A067128 -id:A067128 - cp's OEIS frontend

A216365 Numbers n such that tau(n)*sigma(n) sets a new record.

1, 2, 3, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 48, 60, 72, 84, 90, 96, 108, 120, 144, 168, 180, 240, 300, 336, 360, 420, 480, 504, 540, 600, 630, 660, 720, 840, 1008, 1080, 1200, 1260, 1440, 1680, 2100, 2160, 2520, 3360, 3780, 3960, 4200, 4320, 4620, 4680, 5040

Offset: 1

Arkadiusz Wesolowski, Sep 05 2012

nonn

Positions of record values in A064840.

Not identical to A067128; e.g. a(22) = 144 < 168 = A067128(22).

Charles R Greathouse IV and Donovan Johnson, Table of n, a(n) for n = 1..400 (first 268 terms from Charles R Greathouse IV)

Cf. A000005, A000203, A064840.

lst = {}; k = 0; Do[n = DivisorSigma[0, i]*DivisorSigma[1, i]; If[n > k, AppendTo[lst, i]; k = n], {i, 7!}]; lst

r=0;for(n=1,1e9,t=numdiv(n)*sigma(n);if(t>r,r=t;print1(n", "))) \\ Charles R Greathouse IV, Sep 05 2012

A226462 Smallest number m such that A226460(m) = n.

Original entry on oeis.org

0, 2, 5, 11, 23, 47, 83, 167, 179, 359, 1679, 719, 1439, 2879, 3959, 2159, 2519, 10799, 5039, 9239, 12599, 7559, 20159, 31679, 27719, 37799, 45359, 30239

Offset: 0

Naohiro Nomoto, Jun 08 2013

a(31) = 90719, a(32) = 55439, a(34) = 83159, all others are >= 100000. Most terms are one less than a largely composite number (A067128). - Charlie Neder, Nov 03 2018

Charlie Neder, Python program for computing this sequence (Outputs terms sorted by a(n), not n)

Cf. A048247.

a(9)-a(27) from Charlie Neder, Nov 04 2018

A272984 Positions of zero terms in A272314.

Original entry on oeis.org

163, 186, 248, 339, 366, 543, 558, 590, 617, 623, 651, 676, 691, 695, 722, 756, 764, 798, 848, 854, 974, 1000, 1084, 1103, 1104, 1114, 1129, 1130, 1211, 1235, 1246, 1249, 1254, 1257, 1262, 1265, 1266, 1273, 1342, 1363, 1408, 1433, 1445, 1456, 1475, 1476, 1574

Offset: 1

Peter J. C. Moses and Vladimir Shevelev, May 12 2016

nonn

Peter J. C. Moses, Table of n, a(n) for n = 1..400

Cf. A067128, A272314, A272901.

A273415 The smallest term of A273379 having n primes between two consecutive prime divisors.

Original entry on oeis.org

10, 4680, 6585701522400, 193394747145600, 27377180785991836800, 29378941900252048776672000, 5384823686347760468943298225056000, 404593694258692410380118300618528000, 1714431214566179268370439406441900195214656000, 180656647480221782329653424360823828484237888000

Offset: 1

David A. Corneth, May 22 2016

nonn

Is this sequence infinite?

In the prime factorization of a(n), the 'gap' occurs before the largest prime divisor. For example, 4680 has distinct prime divisors 2, 3, 5 and 13. The gap is before the largest prime 13. All primes up to and including the second largest prime are a divisor of a(n).

P. Erdős, On Highly composite numbers, J. London Math. Soc. 19 (1944), 130--133 MR7,145d; Zentralblatt 61,79.
Vladimir Shevelev, On Erdős constant, arXiv:1605.08884 [math.NT], 2016.

Cf. A067128, A273379.

A309943 Numbers k such that k * d(k) > j * d(j) for all j < k, where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, 72, 84, 90, 96, 108, 120, 144, 168, 180, 210, 216, 240, 288, 300, 336, 360, 420, 480, 504, 540, 600, 630, 660, 672, 720, 840, 1008, 1080, 1200, 1260, 1440, 1680, 1980, 2016, 2100, 2160, 2520, 3120

Offset: 1

Amiram Eldar, Aug 24 2019

nonn

Differs from A002093 for n >= 41.

Nicolas asks if there are infinitely many terms of this sequence that are not largely composite (A067128).

Amiram Eldar, Table of n, a(n) for n = 1..457
Jean-Louis Nicolas, Nombres hautement composés, Acta Arithmetica, Vol. 49 (1988), pp. 395-412, alternative link. See p. 411.

Cf. A000005, A002093, A038040, A067128.

s = {}; dm = 0; Do[d1 = n * DivisorSigma[0, n]; If[d1 > dm, dm = d1; AppendTo[s, n]], {n, 1, 10^4}]; s

A352881 a(n) is the minimal number z having the largest number of solutions to the Diophantine equation 1/z = 1/x + 1/y such that 1 <= x <= y <= 10^n.

Original entry on oeis.org

2, 12, 60, 840, 9240, 55440, 720720, 6126120, 116396280, 232792560, 5354228880, 26771144400, 465817912560, 4813451763120, 24067258815600, 144403552893600, 2671465728531600, 36510031623265200, 219060189739591200, 4709794079401210800, 18839176317604843200, 221360321731856907600

Offset: 1

Darío Clavijo, Apr 06 2022

nonn

Solving for z gives z = (x*y) / (x+y), so x*y == 0 (mod x+y).
All known terms are from A025487:
  a(1) =    2 = 2;
  a(2) =   12 = 2^2 * 3;
  a(3) =   60 = 2^2 * 3 * 5;
  a(4) =  840 = 2^3 * 3 * 5 * 7;
  a(5) = 9240 = 2^3 * 3 * 5 * 7 * 11.
If a solution to the equation 1/z = 1/x + 1/y is found such that gcd(x,y,z) is a square, then x+y, x*y*z, and (x-y)^2 + (2*z)^2 are also squares.
For all solutions, x^2 + y^2 + z^2 is a square.
The sequence is indeed a subsequence of A025487, and likely of A126098 as well. - Max Alekseyev, Mar 01 2023
a(n) < 5*10^(n-1). - Max Alekseyev, Mar 01 2023

			For n=1, we have the following, where r = (x*y) mod (x+y). (In the last four columns, each number marked by an asterisk is a square.)
.
  r  z  x  y  x*y  x+y  x*y*z  x^2+y^2+z^2
  -  -  -  -  ---  ---  -----  -----------
  0  1  2  2    4*   4*     4*           9* (solution)
  2  1  2  4    8    6      8           21
  4  1  2  6   12    8     12           41
  6  1  2  8   16*  10     16*          69
  3  1  3  3    9    6      9*          19
  0  2  3  6   18*   9*    36*          49* (solution)
  3  2  3  9   27   12     54           94
  0  2  4  4   16*   8     32           36* (solution)
  8  2  4  8   32   12     64*          84
  5  2  5  5   25*  10     50           54
  0  3  6  6   36*  12    108           81* (solution)
  7  3  7  7   49*  14    147          107
  0  4  8  8   64*  16*   256*         144* (solution)
  9  4  9  9   81*  18    324*         178
.
z = 2 has the largest number of solutions, so a(1) = 2.
The number of solutions for the resulting z cannot exceed A018892(z).

Max Alekseyev, Table of n, a(n) for n = 1..30
Project Euler, Diophantine reciprocals I, Problem 108.

Cf. A025487, A099829, A126098, A067128, A018892, A126098, A225206.

a(n)=my(bc=0,bk=0,L=10^n);for(k=1,L-1,my(c=0,k2=k^2);for(d=max(1,k2\(L-k)+1),k,if(k2%d==0,c++););if(c>bc,bc=c;bk=k););return(bk); \\ Darío Clavijo, Mar 03 2025

def a(n):
    # k=x*y and d=x+y
    bc, bk, L = 0, None, 10**n
    for k in range(1, L):
        c, k2 = 0, k * k
        for d in range(max(1, k2 // (L - k) + 1), k + 1):
            if k2 % d == 0: c += 1
        if c > bc:
            bc, bk = c, k
    return bk

a(6) from Chai Wah Wu, Apr 10 2022

a(7)-a(22) from Max Alekseyev, Mar 01 2023

A273235 Number of Ramanujan's largely composite numbers having prime(n) as the greatest prime divisor.

Original entry on oeis.org

3, 10, 17, 28, 27, 43, 44, 69, 68, 58, 97, 97, 125, 164, 201, 185, 162, 254, 263, 313, 491, 434, 466, 417, 309, 358, 510, 633, 935, 1148, 454

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 18 2016

Theorem. The sequence is unbounded.
Proof. Since the sequence of highly composite numbers (A002182) is a subsequence of this sequence, it is sufficient to prove that the number M_n of highly composite numbers with the maximal prime divisor p_n is unbounded. Let N be a large highly composite number. Then for the greatest prime divisor p_N of N we have [Erdos] p_N=O(log N). So for all N<=x, p_N=O(log x).
If M_n=O(1), then the number of all highly composite numbers <=x is O(p_n)=O(log x). However, Erdos [Erdos] proved that this number is more than (log x)^(1+c) for a certain c>0.
So we have a contradiction. This means that M_n and this sequence are unbound. QED

P. Erdős, On Highly composite numbers, J. London Math. Soc. 19 (1944), 130--133 MR7,145d; Zentralblatt 61,79.

Cf. A067128, A273015, A273016, A273018, A273057.

A308574 Numbers between a pair of consecutive highly abundant numbers (A002093) having the same sum of divisors as the lesser one.

Original entry on oeis.org

672, 2016, 69300, 146160, 207900, 1627920, 8316000, 9828000, 38253600, 60147360, 105814800, 158004000, 726818400, 95935039200, 191870078400, 2206505901600, 3463953292800, 3800093497200, 4413011803200, 7600186994400, 8826023606400

Offset: 1

Amiram Eldar, Jun 08 2019

Define "largely abundant numbers" to be numbers k such that sigma(k) >= sigma(j) for all j < k. This sequence gives all the largely abundant numbers that are not highly abundant numbers.
Analogous to A244353 as A002093 is analogous to A002182.
No more terms below 10^10.
a(22) > 10^13. - Giovanni Resta, Jul 02 2019

			672 is in the sequence since 660 < 672 < 720, (660, 720) are a pair of consecutive highly abundant numbers, and sigma(672) = sigma(660) = 2016.

Cf. A002093, A002182, A067128, A244353.

s={}; sm=0; Do[s1=DivisorSigma[1,n]; If[s1==sm, AppendTo[s,n]]; If[s1>sm, sm=s1], {n,1,10^5}]; s

a(14)-a(21) from Giovanni Resta, Jul 02 2019

A369183 a(n) = n - A329004(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 3, 4, 5, 0, 1, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6

Offset: 1

Zhicheng Wei, Jan 15 2024

nonn

			a(15) = 3 because the largest Ramanujan's largely composite number below 15 is 12, and 15-12=3.

Cf. A067128, A329004.

dmax = 0; nmax = 1; seq = {}; Do[If[(d = DivisorSigma[0, n]) >= dmax, dmax = d; nmax = n]; AppendTo[seq,n- nmax], {n, 1, 102}];seq (* James C. McMahon, Jan 28 2024 *)

cp's OEIS Frontend

A216365 Numbers n such that tau(n)*sigma(n) sets a new record.

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A226462 Smallest number m such that A226460(m) = n.

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A272984 Positions of zero terms in A272314.

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A273415 The smallest term of A273379 having n primes between two consecutive prime divisors.

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A309943 Numbers k such that k * d(k) > j * d(j) for all j < k, where d(k) is the number of divisors of k (A000005).

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A352881 a(n) is the minimal number z having the largest number of solutions to the Diophantine equation 1/z = 1/x + 1/y such that 1 <= x <= y <= 10^n.

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A273235 Number of Ramanujan's largely composite numbers having prime(n) as the greatest prime divisor.

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A308574 Numbers between a pair of consecutive highly abundant numbers (A002093) having the same sum of divisors as the lesser one.

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Mathematica

Extensions

A369183 a(n) = n - A329004(n).

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