A161344 -id:A161344 - cp's OEIS frontend

A162530 Numbers k whose largest divisor <= sqrt(k) equals 10.

100, 110, 120, 130, 140, 150, 160, 170, 190, 200, 230, 250, 290, 310, 370, 410, 430, 470, 530, 590, 610, 670, 710, 730, 790, 830, 890, 970, 1010, 1030, 1070, 1090, 1130, 1270, 1310, 1370, 1390, 1490, 1510, 1570, 1630, 1670, 1730, 1790, 1810, 1910, 1930

Offset: 1

Omar E. Pol, Jul 05 2009

See A161344 for more information.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A033676, A008578, A161344, A161345, A161424, A161835, A162526, A162527, A162528, A162529, A162531, A162532.

filter:= n -> andmap(t -> t<=10 or t^2 > n, numtheory:-divisors(n)):
select(filter, [seq(n,n=100..10000,10)]); # Robert Israel, Aug 16 2018

ld10Q[n_]:=Last[Select[Divisors[n],#<=Sqrt[n]&]]==10; Select[Range[2000],ld10Q]  (* Harvey P. Dale, Jan 30 2011 *)

Numbers k such that A033676(k) = 10.

More terms from Gerard P. Michon, Jul 12 2009

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

Original entry on oeis.org

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

A161346 a(n) = A161345(n)/3.

Original entry on oeis.org

3, 4, 5, 6, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271

Offset: 1

Omar E. Pol, Jun 20 2009

Union of {4, 6, 9} and all the odd primes. - Amiram Eldar, Apr 17 2024

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A018253, A160811, A160812, A161205, A161344, A161345.

Select[Range[271], Function[{n, s}, Max[TakeWhile[Divisors[n], # <= s &]] == 3] @@ {#, Sqrt@ #} &[3 #] &] (* Michael De Vlieger, Feb 14 2020 *)

Terms beyond a(10) from R. J. Mathar, Jun 24 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

A162348 List of pairs (i,j) of central factors of n, such that i*j = n, where i is the largest divisor of n <= sqrt(n) and j is the smallest divisor of n >= sqrt(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 2, 1, 5, 2, 3, 1, 7, 2, 4, 3, 3, 2, 5, 1, 11, 3, 4, 1, 13, 2, 7, 3, 5, 4, 4, 1, 17, 3, 6, 1, 19, 4, 5, 3, 7, 2, 11, 1, 23, 4, 6, 5, 5, 2, 13, 3, 9, 4, 7, 1, 29, 5, 6, 1, 31, 4, 8, 3, 11, 2, 17, 5, 7, 6, 6, 1, 37, 2, 19, 3, 13, 5, 8, 1, 41, 6, 7, 1, 43, 4, 11, 5, 9, 2, 23, 1, 47, 6, 8, 7

Offset: 1

Omar E. Pol, Jul 04 2009

Note that if n is a square then the square root of n appears repeated: i = j = sqrt(n).

Squarest (least oblong) integral rectangle with area n. This has minimal semiperimeter (A063655), since s = i + j = i + n/i is minimal when ds/di = 1 - n/i^2 = 0, i.e., n = i^2. - Daniel Forgues, Sep 29 2014

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A033676, A033677, A018253, A160812, A161344, A006446, A162190.

f[n_] := Block[{d = Divisors@n}, len = Length[d]/2; {d[[Ceiling@len]], d[[Floor[len + 1]] ]}]; f[1] = {1, 1}; Array[f, 49] // Flatten (* Robert G. Wilson v, Aug 17 2009 *)

a(35) and further terms from Robert G. Wilson v, Aug 17 2009; corrected Aug 18 2009

cp's OEIS Frontend

A162530 Numbers k whose largest divisor <= sqrt(k) equals 10.

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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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A161346 a(n) = A161345(n)/3.

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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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