A002201 -id:A002201 - cp's OEIS frontend

A337993 Numbers k such that L(k) < sigma(k) + k/Pi^2, where L(k) = floor(H(k) + exp(H(k)) * log(H(k))) and H(k) = Sum_{j=1..k} 1/j.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 60, 120, 360, 2520, 5040

Offset: 1

Peter Luschny, Oct 15 2020

Conjecture: This sequence is finite.

S. Ramanujan, Highly composite numbers, Proc. Lond. Math. Soc. 14 (1915), 347-409; reprinted in Collected Papers, Ed. G. H. Hardy et al., Cambridge 1927; Chelsea, NY, 1962.

J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, arXiv:math/0008177 [math.NT], 2000-2001; Am. Math. Monthly 109 (#6, 2002), 534-543.
Peter Luschny, Illustration: Bounds for A057641.
S. Ramanujan, Highly Composite Numbers, Proc. London Math. Soc. 2, 14, 1915, page 15.

Cf. A000203, A002201, A002388, A004490, A057640, A057641.

Cf. A001008, A002805.

A337993Q[n_] := With[{h = HarmonicNumber[n]}, Floor[h + Exp[h]*Log[h]] < DivisorSigma[1, n] + n/Pi^2];
Select[Range[5040], A337993Q] (* Paolo Xausa, Feb 01 2024 *)

k is a term of this sequence <==> A057640(k) < A000203(k) + k/A002388.

A340137 Numbers k in A305056 such that k*A002110(j) is in A004490.

Original entry on oeis.org

1, 2, 4, 12, 24, 48, 144, 720, 1440, 10080, 30240, 60480, 302400, 3326400, 6652800, 19958400, 259459200, 518918400, 3632428800, 61751289600, 1173274502400, 3519823507200, 17599117536000, 35198235072000, 809559406656000, 1619118813312000, 46954445586048000

Offset: 1

Michael De Vlieger, Jan 08 2021

nonn

All terms are in A025487, since all terms m in A004490 are products of primorials P in A002110.

Let Q = A002110(A001221(m)) be the largest primorial divisor Q | m. The terms in this sequence are the primitive quotients k = m/Q for m in A004490.

			a(1) = 1 since there are 2 colossally abundant numbers m that are primorials P, i.e., 2 and 6.
a(2) = 2 since 2 colossally abundant numbers m = 2P, i.e., 12 and 60.
a(3) = 4 since 120 = 4*30 is colossally abundant.
a(4) = 12 since 360 and 2520 = 12P, etc.
Table showing products of primorials in the column heading and terms in this sequence in the row headings that appear in A004490 (and in these cases, also A002201, thereby in their intersection, A224078).
          2   6   30    210    2310    30030      510510
  ------------------------------------------------------
    1:    2   6
    2:       12   60
    4:           120
   12:           360   2520
   24:                 5040   55440   720720
   48:                               1441440
  144:                               4324320
  720:                              21621600   367567200   ...
Textual plot of numbers at (n,k) where row n = a(n) and column k = A002110(k), marking terms (x) in A224078, (*) only in A004490, or (.) only in A002201.
   1: xx
   2:  xx
   3:   x
   4:   xx
   5:    xxx
   6:      x
   7:      x
   8:      xxx*
   9:        .x**
  10:         ..*
  11:          .x***
  12:           ...xx**
  13:               ..x****
  14:                     **
  15:                 ..   **
  16:                  .....***
  17:                      ...**********
  18:                        .....     ***
  19:                            ...     ****
  20:                              .....    ********
The largest term in A224078 = 581442729886633902054768000 = a(13)*A002110(17), so appears at (13,17).

Michael De Vlieger, Table of n, a(n) for n = 1..144
Michael De Vlieger, Annotated color-coded plot (x,y) = (a(n), A002110(j)) highlighting colossally abundant numbers in red. This sequence also can portray many but not all superior highly composite numbers (shown in blue). Terms in A224078 appear in black.
Michael De Vlieger, Simple extended color-coded plot (x,y) = (a(n), A002110(m)) showing 1000 terms of A004490 in red.

Cf. A002110, A004490, A025487, A073751, A301416, A305056, A307327.

Block[{s = Import["https://oeis.org/A073751/b073751.txt", "Data"][[All, -1]], a = 1, b = {}, k, m = 0}, Do[k = a*s[[i]]; If[# > m, m++] &@ PrimePi@ s[[i]]; Set[a, k]; AppendTo[b, k/Product[Prime[j], {j, m}]], {i, 120}]; Union@ b]

A160274 Highly composite numbers A002182(n) with the property that A002182(n+1)/A002182(n) >= A002182(k+1)/A002182(k) for all k>n.

Original entry on oeis.org

1, 2, 6, 12, 60, 360, 2520

Offset: 1

Anonymous, May 07 2009

			2520 is a term of this sequence because 2520 is a highly composite number (A002182(18)), A002182(19)/A002182(18) = 2, and 2 >= A002182(k+1)/A002182(k) for all k>18. (In fact, 2 > A002182(k+1)/A002182(k) for all k>18.)

Cf. A002182, A002201, A072938, A106037, A094348, A003418, A002110.

A309811 (sigma, tau)-superchampion numbers: numbers k for which there is a positive exponent e such that sigma(k)/(ktau(k)^e) >= sigma(j)/(jtau(j)^e) for all j >= 1, where tau(k) is the number of divisors of k (A000005) and sigma(k) is their sum (A000203).

Original entry on oeis.org

1, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 2162160, 4324320, 73513440, 367567200, 6983776800, 160626866400, 321253732800, 9316358251200, 288807105787200, 2021649740510400, 74801040398884800, 224403121196654400, 9200527969062830400, 395622702669701707200

Offset: 1

Amiram Eldar, Aug 25 2019

nonn

Jean-Louis Nicolas, Quelques inégalités effectives entre des fonctions arithmétiques usuelles, Functiones et Approximatio, Vol. 39, No. 2 (2008), pp. 315-334. See section 3.

Cf. A000005, A000203, A002201, A004490.

A332241 Indices of A224078(n) in A025487.

Original entry on oeis.org

2, 4, 6, 13, 17, 27, 55, 67, 138, 264, 314, 406, 582, 1046, 1835, 3609, 16371, 75611, 118893, 342363

Offset: 1

Michael De Vlieger, Feb 07 2020

A224078: Superior highly composite numbers that are colossally abundant. Such numbers are also found in A025487.

Finite and full, since A224078 is finite with 20 terms.

Cf. A002201, A004490, A025487, A224078.

Block[{s = TakeWhile[Import["https://oeis.org/A025487/b025487.txt", "Data"], Length@ # > 0 &][[All, -1]], t = Map[ToExpression[StringSplit[#][[1, -1]] ] &, Rest@ StringSplit[Import["https://oeis.org/A224078/b224078.txt", "Data"], "\n"]]}, Reap[Do[Which[Length@ t == 0, Break[], First[t] == s[[i]], t = Rest@ t; Sow[i]], {i, Length@ s}]][[-1, -1]]]

A353973 Decimal expansion of the sum of the reciprocals of the superior highly composite numbers.

Original entry on oeis.org

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

Offset: 0

Bengt-Göran Persson, May 12 2022

			0.77839341508236478139496374108691827559475820083149...

Cf. A002201, A352418.

Equals Sum_{n>=1} 1/A002201(n).

More terms from Amiram Eldar, Jun 05 2022

A375864 Prime numbers that cannot be written as the sum of a prime number and a superior highly composite number.

Original entry on oeis.org

2, 3, 307, 911, 1201, 1259, 1693, 2179, 2381, 2927, 3191, 3499, 3557, 4201, 4441, 4721, 5573, 6121, 7207, 8219, 8273, 8537, 8627, 8999, 9137, 9203, 9811, 10133, 10357, 11597, 12211, 12343, 13217, 13421, 13921, 15053, 15401, 15551, 15959, 15991, 16411, 16561, 17117, 17207

Offset: 1

Walter Robinson, Aug 31 2024

nonn

			The prime number 37 can be written as the sum of prime number 31 and superior highly composite number 6 and thus is not in this sequence.

Cf. A000040, A002201, A375866.

from sympy import *
SHCN = [2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720]
for x in range(3, 16000, 2):
    waysFound = 0
    if isprime(x):
        iterC = 0
        while iterC < len(SHCN) and SHCN[iterC] < x:
            if isprime(x - SHCN[iterC]):
                waysFound += 1
            iterC += 1
        if waysFound == 0:
            print(x)

cp's OEIS Frontend

A337993 Numbers k such that L(k) < sigma(k) + k/Pi^2, where L(k) = floor(H(k) + exp(H(k)) * log(H(k))) and H(k) = Sum_{j=1..k} 1/j.

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A340137 Numbers k in A305056 such that k*A002110(j) is in A004490.

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A160274 Highly composite numbers A002182(n) with the property that A002182(n+1)/A002182(n) >= A002182(k+1)/A002182(k) for all k>n.

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A309811 (sigma, tau)-superchampion numbers: numbers k for which there is a positive exponent e such that sigma(k)/(k*tau(k)^e) >= sigma(j)/(j*tau(j)^e) for all j >= 1, where tau(k) is the number of divisors of k (A000005) and sigma(k) is their sum (A000203).

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A332241 Indices of A224078(n) in A025487.

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Mathematica

A353973 Decimal expansion of the sum of the reciprocals of the superior highly composite numbers.

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A375864 Prime numbers that cannot be written as the sum of a prime number and a superior highly composite number.

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A309811 (sigma, tau)-superchampion numbers: numbers k for which there is a positive exponent e such that sigma(k)/(ktau(k)^e) >= sigma(j)/(jtau(j)^e) for all j >= 1, where tau(k) is the number of divisors of k (A000005) and sigma(k) is their sum (A000203).