cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A036470 a(n) is the number of distinct possible values of d(k), the number of divisors of k, among numbers k whose binary order (A029837) does not exceed n.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 11, 12, 16, 17, 23, 26, 31, 37, 43, 48, 58, 64, 74, 82, 94, 106, 122, 133, 146, 165, 183, 202, 224, 244, 267, 294, 325, 355, 389, 416, 453, 500, 541, 584, 636, 680, 737, 795, 859, 922, 995, 1068, 1149, 1233, 1324, 1412, 1523, 1616, 1731, 1845
Offset: 0

Views

Author

Labos Elemer

Keywords

Examples

			If 1 <= k <= 128, i.e., the binary order of k is g(k) <= 7, then d(k) takes 12 values {1,2,3,4,5,6,7,8,9,10,12,16}; thus a(7) = 12. The maximal value (16) appears as a(7) in A036451.

Links

Crossrefs

Cf. A000005, A005179, A029837, A036471, A036493.

Extensions

a(20)-a(21) corrected by David A. Corneth, May 12 2018

A036484 a(n) is the minimal number of binary order n which has maximal number of divisors in this interval.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 60, 120, 240, 360, 840, 1680, 2520, 7560, 15120, 27720, 55440, 110880, 221760, 498960, 720720, 1441440, 3603600, 7207200, 14414400, 32432400, 61261200, 122522400, 245044800, 367567200, 735134400, 2095133040
Offset: 0

Views

Author

Labos Elemer

Keywords

Comments

Compare with A007416, where terms of this sequence are present.

Examples

			For n=9, with 256 < k <= 512, d(k) takes 17 distinct values, of which d(k)=24 is the greatest (see A036451 and A036470) and occurs first at k=360, so a(9)=360.

Crossrefs

Cf. A000005, A029837, A005179, A007416, A036470, A036492, A036493.

Programs

  • Mathematica
    Block[{nn = 22, s}, s = TakeList[Array[DivisorSigma[0, # + 1] &, 2^nn - 1], 2^Range[0, nn - 1]]; {1}~Join~Map[2^(#1 - 1) + #2 & @@ FirstPosition[s, #] &, Map[Max, s]]] (* Michael De Vlieger, Nov 04 2020 *)

Extensions

a(22)-a(31) from Sean A. Irvine, Nov 04 2020

A036493 Largest number having binary order n (A029837) and of which the number of divisors is maximal in that range of g(k) = n.

Original entry on oeis.org

1, 2, 4, 8, 12, 30, 60, 120, 240, 504, 840, 1680, 3960, 7560, 15120, 32760, 65520, 131040, 262080, 498960, 997920, 1965600, 3603600, 7207200, 14414400, 32432400, 64864800, 122522400, 245044800, 514594080, 1029188160, 2095133040, 4227022800, 8454045600
Offset: 0

Views

Author

Labos Elemer

Keywords

Comments

This sequence differs from A036451 only at n = 3, 5, 9, 12, and 15, which are the values of n for which there exists more than one k such that g(k) = n and d(k) has the maximum possible value.
a(n) is the largest term k in A067128 such that log_2(k) <= n. - Jon E. Schoenfield, May 13 2018

Examples

			For n = 9, k is in {257, 512}, max(d(k)) = 24 (see A036451); this holds for four different numbers (360, 420, 480, and 504); a(9) = 504 since it is the largest.

Crossrefs

Cf. A000005, A029837, A005179, A007416, A036451, A036470, A036484.

Programs

  • Mathematica
    {1}~Join~Table[Max@ MaximalBy[Range[2^(n - 1) + 1, 2^n], DivisorSigma[0, #] &], {n, 24}] (* Michael De Vlieger, Aug 01 2017 *)

Extensions

a(22)-a(24) from Michael De Vlieger, Aug 01 2017
a(25)-a(33) from Jon E. Schoenfield, May 12 2018

A140864 Smallest odd number with same number of divisors as 3*a(n-1).

Original entry on oeis.org

1, 3, 9, 15, 45, 105, 315, 945, 2835, 3465, 10395, 31185, 45045, 135135, 405405, 675675, 2027025, 3828825, 11486475, 34459425, 72747675, 218243025, 654729075, 1527701175, 4583103525, 11712375675, 35137127025, 105411381075
Offset: 1

Views

Author

J. Lowell, Jul 20 2008

Keywords

Examples

			9*3=27 has 4 divisors, but smallest odd number with 4 divisors is 15.

Links

Crossrefs

Cf. A053624, A019505. d(a(n)) = A036451(n) for first 18 terms.

Programs

  • PARI
    a(nn) = {ia = 1; print1(ia, ", "); for (n = 1, nn - 1, nd = numdiv(3*ia); forstep(i = 1, 3*ia, 2, if (numdiv(i) == nd, ia = i; break;);); print1(ia, ", "););} \\ Michel Marcus, Jun 14 2013
  • PARI
    {/*prints b-file for A140864 - add more for loops for more terms*/ print("#A140864"); print(1" "1); print(2" "3); n = 3; for(p=3,56,tau = numdiv(3*n); exp3n=factor(n)[1,2];delta = bigomega(exp3n+2) - bigomega(exp3n+1); delta = max(delta+1,2); var = exp3n+delta; num = 10^1000; for( n1=1, var, for (n2=0, n1, for( n3=0, n2, for( n4=0, n3, for( n5=0, n4, for( n6=0, n5, for( n7=0, n6, for( n8=0, n7, for( n9=0, n8, for( n10=0, n9, for( n11=0, n10, for( n12=0, n11, for( n13=0, n12, for( n14=0, n13, for( n15=0, n14, if( (n1+1) * (n2+1) * (n3+1) * (n4+1) * (n5+1) * (n6+1) * (n7+1) * (n8+1) * (n9+1) * (n10+1) * (n11+1) * (n12+1) * (n13+1) * (n14+1) * (n15+1) == tau, numtemp = prime(2)^n1 * prime(3)^n2 * prime(4)^n3 * prime(5)^n4 * prime(6)^n5 * prime(7)^n6  * prime(8)^n7 * prime(9)^n8 * prime(10)^n9 * prime(11)^n10 * prime(12)^n11 * prime(13)^n12 * prime(14)^n13 * prime(15)^n14 * prime(16)^n15; if(numtemp < num, num = numtemp); ));););););););) ;);););) ;););); print(p" "num); n=num;)} \\ Dimitri Papadopoulos, May 08 2019

Extensions

a(10) through a(28) from Klaus Brockhaus, Jul 23 2008
a(29) through a(56) from Dimitri Papadopoulos, May 08 2019
Showing 1-4 of 4 results.

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