A180045 -id:A180045 - cp's OEIS frontend

A332770 a(n) is the number of ways to write A180045(n) as (xy+1)(x*z+1) with x > y > z > 1.

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

Offset: 1

Rémy Sigrist, Feb 23 2020

nonn

			The first terms, alongside A180045(n), are:
  n   a(n)  A180045(n)
  --  ----  ---------------------------------------
   1     1   28 = (3*2+1)*(3*1+1)
   2     1   45 = (4*2+1)*(4*1+1)
   3     1   65 = (4*3+1)*(4*1+1)
   4     1   66 = (5*2+1)*(5*1+1)
   5     1   91 = (6*2+1)*(6*1+1)
   6     1   96 = (5*3+1)*(5*1+1)
   7     1  117 = (4*3+1)*(4*2+1)
   8     1  120 = (7*2+1)*(7*1+1)
   9     1  126 = (5*4+1)*(5*1+1)
  10     1  133 = (6*3+1)*(6*1+1)
  11     1  153 = (8*2+1)*(8*1+1)
  12     1  175 = (6*4+1)*(6*1+1)
  13     2  176 = (5*3+1)*(5*2+1) = (7*3+1)*(7*1+1)

Robert Israel, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C program for A332770

Cf. A180045.

C
```
See Links section.
```

N:= 20000: # for a(n) where A180045(n) <= N
V:= Vector(N):
for x from 3 while (2*x+1)*(x+1) <= N do
  for y from 2 to x-1 while (x*y+1)*(x+1) <= N do
    for z from 1 to y-1 do
      v:= (x*y+1)*(x*z+1);
      if v > N then break fi;
      V[v]:= V[v]+1;
od od od:
subs(0=NULL,convert(V,list)); # Robert Israel, Jun 10 2021

A332768 a(n) is the greatest prime factor of A180045(n).

Original entry on oeis.org

7, 5, 13, 11, 13, 3, 13, 5, 7, 19, 17, 7, 11, 19, 31, 5, 11, 29, 19, 23, 7, 3, 11, 13, 11, 7, 31, 43, 41, 37, 7, 31, 17, 17, 29, 7, 41, 23, 19, 37, 31, 19, 19, 5, 11, 7, 17, 31, 7, 13, 29, 5, 43, 31, 61, 7, 41, 37, 73, 23, 53, 31, 13, 71, 23, 11, 61, 67, 11

Offset: 1

Rémy Sigrist, Feb 23 2020

Corvaja and Zannier showed that a(n) tends to infinity as n tends to infinity.

			The first terms, alongside A180045(n), are:
  n   a(n)  A180045(n)
  --  ----  --------------
   1     7   28 =  7 * 2^2
   2     5   45 =  5 * 3^2
   3    13   65 = 13 * 5
   4    11   66 = 11 * 3 * 2
   5    13   91 = 13 * 7
   6     3   96 =  3 * 2^5
   7    13  117 = 13 * 3^2
   8     5  120 =  5 * 3 * 2^3
   9     7  126 =  7 * 3^2 * 2
  10    19  133 = 19 * 7

Rémy Sigrist, Table of n, a(n) for n = 1..10000
P. Corvaja and U. Zannier, On the greatest prime factor of (ab+1)(ac+1), arXiv:math/0205136 [math.NT], 2002.
Rémy Sigrist, PARI program for A332768

Cf. A006530, A180045, A260234.

PARI
```
See Links section.
```

a(n) = A006530(A180045(n)).

A260234 Largest prime factor of the n-th hexagonal number (A000384).

Original entry on oeis.org

3, 5, 7, 5, 11, 13, 5, 17, 19, 11, 23, 13, 7, 29, 31, 17, 7, 37, 13, 41, 43, 23, 47, 7, 17, 53, 11, 29, 59, 61, 7, 13, 67, 23, 71, 73, 19, 13, 79, 41, 83, 43, 29, 89, 23, 47, 19, 97, 11, 101, 103, 53, 107, 109, 37, 113, 29, 59, 17, 61, 41, 7, 127, 43, 131

Offset: 2

Colin Barker, Jul 20 2015

nonn

As A000384(n+1) = (n*2+1)*(n*1+1), A000384(n) belongs to A180045 for n > 3, and a(n) tends to infinity as n tends to infinity. - Rémy Sigrist, Feb 23 2020

			a(3) = 5 because A000384(3) = 15 = 3 * 5.

Colin Barker, Table of n, a(n) for n = 2..1000

Cf. A000384, A006530, A180045, A260233, A260235, A260236.

FactorInteger[#][[-1,1]]&/@PolygonalNumber[6,Range[2,70]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 10 2018 *)

pg(m, n) = (n^2*(m-2)-n*(m-4))/2 \\ n-th m-gonal number
lpf(m) = vecmax(factorint(m)[, 1]) \\ Largest prime factor
a(n) = lpf(pg(6, n))

a(n) = A006530(A000384(n)).

A320883 3-smooth numbers of the form (ab+1)(ac+1), a > b > c > 0.

Original entry on oeis.org

96, 288, 3888, 4608, 31104, 69984, 2654208, 2985984, 4478976, 1088391168, 1528823808, 440301256704

Offset: 1

M. F. Hasler, Nov 19 2018

Subsequence of A320884 = 5-smooth terms of A180045, finite according to Corvaja & Zannier.

Can someone prove that a(12) = 440301256704 = (2359*889 + 1)(2359*89 + 1) = 2^26 * 3^8 is the last term?

P. Corvaja and U. Zannier, On the greatest prime factor of (ab+1)(ac+1), Proceedings of the American Mathematical Society 131 (2003), pp. 1705-1709. See also arXiv:math/0205136 [math.NT], 2002.

Cf. A180045 = {(ab+1)(ac+1); a > b > c > 0}, A320884 (5-smooth terms of A180045), A003586 (3-smooth numbers).

(* This is only a recomputation of the existing sequence. *)
(* Max exponents: *) jmax = 26; kmax = 12;
r[j_, k_] := Reduce[a > b > c > 0 && (a b + 1)(a c + 1) == 2^j*3^k , {a, b, c}, Integers];
Reap[Do[rr = r[j, k]; If[rr =!= False, Print[{j, k, 2^j*3^k}]; Sow[2^j*3^k]], {j, 1, jmax}, {k, 1, kmax}]][[2, 1]] // Union (* Jean-François Alcover, Dec 05 2018 *)

A320883(LIM=35,S=[])={for(m=1,LIM, for(k=0,m, is_A180045(3^k<<(m-k))&& S=setunion(S,[3^k<<(m-k)])));S} \\ Gives all terms up to 2^LIM and possibly some larger terms up to 3^LIM.
is_A320883(n)={vecmax(factor(n,3)[,1])<4 && is_A180045(n)}

Intersection of A180045 = {(ab+1)(ac+1); a > b > c > 0} and A003586 (3-smooth numbers).

A320884 5-smooth numbers of the form (ab+1)(ac+1), a > b > c > 0.

Original entry on oeis.org

45, 96, 120, 225, 288, 540, 640, 1080, 1200, 1920, 2160, 3888, 4000, 4500, 4608, 5760, 6480, 7200, 8640, 9600, 10935, 16875, 18225, 25000, 25600, 27000, 28800, 30720, 31104, 38400, 46080, 48600, 69984, 75000, 81000, 91125, 97200, 102400, 112500, 115200, 164025, 184320

Offset: 1

M. F. Hasler, Nov 19 2018

Corvaja & Zannier show that there are only finitely many p-smooth terms in A180045, for any prime p. This sequences lists these terms for p = 5, and is therefore finite.
Can someone prove that a(163) = 3327916660110655488000000000 = (16775191*16038089 + 1)(16775191*737369 + 1) = 2^42 * 3^18 * 5^9 is the last term? - M. F. Hasler, Nov 19 2018
If a(164) exists it's larger than 10^60. - David A. Corneth, Nov 20 2018

M. F. Hasler, Table of n, a(n) for n = 1..163 (all terms up to 10^30, and up to 10^60 according to _David A. Corneth_)
P. Corvaja and U. Zannier, On the greatest prime factor of (ab+1)(ac+1), Proceedings of the American Mathematical Society 131 (2003), pp. 1705-1709. See also arXiv:math/0205136 [math.NT], 2002.

Cf. A180045 (numbers (ab+1)(ac+1), a>b>c), A320883 (subsequence of 3-smooth terms), A051037 (5-smooth numbers).

(* This is only a recomputation of the existing data section. *)
jmax = 12; kmax = 8; lmax = 5; max = 200000;
r[j_, k_, l_] := r[j, k, l] = If[2^j*3^k*5^l > max, Return[False], Reduce[a > b > c > 0 && (a b + 1)(a c + 1) == 2^j*3^k*5^l, {a, b, c}, Integers]];
rea = Reap[Do[rr = r[j, k, l]; If[rr =!= False, res = {j, k, l, 2^j*3^k*5^l}; Print[res]; Sow[res]], {j, 0, jmax}, {k, 0, kmax}, {l, 0, lmax}]][[2, 1]] //Union;
Print["min = ", Min /@ Transpose[rea], " max = ", Max /@ Transpose[rea]];
Sort[rea[[All, 4]]] (* Jean-François Alcover, Dec 05 2018 *)

is_A320884(n)={vecmax(factor(n,5)[,1])<6 && is_A180045(n)}
A320884=select( is_A180045, A051037_list(1e30))

Intersection of A051037 and A180045.

A320885 7-smooth but not 5-smooth numbers of the form (ab+1)(ac+1), a > b > c > 0.

Original entry on oeis.org

28, 126, 175, 280, 336, 378, 441, 560, 630, 672, 1225, 1470, 1680, 1701, 1792, 2016, 2520, 2835, 3136, 3969, 4200, 5250, 5600, 6860, 7840, 7875, 8400, 8960, 9072, 9408, 11025, 11340, 12096, 13125, 15120, 17640, 19845, 20160, 21000, 23520, 24696, 27440, 30625, 32928, 35000

Offset: 1

M. F. Hasler, Nov 21 2018

Corvaja & Zannier show that there are only finitely many p-smooth terms in A180045, for any prime p. This sequences lists these terms for p = 7 without those for p = 5 (A320884), and is therefore finite.

David A. Corneth, Table of n, a(n) for n = 1..910 (first 671 terms by Maximilian Hasler; terms < 10^35, the largest of which is ~2.6*10^31)
P. Corvaja and U. Zannier, On the greatest prime factor of (ab+1)(ac+1), Proceedings of the American Mathematical Society 131 (2003), pp. 1705-1709. See also arXiv:math/0205136 [math.NT], 2002.

Cf. A080194 (greatest prime factor = 7).

Cf. A180045 (numbers (ab+1)(ac+1), a>b>c>0), A320883 (subsequence of 3-smooth terms), A320884 (subsequence of 5-smooth terms).

Reap[For[k = 7, k <= 35000, k = k+7, If[FactorInteger[k][[-1, 1]] == 7, If[ Reduce[k == (a b + 1)(a c + 1) && a > b > c > 0, {a, b, c}, Integers] =!= False, Print[k]; Sow[k]]]]][[2, 1]] (* Jean-François Alcover, Dec 07 2018 *)

is_A320885(n)={vecmax(factor(n,7)[,1])==7 && is_A180045(n)}
A320885=select( is_A180045, A080194_list(1e20)) \\ Only initial terms, not the complete sequence. For more efficiency, use is_A180045 or a dedicated implementation inside the nested loops in A080194_list().

Intersection of A080194 (gpf(n) = 7) and A180045 ((ab+1)(ac+1)).

A332764 7-smooth numbers of the form (ab+1)*(ac+1), a > b > c > 0.

Original entry on oeis.org

28, 45, 96, 120, 126, 175, 225, 280, 288, 336, 378, 441, 540, 560, 630, 640, 672, 1080, 1200, 1225, 1470, 1680, 1701, 1792, 1920, 2016, 2160, 2520, 2835, 3136, 3888, 3969, 4000, 4200, 4500, 4608, 5250, 5600, 5760, 6480, 6860, 7200, 7840, 7875, 8400, 8640, 8960

Offset: 1

Jon E. Schoenfield, Feb 22 2020

Sequence is finite (see comments at A320884).

David A. Corneth, Table of n, a(n) for n = 1..1072

Cf. A180045 (numbers (ab+1)(ac+1), a > b > c > 0), A320883 (subsequence of 3-smooth terms), A320884 (subsequence of 5-smooth terms), A002473 (7-smooth numbers).

Block[{nn = 9000, nm, m}, nm = Ceiling[(Sqrt[8 nn + 1] - 3)/4]; Union@ Reap[Do[If[a > b > c > 0, Set[m, (a b + 1) (a c + 1)]; If[And[m <= nn, FactorInteger[m][[-1, 1]] <= 7 ], Sow[m]]], {a, nm}, {b, a - 1}, {c, b - 1}]][[2, 1]]] (* Michael De Vlieger, Feb 25 2020, after Jean-François Alcover at A180045 *)

Intersection of A002473 and A180045.

A345215 a(n) is the least positive number k that can be written in exactly k ways as (xy+1)(x*z+1) with x > y > z > 1.

Original entry on oeis.org

1, 28, 176, 561, 2701, 7381, 29161, 51681, 115921, 390241, 260281, 924001, 1334161, 1413721, 1038961, 3178981, 8826301, 3000025, 16597441, 33882241, 12708361, 22589281, 31375081, 63095761, 90336961

Offset: 0

Robert Israel, Jun 10 2021

nonn

For n > 1, a(n) = A180045(k) where A332770(k) = n is the first appearance of n in A332770.

			a(3) = 561 since 176 = (8*4+1)*(8*2+1) = (10*5+1)*(10*1+1) = (16*2+1)*(16*1+1) is the first number that can be obtained in exactly 3 ways.

Cf. A180045, A332770.

N:= 10^8:
V:= Vector(N,datatype=integer[4]):
for x from 3 while (2*x+1)*(x+1) <= N do
  for y from 2 to x-1 while (x*y+1)*(x+1) <= N do
    for z from 1 to y-1 do
      v:= (x*y+1)*(x*z+1);
      if v > N then break fi;
      V[v]:= V[v]+1;
od od od:
R:= Array(0..26):
for j from 1 to N do
  v:= V[j]; if R[v] = 0 then R[v]:= j fi
od:
convert(R[0..24],list);

cp's OEIS Frontend

A332770 a(n) is the number of ways to write A180045(n) as (x*y+1)*(x*z+1) with x > y > z > 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

C

Maple

A332768 a(n) is the greatest prime factor of A180045(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A260234 Largest prime factor of the n-th hexagonal number (A000384).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A320883 3-smooth numbers of the form (ab+1)(ac+1), a > b > c > 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A320884 5-smooth numbers of the form (ab+1)(ac+1), a > b > c > 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A320885 7-smooth but not 5-smooth numbers of the form (ab+1)(ac+1), a > b > c > 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A332764 7-smooth numbers of the form (ab+1)*(ac+1), a > b > c > 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A345215 a(n) is the least positive number k that can be written in exactly k ways as (x*y+1)*(x*z+1) with x > y > z > 1.

Views

Author

A332770 a(n) is the number of ways to write A180045(n) as (xy+1)(x*z+1) with x > y > z > 1.

A345215 a(n) is the least positive number k that can be written in exactly k ways as (xy+1)(x*z+1) with x > y > z > 1.