A191973 -id:A191973 - cp's OEIS frontend

A127730 Triangle read by rows: row n consists of the positive integers m where m+n divides m*n.

2, 6, 4, 12, 20, 3, 6, 12, 30, 42, 8, 24, 56, 18, 72, 10, 15, 40, 90, 110, 4, 6, 12, 24, 36, 60, 132, 156, 14, 35, 84, 182, 10, 30, 60, 210, 16, 48, 112, 240, 272, 9, 18, 36, 63, 90, 144, 306, 342, 5, 20, 30, 60, 80, 180, 380, 28, 42, 126, 420, 22, 99, 220, 462

Offset: 2

Leroy Quet, Jan 26 2007

The maximum term of the n-th row, for n >= 2, is n*(n-1). The minimum term of row n is A063427(n). 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*(p^j -1), 1 <= j <= k.)

			Row 6 is (3,6,12,30) because 6+3 = 9 divides 6*3 = 18, 6+6 = 12 divides 6*6 = 36, 6+12 = 18 divides 6*12 = 72 and 6+30 = 36 divides 6*30 = 180.

Nathaniel Johnston, Rows n = 2..500, flattened

Cf. A063427, A063647, A127731, A191973.

for n from 2 to 20 do for m from 1 to n*(n-1) do if(m*n mod (m+n) = 0)then printf("%d, ",m): fi: od: od: # Nathaniel Johnston, Jun 22 2011

f[n_] := Select[Range[n^2], Mod[n*#, n + # ] == 0 &];Table[f[n], {n, 2, 24}] // Flatten (* Ray Chandler, Feb 13 2007 *)

arow(n)=local(d,m);d=divisors(n^2);vector(#d\2,k,m=d[ #d\2-k+1];n*(n-m)/m) \\ Franklin T. Adams-Watters, Aug 07 2009

Let d_n be the sequence of divisors of n^2 that are less than n, in reverse order. Then T(n,k) = n*(n-d_n(k))/d_n(k). - Franklin T. Adams-Watters, Aug 07 2009

Extended by Ray Chandler, Feb 13 2007

A146564 a(n) is the number of solutions of the equation k*n/(k-n) = c. k,c integers.

Original entry on oeis.org

1, 4, 4, 7, 4, 13, 4, 10, 7, 13, 4, 22, 4, 13, 13, 13, 4, 22, 4, 22, 13, 13, 4, 31, 7, 13, 10, 22, 4, 40, 4, 16, 13, 13, 13, 37, 4, 13, 13, 31, 4, 40, 4, 22, 22, 13, 4, 40, 7, 22, 13, 22, 4, 31, 13, 31, 13, 13, 4, 67, 4, 13, 22, 19, 13, 40, 4, 22, 13, 40, 4, 52

Offset: 1

Ctibor O. Zizka, Nov 01 2008

In general, if n is a prime p then a(p)=4, and k is from {p-1, p+1, 2*p, p^2+p}.
In general, if n is a squared prime p^2 then a(p^2)=7, and k is from {p^2-p, p^2-1, p^2+1, p^2+p, p^3-p^2, p^3+p^2, p^4+p^2}.
The sequence counts solutions with k>0 and any sign of c, or, alternatively, solutions with c>0 and any sign of k. If solutions were constrained to k>0 and c>0, A048691 would result. - R. J. Mathar, Nov 21 2008

			For n=7 we search the number of integer solutions of the equation 7*k/(k-7). This holds for k from {6,8,14,56}. Then a(7)=4. For n=10 we search the number of integer solutions of the equation 10*k/(k-10). This holds for k from {5,6,8,9,11,12,14,15,20,30,35,60,110}. Then a(10)=13.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Umberto Cerruti, Percorsi tra i numeri (in Italian), pages 2-4.

Cf. A191973.

[# [k:k in {1..n^2+n} diff {n}| IsIntegral(k*n/(k-n))]:n in [1..75]]; // Marius A. Burtea, Oct 18 2019

A146564 := proc(n) local b,d,k,c ; b := numtheory[divisors](n^2) ; kbag := {} ; for d in b do k := d+n ; if k > 0 then kbag := kbag union {k} ; fi ; k := -d+n ; if k > 0 then kbag := kbag union {k} ; fi; end do; RETURN(nops(kbag)) ; end: for n from 1 to 800 do printf("%d,",A146564(n)) ; od: # R. J. Mathar, Nov 21 2008

psi[n_] := Module[{pp, ee}, {pp, ee} = Transpose[FactorInteger[n]]; If[Max[pp] == 3, n, Times@@(pp+1) * Times@@(pp^(ee-1))]];
a[n_] := Sum[psi[2^PrimeNu[d]], {d, Divisors[n]}]-1;
a /@ Range[72] (* Jean-François Alcover, Jan 18 2020 *)

jordantot(n,k)=sumdiv(n,d,d^k*moebius(n/d));
dedekindpsi(n)=jordantot(n,2)/eulerphi(n);
A146564(n)=sumdiv(n, d, dedekindpsi(2^omega(d)));
for(n=1, 200, print(n" "A146564(n))) \\ Enrique Pérez Herrero, Apr 14 2012

Conjecture: a(n) = A048691(n)+A063647(n). - R. J. Mathar, Nov 21 2008 (See Corollary 4 in Cerruti's paper.)

a(n) = Sum_{d|n} psi(2^omega(d)), where psi is A001615 and omega is A001221. - Enrique Pérez Herrero, Apr 13 2012

Extended beyond a(11) by R. J. Mathar, Nov 21 2008

A162821 Positive numbers k such that 30*k/(30-k) are integers.

Original entry on oeis.org

5, 10, 12, 15, 18, 20, 21, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 39, 40, 42, 45, 48, 50, 55, 60, 66, 75, 80, 90, 105, 120, 130, 180, 210, 255, 330, 480, 930

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 13 2009

The number k=30 is explicitly included, treating the result of division through zero as an integer.

Row 30 of A191973.

Cf. A162688, A162689, A162690, A162691, A162692, A162693, A162694, A162817, A162818, A162819, A162820, A191973.

for m from 1 to 930 do if(m=30 or m*30 mod (m-30) = 0)then printf("%d, ", m): fi: od: # Nathaniel Johnston, Jun 22 2011

f[a_,b_]:=(a*b)/(a-b); a=30;lst={};Do[If[f[a,n]==IntegerPart[f[a,n]], AppendTo[lst,n]],{n,9!}];lst

Keywords fini,full added by R. J. Mathar, Jul 31 2009

A162822 Positive numbers k such that 36*k/(36-k) are integers.

Original entry on oeis.org

9, 12, 18, 20, 24, 27, 28, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 44, 45, 48, 52, 54, 60, 63, 72, 84, 90, 108, 117, 144, 180, 198, 252, 360, 468, 684, 1332

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 13 2009

The number k=36 is explicitly included, treating the result of division through zero as an integer.

Row 36 of A191973.

Cf. A162688, A162689, A162690, A162691, A162692, A162693, A162694, A162817, A162818, A162819, A162820, A162821, A191973.

for m from 1 to 1332 do if(m=36 or m*36 mod (m-36) = 0)then printf("%d, ", m): fi: od: # Nathaniel Johnston, Jun 22 2011

f[a_,b_]:=(a*b)/(a-b); a=36;lst={};Do[If[f[a,n]==IntegerPart[f[a,n]], AppendTo[lst,n]],{n,9!}];lst
Sort[Join[{36},Select[Range[1500],IntegerQ[(36#)/(36-#)]&]]]  (* Harvey P. Dale, Mar 23 2011 *)

Keywords fini,full added by R. J. Mathar, Jul 31 2009

A162823 Positive numbers k such that 42*k/(42-k) are integers.

Original entry on oeis.org

6, 14, 21, 24, 28, 30, 33, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 51, 54, 56, 60, 63, 70, 78, 84, 91, 105, 126, 140, 168, 189, 238, 294, 336, 483, 630, 924, 1806

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 13 2009

The number k=42 is explicitly included, treating the result of division through zero as an integer.

Row 42 of A191973.

Cf. A162688, A162689, A162690, A162691, A162692, A162693, A162694, A162817, A162818, A162819, A162820, A162821, A162822, A191973.

for m from 1 to 1806 do if(m=42 or m*42 mod (m-42) = 0)then printf("%d, ", m): fi: od: # Nathaniel Johnston, Jun 22 2011

f[a_,b_]:=(a*b)/(a-b); a=42;lst={};Do[If[f[a,n]==IntegerPart[f[a,n]], AppendTo[lst,n]],{n,9!}];lst
Join[{42},Select[Range[2000],IntegerQ[(42#)/(42-#)]&]]//Quiet//Sort (* Harvey P. Dale, Mar 14 2020 *)

Keywords fini,full added by R. J. Mathar, Jul 31 2009

A162824 Positive numbers k such that 48*k/(48-k) are integers.

Original entry on oeis.org

12, 16, 24, 30, 32, 36, 39, 40, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 56, 57, 60, 64, 66, 72, 80, 84, 96, 112, 120, 144, 176, 192, 240, 304, 336, 432, 624, 816, 1200, 2352

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 13 2009

The number k=48 is explicitly included.

Row 48 of A191973.

Cf. A162688, A162689, A162690, A162691, A162692, A162693, A162694, A162817, A162818, A162819, A162820, A162821, A162822, A162823, A191973.

for m from 1 to 2352 do if(m=48 or m*48 mod (m-48) = 0)then printf("%d, ", m): fi: od: # Nathaniel Johnston, Jun 22 2011

f[a_,b_]:=(a*b)/(a-b); a=48;lst={};Do[If[f[a,n]==IntegerPart[f[a,n]], AppendTo[lst,n]],{n,9!}];lst

Keywords fini,full added by R. J. Mathar, Jul 31 2009

A162825 Positive numbers k such that 60*k/(60-k) are integers.

Original entry on oeis.org

10, 12, 15, 20, 24, 30, 35, 36, 40, 42, 44, 45, 48, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 72, 75, 76, 78, 80, 84, 85, 90, 96, 100, 105, 108, 110, 120, 132, 135, 140, 150, 160, 180, 204, 210, 240, 260, 285, 300, 360, 420

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 13 2009

The number k=60 is explicitly included. The last entry in the sequence is a(68) = 3660.

Row 60 of A191973.

Nathaniel Johnston, Table of n, a(n) for n = 1..68 (full sequence)

Cf. A162688, A162689, A162690, A162691, A162692, A162693, A162694, A162817, A162818, A162819, A162820, A162821, A162822, A162823, A162824, A191973.

for n from 1 to 3660 do if(n=60 or type(60*n/(60-n),integer))then printf("%d, ",n): fi: od: # Nathaniel Johnston, Jun 22 2011

f[a_,b_]:=(a*b)/(a-b); a=60;lst={};Do[If[f[a,n]==IntegerPart[f[a,n]], AppendTo[lst,n]],{n,9!}];lst
Select[Range[500],If[#==60,True,IntegerQ[(60#)/(60-#)]]&] (* Harvey P. Dale, Sep 26 2023 *)

Keyword fini added by R. J. Mathar, Jul 31 2009

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A127730 Triangle read by rows: row n consists of the positive integers m where m+n divides m*n.

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A146564 a(n) is the number of solutions of the equation k*n/(k-n) = c. k,c integers.

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A162821 Positive numbers k such that 30*k/(30-k) are integers.

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A162822 Positive numbers k such that 36*k/(36-k) are integers.

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A162823 Positive numbers k such that 42*k/(42-k) are integers.

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A162824 Positive numbers k such that 48*k/(48-k) are integers.

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A162825 Positive numbers k such that 60*k/(60-k) are integers.

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