A063428 -id:A063428 - cp's OEIS frontend

A066243 Superseded by A063428.

2, 2, 2, 4, 2, 6, 4, 6, 5, 10, 3, 12, 7, 6, 8, 16, 9, 18, 4, 12, 11, 22, 6, 20, 13, 18, 12, 28, 5

Offset: 2

dead

A063647 Number of ways to write 1/n as a difference of exactly 2 unit fractions.

0, 1, 1, 2, 1, 4, 1, 3, 2, 4, 1, 7, 1, 4, 4, 4, 1, 7, 1, 7, 4, 4, 1, 10, 2, 4, 3, 7, 1, 13, 1, 5, 4, 4, 4, 12, 1, 4, 4, 10, 1, 13, 1, 7, 7, 4, 1, 13, 2, 7, 4, 7, 1, 10, 4, 10, 4, 4, 1, 22, 1, 4, 7, 6, 4, 13, 1, 7, 4, 13, 1, 17, 1, 4, 7, 7, 4, 13, 1, 13, 4, 4, 1, 22, 4, 4, 4, 10, 1, 22, 4, 7, 4, 4, 4

Offset: 1

Henry Bottomley, Jul 23 2001

Also number of ways to write 1/n as sum of exactly two distinct unit fractions. - Thomas L. York, Jan 11 2014
Also number of positive integers m such that 1/n + 1/m is a unit fraction. - Jon E. Schoenfield, Apr 17 2018
If 1/n = 1/b - 1/c then n = bc/(c-b) and 1/n = 1/(2n-b) + 1/(c+2n) (though it is also the case that 1/n = 1/(2n) + 1/(2n) equivalent to b = c = 0).
Also number of divisors of n^2 less than n. - Vladeta Jovovic, Aug 13 2001
Number of elements in the set {(x,y): x|n, y|n, xVladeta Jovovic, May 03 2002
Also number of positive integers of the form k*n/(k+n). - Benoit Cloitre, Jan 04 2002
This is similar to A062799, having the same first 29 terms. But they are different sequences.
If A001221(n) = omega(n) <= 2, then a(n) = A062799(n); if A001221(n) > 2, then a(n) > A062799(n). - Matthew Vandermast, Aug 25 2004
Number of r X s integer-sided rectangles such that r + s = 4n, r < s and (s - r) | (s * r). - Wesley Ivan Hurt, Apr 24 2020
Also number of integer-sided right triangles with 2n as a leg. Equivalent to the even indices of A046079. - Nathaniel C Beckman, May 14 2020; Jun 26 2020
a(n) is the number of positive integers k such that k+n divides k*n. - Thomas Ordowski, Dec 02 2024

			a(10) = 4 since 1/10 = 1/5 - 1/10 = 1/6 - 1/15 = 1/8 - 1/40 = 1/9 - 1/90.
a(12) = 7: the divisors of 12 are 1, 2, 3, 4, 6 and 12 and the decompositions are (1, 2), (1, 3), (1, 4), (1, 6), (1, 12), (2, 3), (3, 4).

T. D. Noe, Table of n, a(n) for n = 1..10000
Christopher J. Bradley, Solution to Problem 2175, Crux Mathematicorum, Vol. 23, No. 7, (Nov 1997), pp. 443-444.
Umberto Cerruti, Percorsi tra i numeri (in Italian), pages 3-4.
Roger B. Eggleton, Unitary Fractions: 10501, The American Mathematical Monthly, Vol. 105, No. 4 (Apr., 1998), p. 372.
Amiram Eldar, Plot of (Sum_{i=1..n} a(i))/f(n) - 1, where f(n) is an asymptotic function, for n = 2^(1..28).
Amiram Eldar, Plot of Sum_{i=1..n} a(i)/(n*log(n)*log(log(n))) for n = 2^(1..28).
Amarnath Murthy, Decomposition of the divisors of a natural number into pairwise coprime sets, Smarandache Notions Journal, Vol. 12, No. 1-2-3, Spring 2001, pp. 303-306.

Cf. A018892, A063427, A063428, A129296.
First twenty-nine terms identical to those of A062799.
Cf. A001221, A046079, A048691, A063717, A063718.
Cf. A001620, A082633, A013661, A201994, A306016.

[(NumberOfDivisors(n^2)-1)/2 : n in [1..100]]; // Vincenzo Librandi, Apr 18 2018

Table[(Length[Divisors[n^2]] - 1)/2, {n, 1, 100}]
(DivisorSigma[0,Range[100]^2]-1)/2 (* Harvey P. Dale, Apr 15 2013 *)

for(n=1,100,print1(sum(i=1,n^2,if((n*i)%(i+n),0,1)),","))

a(n)=numdiv(n^2)\2 \\ Charles R Greathouse IV, Oct 03 2016

a(n) = (tau(n^2)-1)/2.
a(n) = A018892(n)-1. If n = (p1^a1)(p2^a2)...(pt^at), a(n) = ((2*a1+1)(2*a2+1)...(2*at+1)-1)/2.
If n is prime a(n)=1. Conjecture: (1/n)*Sum_{i=1..n} a(i) = C*log(n)*log(log(n)) + o(log(n)) with C=0.7... [The conjecture is false. See the plot and the asymptotic formula below. - Amiram Eldar, Oct 03 2024]
Bisection of A046079. - Lekraj Beedassy, Jul 09 2004
a(n) = Sum_{i=1..2*n-1} (1 - ceiling(i*(4*n-i)/(4*n-2*i)) + floor(i*(4*n-i)/(4*n-2*i))). - Wesley Ivan Hurt, Apr 24 2020
Sum_{k=1..n} a(k) ~ (n/(2*zeta(2)))*(log(n)^2/2 + log(n)*(3*gamma - 1) + 1 - 3*gamma + 3*gamma^2 - 3*gamma_1 - zeta(2) + (2 - 6*gamma - 2*log(n))*zeta'(2)/zeta(2) + (2*zeta'(2)/zeta(2))^2 - 2*zeta''(2)/zeta(2)), where gamma is Euler's constant (A001620) and gamma_1 is the first Stieltjes constant (A082633). - Amiram Eldar, Oct 03 2024

A063427 a(n) is the smallest positive integer k such that n*k/(n+k) is an integer.

Original entry on oeis.org

2, 6, 4, 20, 3, 42, 8, 18, 10, 110, 4, 156, 14, 10, 16, 272, 9, 342, 5, 28, 22, 506, 8, 100, 26, 54, 21, 812, 6, 930, 32, 66, 34, 14, 12, 1332, 38, 78, 10, 1640, 7, 1806, 44, 30, 46, 2162, 16, 294, 50, 102, 52, 2756, 27, 66, 8, 114, 58, 3422, 12, 3660, 62, 18, 64, 104

Offset: 2

Henry Bottomley, Jul 19 2001

nonn

This produces the smallest positive integer value for n*k/(n+k).
Equivalently, smallest c such that 1/n + 1/c = 1/b has integer solutions.
Largest c is 1/(n(n-1)) since 1/n + 1/(n(n-1)) = 1/(n-1).
Let L(x,y)=x+y be the "basic" linear form. Let Q(x,y) = x^2 + x*y + y^2 be the "basic" quadratic form. Let C(x,y) = x^3 + y^3 + x^2*y + x*y^2 + x*y + x^2 + y^2 + x + y be the "basic" cubic form. Then a(n) = min(x/Q(x,n)=0 mod L(x,n)) = min(x/C(x,n) = 0 mod L(x,n)). - Benoit Cloitre, Jan 02 2002
For p=prime, a(p^k) = p^k*(p-1). - Leroy Quet, Jan 25 2007
a(n) = n*(d(i)-d(i-1))/d(i-1), where d(i) is the i-th divisor of n that minimizes (d(i)-d(i-1))/d(i-1) with i>=2. In general, let f(n) be an integer function, then n*f(n)/(n+f(n))=c, c positive integer, has a solution only if f(n) >= n*(d(i)-d(i-1))/d(i-1). - Ctibor O. Zizka, Sep 17 2015

			a(6) = 3 because 6*3/(6+3)=2 is the smallest integer of the form 6*k/(6+k).
a(10) = 10 since 1/10 + 1/10 = 1/5, 1/10 + 1/15 = 1/6, 1/10 + 1/40 = 1/8, 1/10 + 1/90 = 1/9 and so the first sum provides the value.

Harry J. Smith, Table of n, a(n) for n = 2..1000

Cf. A063428, A127730.

Table[k=1;While[!IntegerQ[(k n)/(k+n)],k++];k,{n,2,70}] (* Harvey P. Dale, Jun 24 2011 *)

a(n) = { my(k=1); while (n*k%(n + k), k++); k } \\ Harry J. Smith, Aug 20 2009

a(n) = n*A063428(n)/(n-A063428(n)).

New description from Benoit Cloitre, Dec 30 2001
Entry revised by N. J. A. Sloane, Feb 13 2007
Definition revised by Franklin T. Adams-Watters, Aug 07 2009

A339144 a(n) is the smallest positive integer such that n*a(n) contains n+a(n) as a substring. If no such number exists then a(n) = -1.

Original entry on oeis.org

-1, 2, -1, 68, -1, -1, -1, 44, -1, 890, 110, 60, 44, 35, 30, 27, 25, 23, 22, 20, 929, 19, 18, 88, 17, -1, 16, 16, 68, 15, 15, 60, 58, 56, 14, 14, 14, 371, 48, 360, 336, 562, 9104, 8, 13, 13, 283, 39, 269, 450, 37, 452, 245, 18, 679, 34, 225, 33, 2053, 12, 12, 12, 12, 12, 30, 369, 889, 4, 16961

Offset: 1

Scott R. Shannon, Nov 25 2020

For n = 1, 3, 5, 6, 7, 9, 26 no value has been found for which n*a(n) contains n + a(n) as a substring (obviously true for n = 1) for a(n) up to 5x10^10. It is likely, although unproven, that this is the complete list of values for which a(n) = -1.

The sequence values display erratic behavior. Most of the term values appear random but there are ranges of n values with the same value. The largest such range for the first one million terms is a(501000) to a(501499), 500 terms, all of which have a(n) = 1002. Patterns also appear for n value corresponding to multiples of powers-of-ten. For example if n=10^k then a(n) = 89*10^k. The largest value in the first one million terms is a(554635) = 879948670.

			a(2) = 2 as 2*2 = 4 which contains 2 + 2 = 4 as a substring.
a(4) = 68 as 4*68 = 272 which contains 4+68 = 72 as a substring.
a(69) = 16961 as 69*16961 = 1170309 which contains 69+16961 = 17030 as a substring.
a(501000) = 1002 as 501000*1002 = 502002000 which contains 501000+1002 = 502002 as a substring. This is the first of 500 consecutive terms with a(n) = 1002.
a(554635) = 879948670 as 554635*879948670 = 488050330585450 which contain 554635+879948670 = 880503305 as a substring. This is the largest value of a(n) for the first one million terms.

Scott R. Shannon, Table of n, a(n) for n = 1..20000.

Cf. A339107, A063427, A063428, A328095, A333923, A332580.

isok(n, k) = #strsplit(Str(n*k), Str(n+k)) > 1;
a(n) = {if (vecsearch([1, 3, 5, 6, 7, 9, 26], n), return (-1)); my(k=1); while (! isok(k, n), k++); k;} \\ Michel Marcus, Dec 02 2020 and Jan 23 2021

A191973 Irregular triangle read by rows: row n consists of n and the positive integers m where m-n divides m*n.

Original entry on oeis.org

1, 2, 1, 2, 3, 4, 6, 2, 3, 4, 6, 12, 2, 3, 4, 5, 6, 8, 12, 20, 4, 5, 6, 10, 30, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 18, 24, 42, 6, 7, 8, 14, 56, 4, 6, 7, 8, 9, 10, 12, 16, 24, 40, 72, 6, 8, 9, 10, 12, 18, 36, 90, 5, 6, 8, 9, 10, 11, 12, 14, 15, 20, 30, 35, 60

Offset: 1

Nathaniel Johnston, Jun 22 2011

The maximum term of the n-th row is n*(n+1). The minimum term of the n-th row seems to be A063428(n) if n>=2. The length of row n is A146564(n) + 1.

			The triangle begins:
1 2
1 2 3 4  6
2 3 4 6  12
2 3 4 5  6  8  12 20
4 5 6 10 30
2 3 4 5  6  7  8  9  10 12 15 18 24 42
6 7 8 14 56
...

Nathaniel Johnston, Rows 1..250, flattened

Cf. A162821 (row 30), A162822 (row 36), A162823 (row 42), A162824 (row 48), A162825 (row 60), A127730.

for n from 1 to 10 do for m from 1 to n*(n+1) do if(n=m or m*n mod (m-n) = 0)then printf("%d, ",m): fi: od: od:

A339403 a(n) is the smallest positive integer such that n*a(n) contains the string n+a(n) in reverse as a substring. If no such number exists then a(n) = -1.

Original entry on oeis.org

0, -1, 2, 24, 37, 26, 34, 35, 57, 9, -1, 12, 11, 45, 193, 228, 28, 51, 23, 44, 841, 11, 27, 18, 3, 626, 5, 22, 16, 46716, 56, 41, 33, 32, 6, 7, 21, 4, 3, 24, 592, 31, 7, 619, 19, 13, 38, 2, 117, 5, 463, 17, 34, 308, 33, 36, 30, 8, 31, 4, 23, 21, 648, 124, 921, 903, 386, 395, 4, 334, 755, 31, 563

Offset: 0

Scott R. Shannon, Dec 03 2020

This is a variation of A339144 where, instead of the n*a(n) containing n+a(n) as a substring, it contains the reverse of the string n+a(n), including any leading zeros.
Based on a search limit of 5x10^9 up to n = 100000 the values of n for which no a(n) is found are n = 10^k, with k>=0, and 17500. A test of 175000 and 1750000 also found no a(n) indicating that all values of the form 17500*10^k may have no term for a(n).
It is found that when n = 200*10^k, with k>=0, the corresponding value for a(n) is significantly larger than neighboring terms. As an example a(20000) = 666843331, which is the largest term up to n = 100000.
Unlike A339144, which contains multiple consecutive terms with the same value of a(n), in this sequence the largest consecutive run of the same a(n) in the first 100000 terms is only two. The first term of these pairs occurs at n = 110, 121, 2717, 4368, 7916, 10100, 11211, 13231, 17271, 44573, 63529.

			a(3) = 24 as 3*24 = 72 which contains reverse(3+24) = reverse(27) = 72 as a substring.
a(6) = 34 as 6*34 = 204 which contains reverse(6+34) = reverse(40) = 04 as a substring. Note the leading zero is included.
a(29) = 46716 as 29*46716 = 1354764 which contains reverse(29+4671) = reverse(46745) = 54764 as a substring.
a(110) = 11 as 110*11 = 1210 which contains reverse(110+11) = reverse(121) = 121 as a substring. This is the first of two consecutive terms with a(n) = 11.
a(20000) = 666843331 as 20000*666843331 = 13336866620000 which contains reverse(20000+666843331) = reverse(666863331) = 133368666 as a substring. This is the largest value in the first 100000 terms.

Scott R. Shannon, Table of n, a(n) for n = 0..10000

Cf. A339144, A339107, A063427, A063428, A328095, A333923, A332580.

isok(n, k) = #strsplit(Str(n*k), concat(Vecrev(Str(n+k)))) > 1;
ispt(n) = my(t); ispower(n,,&t) && (t==10);
a(n) = {if ((n==1) || (n==10) || ispt(n), return (-1)); my(k=0); while (! isok(n, k), k++); k;} \\ Michel Marcus, Jan 22 2021

A063649 Largest b such that 1/n=1/c+1/b has integer solutions with c>b.

Original entry on oeis.org

3, 4, 6, 6, 10, 8, 12, 12, 15, 12, 21, 14, 21, 24, 24, 18, 30, 20, 36, 30, 33, 24, 42, 30, 39, 36, 44, 30, 55, 32, 48, 44, 51, 60, 63, 38, 57, 52, 72, 42, 78, 44, 66, 72, 69, 48, 84, 56, 75, 68, 78, 54, 90, 80, 105, 76, 87, 60, 110, 62, 93, 112, 96, 90, 110, 68, 102, 92, 120

Offset: 2

Henry Bottomley, Jul 23 2001

Smallest b is (n+1) since 1/n = 1/(n(n+1))+1/(n+1).

			a(10)=15 since 1/10=1/20+1/20=1/30+1/15=1/35+1/14=1/60+1/12=1/110+1/11, but the first sum does not have c>b, leaving the second sum to provide the value.

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A018892, A063427, A063428, A063647, A063648.

f:= proc(n) local b;
  for b from 2*n-1 by -1 do
     if n*b mod (b-n) = 0 then return b fi
od
end proc:
map(f, [$2..100]); # Robert Israel, Dec 01 2019

a[n_] := n + SelectFirst[Divisors[n^2] // Reverse, #Jean-François Alcover, Jun 07 2020 *)

a(n) = n*A063648(n)/(A063648(n)-n) = 2n-A063428(n).
From Robert Israel, Dec 01 2019: (Start)
a(n) = n + A063717(n).
a(n) = n + 1 if and only if n is prime. (End)

A063717 a(n) is the greatest divisor of n^2 that is less than n.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 4, 3, 5, 1, 9, 1, 7, 9, 8, 1, 12, 1, 16, 9, 11, 1, 18, 5, 13, 9, 16, 1, 25, 1, 16, 11, 17, 25, 27, 1, 19, 13, 32, 1, 36, 1, 22, 27, 23, 1, 36, 7, 25, 17, 26, 1, 36, 25, 49, 19, 29, 1, 50, 1, 31, 49, 32, 25, 44, 1, 34, 23, 50, 1, 64, 1, 37, 45, 38, 49, 52, 1, 64, 27, 41

Offset: 2

Vladeta Jovovic, Aug 12 2001

nonn

Smaller of two distinct numbers with minimum sum whose geometric mean is n. E.g., a(12) = 9 as 12^2 = 144 = 1*144 = 2*72 = 3*48 = 4*36 = 6*24 = 8*18 = 9*16, etc. - Amarnath Murthy, Feb 15 2003

			a(45)=27 because set of divisors of 45^2 is {1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 135, 225, 405, 675, 2025} and the greatest element of the set less than 45 is 27.

Harry J. Smith, Table of n, a(n) for n = 2..1000

A063649(n) = n + a(n), A063718(n) = n^2/A063717(n), A063428(n) = n - a(n).

Cf. A063718.

with(numtheory): for n from 2 to 200 do a := divisors(n^2): b := a[(tau(n^2)-1)/2]: printf(`%d,`,b); od:

f[n_] := Module[{dn2 = Divisors[n^2]}, Last[Take[dn2, {1, Flatten[Position[dn2, n]][[ 1]] - 1}]]]; Table[f[i], {i, 2, 85}]
Table[Select[Divisors[n^2],#Harvey P. Dale, Apr 23 2016 *)

{ for (n=2, 1000, d=divisors(n^2); write("b063717.txt", n, " ", d[length(d)\2]) ) } \\ Harry J. Smith, Aug 28 2009

A063648 Smallest c such that 1/n=1/c+1/b has integer solutions with c>b.

Original entry on oeis.org

6, 12, 12, 30, 15, 56, 24, 36, 30, 132, 28, 182, 42, 40, 48, 306, 45, 380, 45, 70, 66, 552, 56, 150, 78, 108, 77, 870, 66, 992, 96, 132, 102, 84, 84, 1406, 114, 156, 90, 1722, 91, 1892, 132, 120, 138, 2256, 112, 392, 150, 204, 156, 2862, 135, 176, 120, 228, 174

Offset: 2

Henry Bottomley, Jul 23 2001

nonn

Largest c is n(n+1) since 1/n=1/(n(n+1))+1/(n+1)

			a(10)=30 since 1/10=1/20+1/20=1/30+1/15=1/35+1/14=1/60+1/12=1/110+1/11, but the first sum does not have c>b, leaving the second sum to provide the value.

Cf. A018892, A063427, A063428, A063647, A063649.

a(n) =n*A063649(n)/(A063649(n)-n) =A063427(n)+2n.

A127731 Triangle read by rows, where row n consists of the r's where r = (nm)/(n+m) and the m's are positive integers such that (n+m) divides (nm).

Original entry on oeis.org

1, 2, 2, 3, 4, 2, 3, 4, 5, 6, 4, 6, 7, 6, 8, 5, 6, 8, 9, 10, 3, 4, 6, 8, 9, 10, 11, 12, 7, 10, 12, 13, 6, 10, 12, 14, 8, 12, 14, 15, 16, 6, 9, 12, 14, 15, 16, 17, 18, 4, 10, 12, 15, 16, 18, 19, 12, 14, 18, 20, 11, 18, 20, 21, 22, 6, 8, 12, 15, 16, 18, 20, 21, 22, 23, 20, 24, 13, 22, 24, 25

Offset: 2

Leroy Quet, Jan 26 2007

The maximum term of the n-th row, for n >= 2, is (n-1). The minimum term of row n is A063428(n), for n >= 3. Row n contains A063647(n) terms (according to a comment by Benoit Cloitre). For p prime, row p^k has k terms. (Each term in row p^k is of the form p^(k-j)*(p^j -1), 1<=j<=k.)

			Row 6 is (2,3,4,5) because row 6 of irregular array A127730 is (3,6,12,30); and (6*3)/(6+3) = 2, (6*6)/(6+6) = 3, (6*12)/(6+12) = 4 and (6*30)/(6+30) = 5.

Cf. A063428, A063647, A127730.

f[n_] := Select[Table[n*m/(n + m), {m, n^2}], IntegerQ];Table[f[n], {n, 2, 26}] // Flatten (* Ray Chandler, Feb 13 2007 *)

Extended by Ray Chandler, Feb 13 2007

cp's OEIS Frontend

A066243 Superseded by A063428.

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A063647 Number of ways to write 1/n as a difference of exactly 2 unit fractions.

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A063427 a(n) is the smallest positive integer k such that n*k/(n+k) is an integer.

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A339144 a(n) is the smallest positive integer such that n*a(n) contains n+a(n) as a substring. If no such number exists then a(n) = -1.

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A191973 Irregular triangle read by rows: row n consists of n and the positive integers m where m-n divides m*n.

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A339403 a(n) is the smallest positive integer such that n*a(n) contains the string n+a(n) in reverse as a substring. If no such number exists then a(n) = -1.

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A063649 Largest b such that 1/n=1/c+1/b has integer solutions with c>b.

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A063717 a(n) is the greatest divisor of n^2 that is less than n.

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A127731 Triangle read by rows, where row n consists of the r's where r = (nm)/(n+m) and the m's are positive integers such that (n+m) divides (nm).