 16, 36, 10, 66, 16, 22, 6, 70, 12, 72, 0, 24, 18, 50, 12, 78, 16, 36, 0, 82, 12, 48, 0, 28, 20, 88, 8, 60, 22, 30, 0, 54, 16

Offset: 1

Antti Karttunen, Mar 11 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A008683, A060681, A300236, A300722, A322873 (ordinal transform).

A060681[n_] := n - n/FactorInteger[n][[1, 1]];
a[n_] := Sum[MoebiusMu[n/d]*A060681[d], {d, Divisors[n]}];
Array[a, 100] (* Jean-François Alcover, Jan 07 2022 *)

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A060681(n) = (n-A032742(n));
A300721(n) = sumdiv(n,d,moebius(n/d)*A060681(d));

a(n) = Sum_{d|n} A008683(n/d)*A060681(d).
a(n) = A060681(n) - A300722(n).
a(n) = A000010(n) - A300236(n).

A305808 Dirichlet convolution of A032742 (the largest proper divisor of n) with itself.

Original entry on oeis.org

1, 2, 2, 5, 2, 8, 2, 12, 7, 12, 2, 22, 2, 16, 12, 28, 2, 30, 2, 34, 16, 24, 2, 56, 11, 28, 24, 46, 2, 56, 2, 64, 24, 36, 16, 87, 2, 40, 28, 88, 2, 76, 2, 70, 46, 48, 2, 136, 15, 70, 36, 82, 2, 108, 24, 120, 40, 60, 2, 172, 2, 64, 62, 144, 28, 116, 2, 106, 48, 108, 2, 228, 2, 76, 70, 118, 24, 136, 2, 216, 81, 84, 2, 236, 36, 88, 60, 184, 2, 228, 28

Offset: 1

Antti Karttunen, Jun 13 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A032742, A300236, A300239, A305807.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A305808(n) = sumdiv(n,d,A032742(n/d)*A032742(d));

a(n) = Sum_{d|n} A032742(n/d)*A032742(d).

cp's OEIS Frontend

A300239 Difference between A032742 (the largest proper divisor of n) and its Möbius transform (A300236).

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A032742 a(1) = 1; for n > 1, a(n) = largest proper divisor of n (that is, for n>1, maximum divisor d of n in range 1 <= d < n).

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A305807 Dirichlet inverse of A032742 (the largest proper divisor of n).

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A143112 A051731 * A032742 = sum of largest proper divisors of the divisors of n.

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A300721 Möbius transform of A060681, the largest difference between consecutive divisors of n (ordered by size).

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A305808 Dirichlet convolution of A032742 (the largest proper divisor of n) with itself.

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