A004490 -id:A004490 - cp's OEIS frontend

A073751 Prime numbers that when multiplied in order yield the sequence of colossally abundant numbers A004490.

2, 3, 2, 5, 2, 3, 7, 2, 11, 13, 2, 3, 5, 17, 19, 23, 2, 29, 31, 7, 3, 37, 41, 43, 2, 47, 53, 59, 5, 61, 67, 71, 73, 11, 79, 2, 83, 3, 89, 97, 13, 101, 103, 107, 109, 113, 127, 131, 137, 139, 2, 149, 151, 7, 157, 163, 167, 17, 173, 179, 181, 191, 193, 197, 199, 19, 211, 3

Offset: 1

T. D. Noe, Aug 07 2002

nonn

The Mathematica program presents a very fast method of computing the factors of colossally abundant numbers. The 100th number has a sigma(n)/n ratio of 10.5681.
This calculation assumes that the ratio of consecutive colossally abundant numbers is always prime, which is implied by a conjecture mentioned in Lagarias's paper.
The ratio of consecutive colossally abundant numbers is prime for at least the first 10^7 terms. The (10^7)-th term is a 77908696-digit number which has a sigma(n)/n value of 33.849.
Alaoglu and Erdős's paper proves that the quotient of two consecutive colossally abundant numbers is either a prime or the product of two distinct primes.
From Robert G. Wilson v, May 30 2014: (Start)
First occurrence of the n-th prime: 1, 2, 4, 7, 9, 10, 14, 15, 16, 18, 19, 22, 23, 24, 26, 27, 28, 30, 31, 32, ..., .
Positions of 2: 1, 3, 5, 8, 11, 17, 25, 36, 51, 77, 114, 178, 282, 461, 759, 1286, 2200, 3812, 6664, ..., .
Positions of 3: 2, 6, 12, 21, 38, 68, 132, 271, 595, 1356, 3191, 7775, ..., . (End)

T. D. Noe, Table of n, a(n) for n = 1..10000
L. Alaoglu and P. Erdos, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469. Errata
Keith Briggs, Abundant numbers and the Riemann Hypothesis, Experimental Math., Vol. 16 (2006), p. 251-256.
Young Ju Choie; Nicolas Lichiardopol; Pieter Moree; Patrick Solé, On Robin's criterion for the Riemann hypothesis, Journal de théorie des nombres de Bordeaux, 19 no. 2 (2007), pp. 357-372.
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.
T. Schwabhäuser, Preventing Exceptions to Robin's Inequality, arXiv preprint arXiv:1308.3678 [math.NT], 2013.
Eric Weisstein's World of Mathematics, Colossally Abundant Number

Cf. A004490.

pFactor[f_List] := Module[{p=f[[1]], k=f[[2]]}, N[Log[(p^(k+2)-1)/(p^(k+1)-1)]/Log[p]]-1]; maxN=100; f={{2, 1}, {3, 0}}; primes=1; lst={2}; x=Table[pFactor[f[[i]]], {i, primes+1}]; For[n=2, n<=maxN, n++, i=Position[x, Max[x]][[1, 1]]; AppendTo[lst, f[[i, 1]]]; f[[i, 2]]++; If[i>primes, primes++; AppendTo[f, {Prime[i+1], 0}]; AppendTo[x, pFactor[f[[ -1]]]]]; x[[i]]=pFactor[f[[i]]]]; lst

A189228 Superabundant numbers (A004394) that are not colossally abundant (A004490).

Original entry on oeis.org

1, 4, 24, 36, 48, 180, 240, 720, 840, 1260, 1680, 10080, 15120, 25200, 27720, 110880, 166320, 277200, 332640, 554400, 665280, 2162160, 3603600, 7207200, 8648640, 10810800, 36756720, 61261200, 73513440, 122522400, 147026880, 183783600, 698377680, 735134400

Offset: 1

Jonathan Sondow, Jun 07 2011

nonn

All colossally abundant (CA) numbers are also superabundant (SA). (Proof. If n is CA and k < n, then sigma(n)/n = n^{epsilon}*sigma(n)/n^{1+epsilon} >= n^{epsilon}*sigma(k)/k^{1+epsilon} > k^{epsilon}*sigma(k)/k^{1+epsilon} = sigma(k)/k, and so n is SA.)

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A112974 (Number of superabundant numbers between two consecutive colossally abundant numbers) and A166735 (Superabundant numbers that are not highly composite).

A342012 Primorial deflation of the n-th colossally abundant number: the unique integer k such that A108951(k) = A004490(n).

Original entry on oeis.org

2, 3, 6, 10, 20, 30, 42, 84, 132, 156, 312, 468, 780, 1020, 1140, 1380, 2760, 3480, 3720, 5208, 7812, 9324, 10332, 10836, 21672, 23688, 26712, 29736, 49560, 51240, 56280, 59640, 61320, 96360, 104280, 208560, 219120, 328680, 352440, 384120, 453960, 472680, 482040, 500760, 510120, 528840, 594360, 613080, 641160, 650520, 1301040

Offset: 1

Antti Karttunen, Mar 08 2021

nonn

In contrast to A329902, this sequence is monotonic, because each term is obtained from the previous, either by multiplying it by 2, or by "bumping" one [or hypothetically: two] of its prime factors one step up (i.e., replacing it with the next larger prime), and both operations are guaranteed to make the number larger.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A004490, A005940, A073751, A108951, A319626, A329900, A342010 [= A001222(a(n))], A342011, A342013.

Cf. also A217867, A329902.

v073751 = readvec("b073751_to.txt");
A073751(n) = v073751[n];
A004490list(v073751) = { my(v=vector(#v073751)); v[1] = 2; for(n=2,#v,v[n] = v073751[n]*v[n-1]); (v); };
v004490 = A004490list(v073751);
A004490(n) = v004490[n];
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A319626(n) = (n / gcd(n, A064989(n)));
A342012(n) = A319626(A004490(n));

a(n) = A319626(A004490(n)) = A329900(A004490(n)).

a(n) = A005940(1+A342013(n)).

A342011 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), for all i, j >= 1, with f(1) = 2 and f(n) = A004490(n)/A004490(n-1) when n > 1, where A004490(n) is the n-th colossally abundant number.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 4, 1, 5, 6, 1, 2, 3, 7, 8, 9, 1, 10, 11, 4, 2, 12, 13, 14, 1, 15, 16, 17, 3, 18, 19, 20, 21, 5, 22, 1, 23, 2, 24, 25, 6, 26, 27, 28, 29, 30, 31, 32, 33, 34, 1, 35, 36, 4, 37, 38, 39, 7, 40, 41, 42, 43, 44, 45, 46, 8, 47, 2, 48, 49, 50, 3, 51, 52, 53, 54, 1, 55, 56, 57, 58, 59, 60, 61, 62, 9, 63, 64

Offset: 1

Antti Karttunen, Mar 08 2021

nonn

This is also the restricted growth sequence transform of A073751, provided that quotient A004490(1+n)/A004490(n) is always prime, which is implied by a conjecture mentioned in Lagarias' paper. Note that the b-file of A073751 is computed based on the knowledge that the conjecture holds at least for the first 10^7 quotients.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (computed from the b-file of A073751 provided by T. D. Noe)
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.

Cf. A000720, A004490, A073751, A342010, A342012, A342013.

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
v073751 = readvec("b073751_to.txt"); \\ Prepared with gawk '{ print $2 }' < b073751.txt > b073751_to.txt
v342011 = rgs_transform(v073751);
A342011(n) = v342011[n];
for(n=1,#v342011,write("b342011.txt", n, " ", A342011(n)));

a(n) = A000720(A073751(n)), up to the first n where A004490(n)/A004490(n-1) is not a prime.

A353561 Prime shadows of colossally abundant numbers, A004490.

Original entry on oeis.org

2, 4, 6, 12, 20, 30, 60, 84, 168, 336, 528, 880, 1320, 2640, 5280, 10560, 12480, 24960, 49920, 74880, 104832, 209664, 419328, 838656, 1096704, 2193408, 4386816, 8773632, 14622720, 29245440, 58490880, 116981760, 233963520, 350945280, 701890560, 784465920, 1568931840, 2465464320, 4930928640, 9861857280, 14792785920

Offset: 1

Antti Karttunen, Apr 29 2022

nonn

Sequence is injective (no duplicate values occur) because A004490 is a subsequence of A025487.

Amiram Eldar, Table of n, a(n) for n = 1..3367

Cf. A004490, A025487, A122111, A181819, A342012.

Cf. also A353562, A353568.

v004490 = readvec("b004490_to.txt"); \\ Prepared from b-file of A004490 with gawk ' { print $2 } '
A004490(n) = v004490[n];
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A353561(n) = A181819(A004490(n));

a(n) = A181819(A004490(n)).

a(n) = A122111(A342012(n)).

A110443 Index k of the least colossally abundant number c=A004490(k) with sigma(c)/c >= n.

Original entry on oeis.org

2, 5, 9, 14, 19, 27, 40, 56, 82, 119, 180, 275, 426, 673, 1072, 1728, 2807, 4596, 7558, 12497, 20741, 34544, 57704, 96638, 162204, 272802, 459637, 775678, 1310959, 2218651, 3759507, 6377858

Offset: 2

T. D. Noe, Jul 20 2005

For the numbers, instead of the indices, see A110442. The last term, a(33), yields a 48361012-digit number having 6375908 prime factors.

			a(4)=9 because the 9th colossally abundant number is c=55440 and sigma(c)/c = 4.18701 > 4, while the 8th colossally abundant number is c'=5040 and sigma(c')/c' = 3.838 < 4.

Cf. A004490, A110442.

Definition and example clarified by Jonathan Sondow, Dec 08 2011

A328549 1, together with the numbers that are simultaneously superior highly composite (A002201), colossally abundant (A004490), deeply composite (A095848), and miserable average divisor numbers (A263572).

Original entry on oeis.org

1, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440

Offset: 1

N. J. A. Sloane, Oct 20 2019

Presumably there are no further terms.
From Hal M. Switkay, Nov 04 2019: (Start)
1. a(n+1) is the product of the first n terms of A328852.
2. This sequence is most rapidly constructed as the intersection of A095849 and A224078. It is designed to list all potential solutions to a question. Let n be a natural number, k real <= 0, e real > 0. Let P(n,k,e) state: on the domain of natural numbers, sigma_k(x)/x^e reaches a maximum at x = n. This implies Q(n,k): sigma_k(n) > sigma_k(m) for m < n a natural number. We ask: for which natural numbers n is it true for all real k <= 0 that there is a real e > 0 such that P(n,k,e)?
If any such n exist, they must belong to the present sequence. A095849 consists of all natural numbers n such that for all real k <= 0, Q(n,k) holds. A224078 consists of all natural numbers n such that for some real e0 and e1 both > 0, P(n,0,e0) and P(n,-1,e1) hold. It would be interesting to see the list of n for which there is an e2 > 0 such that P(n,-2,e2) holds.
Conjecture: the solutions to this problem, if any, form an initial sequence of the present sequence. (End)
Every term of this sequence is also in A065385: a record for the cototient function. - Hal M. Switkay, Feb 27 2021
Every term of this sequence, except the first, is also in A210594: factor-dense numbers. - Hal M. Switkay, Mar 29 2021

Hal M. Switkay, Email to N. J. A. Sloane, Oct 20 2019

1 together with the intersection of A002201, A004490, A095848, A263572.

Cf. A095849, A224078, A328852, A210594.

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]

A259632 a(n) = floor(exp(H_k)*log(H_k)) - sigma(k) where k is the n-th colossally abundant number (Sequence A079526 applied to the colossally abundant numbers (A004490).)

Original entry on oeis.org

-2, -2, -3, -2, 0, 28, 199, 483, 9040, 143814, 306295, 963844, 5155067, 81053615, 1334916470, 29106956400, 58655156000, 1817551640000, 56466287000000, 376943530000000, 1144451930000000, 41803527000000000

Offset: 1

Gene Ward Smith, Dec 17 2016

sign

It follows easily from the work of Lagarias that the Riemann hypothesis is equivalent to this sequence's being nonnegative except for the first four terms.

J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (#6, 2002), 534-543.

Cf. A004490, A079526.

A279609 a(n) = floor(H(k) + exp(H(k))*log(H(k))) - sigma(k) where H(k) is the k-th harmonic number Sum_{j=1..k} 1/j and k is the n-th colossally abundant number A004490(n).

Original entry on oeis.org

0, 0, 0, 2, 6, 34, 207, 492, 9051, 143828, 306310, 963859, 5155084, 81053635, 1334916490, 29106956400, 58655156200, 1817551636000, 56466287472000, 376943525488000, 1144451930851200, 41803526752345600

Offset: 2

Gene Ward Smith, Dec 15 2016

nonn

By a theorem of J. C. Lagarias, the Riemann hypothesis is equivalent to the proposition that this sequence never takes a negative value. In fact, by inspection it appears to be monotone increasing; this conjecture implies the Riemann hypothesis but is not in any obvious way implied by it. Stronger conjectures are easy to formulate--for example, if F(n) is the function defined by this sequence, then F(n)/2^n also appears to be monotone increasing.

J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (#6, 2002), 534-543.

Cf. A004490, A057641.

cp's OEIS Frontend

A073751 Prime numbers that when multiplied in order yield the sequence of colossally abundant numbers A004490.

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A189228 Superabundant numbers (A004394) that are not colossally abundant (A004490).

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A342012 Primorial deflation of the n-th colossally abundant number: the unique integer k such that A108951(k) = A004490(n).

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A342011 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), for all i, j >= 1, with f(1) = 2 and f(n) = A004490(n)/A004490(n-1) when n > 1, where A004490(n) is the n-th colossally abundant number.

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A353561 Prime shadows of colossally abundant numbers, A004490.

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A110443 Index k of the least colossally abundant number c=A004490(k) with sigma(c)/c >= n.

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A328549 1, together with the numbers that are simultaneously superior highly composite (A002201), colossally abundant (A004490), deeply composite (A095848), and miserable average divisor numbers (A263572).

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

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A259632 a(n) = floor(exp(H_k)*log(H_k)) - sigma(k) where k is the n-th colossally abundant number (Sequence A079526 applied to the colossally abundant numbers (A004490).)

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A279609 a(n) = floor(H(k) + exp(H(k))*log(H(k))) - sigma(k) where H(k) is the k-th harmonic number Sum_{j=1..k} 1/j and k is the n-th colossally abundant number A004490(n).