A161345 -id:A161345 - cp's OEIS frontend

A162532 Numbers k whose largest divisor <= sqrt(k) equals 12.

144, 156, 168, 180, 192, 204, 216, 228, 264, 276, 348, 372, 444, 492, 516, 564, 636, 708, 732, 804, 852, 876, 948, 996, 1068, 1164, 1212, 1236, 1284, 1308, 1356, 1524, 1572, 1644, 1668, 1788, 1812, 1884, 1956, 2004, 2076, 2148, 2172, 2292, 2316, 2364

Offset: 1

Omar E. Pol, Jul 05 2009

See A161344 for more information.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A033676, A008578, A161344, A161345, A161424, A161835, A162526, A162527, A162528, A162529, A162530, A162531.

A033676 := proc(n) local dvs; dvs := sort(convert(numtheory[divisors](n),list)) ; op(floor((nops(dvs)+1)/2) ,dvs) ; end: for n from 1 to 3500 do if A033676(n) = 12 then printf("%d,",n) ; fi; od: # R. J. Mathar, Jul 13 2009

ld12Q[n_]:=First[Select[Reverse[Divisors[n]],#<=Sqrt[n]&]]==12;Select[ 12*Range[ 200], ld12Q] (* Harvey P. Dale, Mar 29 2013 *)

Numbers k such that A033676(k)=12.

More terms from R. J. Mathar, Jul 13 2009

A162529 Numbers k whose largest divisor <= sqrt(k) equals 9.

Original entry on oeis.org

81, 90, 99, 108, 117, 126, 135, 153, 162, 171, 189, 207, 243, 261, 279, 333, 369, 387, 423, 477, 531, 549, 603, 639, 657, 711, 747, 801, 873, 909, 927, 963, 981, 1017, 1143, 1179, 1233, 1251, 1341, 1359, 1413, 1467, 1503, 1557, 1611, 1629, 1719, 1737, 1773

Offset: 1

Omar E. Pol, Jul 05 2009

See A161344 for more information.

Cf. A033676, A008578, A161344, A161345, A161424, A161835, A162526, A162527, A162528, A162530, A162531, A162532.

A033676 := proc(n) local dvs; dvs := sort(convert(numtheory[divisors](n),list)) ; op(floor((nops(dvs)+1)/2) ,dvs) ; end: for n from 1 to 2500 do if A033676(n) = 9 then printf("%d,",n) ; fi; od: # R. J. Mathar, Jul 13 2009

lst = {}; For[n = 1, n <= 5000, n++, If[Last[Select[Divisors[n], # <= Sqrt@n &]] == 9, PrependTo[lst, n]]]; Reverse@lst (* Jasper Mulder (jasper.mulder(AT)planet.nl), Jul 14 2009 *)

Numbers k such that A033676(k)=9.

More terms from R. J. Mathar and Jasper Mulder (jasper.mulder(AT)planet.nl), Jul 13 2009

A162531 Numbers k whose largest divisor <= sqrt(k) is 11.

Original entry on oeis.org

121, 132, 143, 154, 165, 176, 187, 198, 209, 220, 231, 242, 253, 275, 297, 319, 341, 363, 385, 407, 451, 473, 517, 539, 583, 605, 649, 671, 737, 781, 803, 847, 869, 913, 979, 1067, 1111, 1133, 1177, 1199, 1243, 1331, 1397, 1441, 1507, 1529, 1639, 1661

Offset: 1

Omar E. Pol, Jul 05 2009

See A161344 for more information.

Cf. A033676, A008578, A161344, A161345, A161424, A161835, A162526, A162527, A162528, A162529, A162530, A162532.

A033676 := proc(n) local dvs; dvs := sort(convert(numtheory[divisors](n),list)) ; op(floor((nops(dvs)+1)/2) ,dvs) ; end: for n from 1 to 2500 do if A033676(n) = 11 then printf("%d,",n) ; fi; od: # R. J. Mathar, Jul 13 2009

ld = 11;
selQ[n_] := AllTrue[Divisors[n], # <= ld || #^2 > n&];
Select[ Range[ld, 200] ld, selQ] (* Jean-François Alcover, Apr 14 2020 *)

Numbers k such that A033676(k)=11.

More terms from R. J. Mathar and Jasper Mulder (jasper.mulder(AT)planet.nl), Jul 13 2009

A162526 Numbers k whose largest divisor <= sqrt(k) equals 6.

Original entry on oeis.org

36, 42, 48, 54, 60, 66, 78, 102, 114, 138, 174, 186, 222, 246, 258, 282, 318, 354, 366, 402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 762, 786, 822, 834, 894, 906, 942, 978, 1002, 1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1338, 1362

Offset: 1

Omar E. Pol, Jul 05 2009

See A161344 for more information.

Cf. A033676, A008578, A161344, A161345, A161424, A161835, A162527, A162528, A162529, A162530, A162531, A162532.

A033676 := proc(n) local dvs; dvs := sort(convert(numtheory[divisors](n),list)) ; op(floor((nops(dvs)+1)/2) ,dvs) ; end: for n from 1 to 2000 do if A033676(n) = 6 then printf("%d,",n) ; fi; od: # R. J. Mathar, Jul 13 2009

ld6Q[n_]:=Last[Select[Divisors[n],#<=Sqrt[n]&]]==6; Select[Range[ 1400],ld6Q] (* Harvey P. Dale, Mar 08 2012 *)

Numbers k such that A033676(k)=6.

More terms from R. J. Mathar, Jul 13 2009

A164000 Main diagonal of array in A163280.

Original entry on oeis.org

1, 6, 15, 28, 45, 66, 91, 128, 162, 200, 231, 372, 325, 406, 495, 656, 561, 954, 703, 1180, 987, 1078, 1035, 1896, 1375, 1534, 1701, 2324, 1653, 3090, 1891, 3104, 2541, 2686, 3045, 5004, 2701, 3382, 3627, 5560, 3321, 6846, 3655, 6028, 6165, 5014, 4371

Offset: 1

Omar E. Pol, Aug 08 2009

nonn

Cf. A008578, A161344, A161345, A163280.

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: A163280 := proc(n, k) local r, T ; r := 0 ; for T from k^2 by k do if A033676(T) = k then r := r+1 ; if r = n then RETURN(T) ; end if; end if; end do: end proc: A164000 := proc(n) A163280(n,n) ; end proc: seq(A164000(n),n=1..60) ; # R. J. Mathar, Feb 16 2010

nmax = 50;
pm = Prime[nmax];
selDiv[n_] := Select[Divisors[n], #^2 <= n&][[-1]];
Clear[col];
col[k_] := col[k] = Select[Range[k pm], selDiv[#] == k&];
a[n_] := col[n][[n]];
Array[a, nmax] (* Jean-François Alcover, Mar 24 2020 *)

lista(nn) = my(v = apply(f, [1..(2*nn-1)^2]), cols = vector(nn, i, select(x->(x==i), v, 1))); vector(nn, i, cols[i][i]); \\ Michel Marcus, Jan 23 2023

Terms from a(13) on by R. J. Mathar, Feb 16 2010

A147861 Triangle read by rows: T(n,k)=min(k, n/k) if k divides n, T(n,k)=0 otherwise (n >=1, 1<=k<=n).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 0, 0, 0, 1, 1, 2, 2, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 2, 0, 2, 0, 0, 0, 1, 1, 0, 3, 0, 0, 0, 0, 0, 1, 1, 2, 0, 0, 2, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 3, 0, 2, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Omar E. Pol, Nov 16 2008, Jul 20 2009

			Triangle begins:
1;
1,1;
1,0,1;
1,2,0,1;
1,0,0,0,1;
1,2,2,0,0,1;
1,0,0,0,0,0,1;
1,2,0,2,0,0,0,1;
1,0,3,0,0,0,0,0,1;
1,2,0,0,2,0,0,0,0,1;
1,0,0,0,0,0,0,0,0,0,1;
1,2,3,3,0,2,0,0,0,0,0,1;
1,0,0,0,0,0,0,0,0,0,0,0,1;
1,2,0,0,0,0,2,0,0,0,0,0,0,1;
1,0,3,0,3,0,0,0,0,0,0,0,0,0,1;
1,2,0,4,0,0,0,2,0,0,0,0,0,0,0,1;
1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
1,2,3,0,0,3,0,0,2,0,0,0,0,0,0,0,0,1;
1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
1,2,0,4,4,0,0,0,0,2,0,0,0,0,0,0,0,0,0,1;
1,0,3,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
1,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1;
1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
1,2,3,4,0,4,0,3,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1;
1,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
1,2,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,1;
1,0,3,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
1,2,0,4,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
1,2,3,0,5,5,0,0,0,3,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
...
Row sums: A117004.

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos [From _Omar E. Pol_, Jul 20 2009]
Omar E. Pol, Illustration: Divisors and pi(x) [From _Omar E. Pol_, Jul 20 2009]

Cf. A000005, A008578, A027750, A051731, A117004, A127093, A161345, A161424, A161835, A162526, A162527, A162528, A162529, A162530, A162531, A162532. [From Omar E. Pol, Jul 20 2009]

Submitted without a definition, which was supplied by Jon E. Schoenfield, Dec 13 2008

A163100 Triangle giving positive values of A147861.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 1, 2, 3, 3, 2, 1, 1, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 4, 2, 1, 1, 1, 1, 2, 3, 3, 2, 1, 1, 1, 1, 2, 4, 4, 2, 1, 1, 3, 3, 1, 1, 2, 2, 1, 1, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 5, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 4, 4, 2, 1, 1, 1, 1, 2

Offset: 1

Omar E. Pol, Jul 20 2009

			Triangle begins:
1;
1,1;
1,..1;
1,2,..1;
1,......1;
1,2,2,....1;
1,..........1;
1,2,..2,......1;
1,..3,..........1;
1,2,....2,........1;
1,..................1;
1,2,3,3,..2,..........1;
1,......................1;
1,2,........2,............1;
1,..3,..3,..................1;
1,2,..4,......2,..............1;
1,..............................1;
1,2,3,....3,....2,................1;
1,..................................1;
1,2,..4,4,........2,..................1;
1,..3,......3,..........................1;
1,2,................2,....................1;
1,..........................................1;
1,2,3,4,..4,..3,......2,......................1;
1,......5,......................................1;
1,2,....................2,........................1;
1,..3,..........3,..................................1;
1,2,..4,....4,............2,..........................1;
1,......................................................1;
1,2,3,..5,5,......3,........2,............................1;
...
Row sums: A117004.

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos
Omar E. Pol, Illustration: Divisors and pi(x)

Cf. A000005, A008578, A027750, A051731, A117004, A127093, A147861, A161345, A161424, A161835, A162526, A162527, A162528, A162529, A162530, A162531, A162532.

A147861 := proc(n,k) if k<=0 or k > n then 0; else if n mod k = 0 then min(k,n/k) ; else 0; fi; fi; end proc: A163100 := proc(n,k) local dvs; dvs := sort(convert(numtheory[divisors](n),list)) ; min( op(k,dvs),n/op(k,dvs)) ; end: for n from 1 to 60 do for k from 1 to numtheory[tau](n) do printf("%d,",A163100(n,k) ) ; end do; end do: # R. J. Mathar, Aug 01 2009

Extended beyond row 12 by R. J. Mathar, Aug 01 2009

A163990 Square array read by antidiagonals where the row n lists the numbers k such that their largest divisor <= sqrt(k) equals n.

Original entry on oeis.org

1, 4, 2, 9, 6, 3, 16, 12, 8, 5, 25, 20, 15, 10, 7, 36, 30, 24, 18, 14, 11, 49, 42, 35, 28, 21, 22, 13, 64, 56, 48, 40, 32, 27, 26, 17, 81, 72, 63, 54, 45, 44, 33, 34, 19, 100, 90, 80, 70, 60, 50, 52, 39, 38, 23, 121, 110, 99, 88, 77, 66, 55, 68, 51, 46, 29, 144, 132, 120, 108

Offset: 1

Omar E. Pol, Aug 11 2009

This sequence is a permutation of the natural numbers.
Note that the first row is formed by 1 together the prime numbers and the first column are the squares of the natural numbers.
For more information see A163280, the main entry for this sequence. (See also A161344).

			Array begins:
1, 2, 3, 5, 7, 11,
4, 6, 8, 10, 14,
9, 12, 15, 18,
16, 20, 24,
25, 30,
36,
See also the array in A163280.

Index entries for sequences that are permutations of the natural numbers
Omar E. Pol, Illustration for A008578, the first row of the array [From _Omar E. Pol_, Oct 24 2009]
Omar E. Pol, Illustration for A161344, the second row of the array [From _Omar E. Pol_, Oct 24 2009]
Omar E. Pol, Illustration for A008578, A161344, A161345 and A161424 [From _Omar E. Pol_, Oct 24 2009]

Cf. A000027, A000040, A033676, A147861, A163100, A163280, A163281, A163991, A164000.
Cf. Rows: 1=A008578, 2=A161344, 3=A161345, 4=A161424, 5=A161835, 6=A162526, 7=A162527, 8=A162528, 9=A162529, 10=A162530, 11=A162531, 12=A162532.
Cf. Columns: 1=A000290, 2=A002378, 4=A164004, 6=A164006, 7=A164007, 8=A164008, 9=A164009, 10=A164010, 11=A164011, 12=A164012.

Row n lists the numbers k such that A033676(k)=n.

A164004 Zero together with row 4 of the array in A163280.

Original entry on oeis.org

0, 5, 10, 18, 28, 40, 54, 70, 88, 108, 130, 154, 180, 208, 238, 270, 304, 340, 378, 418, 460, 504, 550, 598, 648, 700, 754, 810, 868, 928, 990, 1054, 1120, 1188, 1258, 1330, 1404, 1480, 1558, 1638, 1720, 1804, 1890, 1978, 2068, 2160, 2254, 2350, 2448, 2548

Offset: 0

Omar E. Pol, Aug 08 2009

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).

Cf. A008578, A161344, A161345, A163280, A164000, A005563, A164005.

Cf. A028552.

A033676 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do if d^2 <= n then a := max(a,d) ; fi; od: a; end: A163280 := proc(n,k) local r,T ; r := 0 ; for T from k^2 by k do if A033676(T) = k then r := r+1 ; if r = n then RETURN(T) ; fi; fi; od: end: A164004 := proc(n) if n = 0 then 0; else A163280(4,n) ; fi; end: seq(A164004(n),n=0..80) ; # R. J. Mathar, Aug 09 2009

Join[{0, 5}, Table[n*(n + 3), {n, 2, 50}]] (* G. C. Greubel, Aug 28 2017 *)

x='x+O('x^50); concat([0], Vec(x*(x^3 -3*x^2 +5*x -5)/(x-1)^3)) \\ G. C. Greubel, Aug 28 2017

Conjectures from Colin Barker, Apr 07 2015: (Start)
a(n) = n*(3+n) = A028552(n) for n > 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4.
G.f.: x*(x^3 - 3*x^2 + 5*x - 5) / (x-1)^3. (End)
E.g.f.: x*(x+4)*exp(x) + x. - G. C. Greubel, Aug 28 2017

Extended beyond a(12) by R. J. Mathar, Aug 09 2009

A164006 Zero together with row 6 of the array in A163280.

Original entry on oeis.org

0, 11, 22, 27, 44, 50, 66, 84, 104, 126, 150, 176, 204, 234, 266, 300, 336, 374, 414, 456, 500, 546, 594, 644, 696, 750, 806, 864, 924, 986, 1050, 1116, 1184, 1254, 1326, 1400, 1476, 1554, 1634, 1716, 1800, 1886, 1974, 2064, 2156, 2250, 2346, 2444, 2544, 2646

Offset: 0

Omar E. Pol, Aug 08 2009

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A008578, A161344, A161345, A163280, A164000, A164005, A164007.

Cf. A028557 for n > 4. - R. J. Mathar, Aug 09 2009

A033676 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do if d^2 <= n then a := max(a,d) ; fi; od: a; end: A163280 := proc(n,k) local r,T ; r := 0 ; for T from k^2 by k do if A033676(T) = k then r := r+1 ; if r = n then RETURN(T) ; fi; fi; od: end: A164006 := proc(n) if n = 0 then 0; else A163280(6,n) ; fi; end: seq(A164006(n),n=0..80) ; # R. J. Mathar, Aug 09 2009

Join[{0,11,22,27}, Table[n*(n + 5), {n, 4, 50}]] (* G. C. Greubel, Aug 28 2017 *)

concat(0, Vec(x*(8*x^6-21*x^5+23*x^4-18*x^3+6*x^2+11*x-11)/(x-1)^3 + O(x^100))) \\ Colin Barker, Nov 24 2014

From Colin Barker, Nov 24 2014: (Start)
a(n) = n*(n+5) for n > 4.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 7.
G.f.: x*(8*x^6 - 21*x^5 + 23*x^4 - 18*x^3 + 6*x^2 + 11*x - 11) / (x-1)^3. (End)
E.g.f.: (x/2)*(10 + 8*x + x^2 + 2*(x + 6)*exp(x)). - G. C. Greubel, Aug 28 2017

Extended beyond a(12) by R. J. Mathar, Aug 09 2009

cp's OEIS Frontend

A162532 Numbers k whose largest divisor <= sqrt(k) equals 12.

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A162529 Numbers k whose largest divisor <= sqrt(k) equals 9.

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A162531 Numbers k whose largest divisor <= sqrt(k) is 11.

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A162526 Numbers k whose largest divisor <= sqrt(k) equals 6.

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A164000 Main diagonal of array in A163280.

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A147861 Triangle read by rows: T(n,k)=min(k, n/k) if k divides n, T(n,k)=0 otherwise (n >=1, 1<=k<=n).

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A163100 Triangle giving positive values of A147861.

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A163990 Square array read by antidiagonals where the row n lists the numbers k such that their largest divisor <= sqrt(k) equals n.

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