A005934 -id:A005934 - cp's OEIS frontend

A364990 Coreful triperfect numbers: numbers k such that csigma(k) = 3*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

3600, 25200, 28224, 39600, 46800, 61200, 68400, 82800, 104400, 111600, 133200, 141120, 147600, 154800, 169200, 190800, 212400, 219600, 241200, 255600, 262800, 277200, 284400, 298800, 310464, 320400, 327600, 349200, 363600, 366912, 370800, 385200, 392400, 406800

Offset: 1

Amiram Eldar, Aug 15 2023

nonn

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947) (the term "coreful divisor" was used by Hardy and Subbarao, 1983).
If k is a term, then also m*k is, for any squarefree m coprime to k. Thus there are infinitely many coreful triperfect numbers, and all of them can be generated from the sequence of primitive terms, which is the subsequence of powerful terms of this sequence. This sequence is c(n) = A064549(A005820(n)), i.e., the triperfect numbers (A005820), multiplied by their squarefree kernel (A007947): 3600, 28224, 1071645696, 1651818858099200, 532098668445696, 317519577357516800, ...
The asymptotic density of this sequence is Sum_{i>=1} beta(c(i))/c(i), where beta(n) = (6/Pi^2) * Product_{p|n} (p/(p+1)) = 0.0000797856... . If there are only 6 triperfect numbers, then the exact value of this density is 18575679807276818039685539/(23589576231586703755916083200 * Pi^2).

			3600 is in the sequence since its coreful divisors are {30, 60, 90, 120, 150, 180, 240, 300, 360, 450, 600, 720, 900, 1200, 1800, 3600}, whose sum is 10800 = 3 * 3600.

Amiram Eldar, Table of n, a(n) for n = 1..10000
G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer., Vol. 37 (1983), pp. 277-307. (Annotated scanned copy)

Cf. A005820, A007947, A057723, A064549, A307958.

f[p_, e_] := (p^(e+1)-1)/(p-1)-1; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[500000], s[#] == 3*# &]

s(n) = {my(f = factor(n)); prod(i = 1, #f~, sigma(f[i, 1]^f[i, 2]) - 1);}
is(n) = s(n) == 3*n;

A379593 Numbers that set records in A379592.

Original entry on oeis.org

8, 32, 128, 512, 2048, 8192, 20736, 41472, 82944, 165888, 186624, 373248, 746496, 1492992, 2985984, 5971968, 6718464, 11943936, 23887872, 26873856, 53747712, 107495424, 214990848, 241864704, 429981696, 859963392, 967458816, 1719926784, 3439853568, 3869835264, 7464960000

Offset: 1

Michael De Vlieger, Dec 30 2024

nonn

Proper subset of the intersection of A025487 and A320966.
Let k be a powerful number (in A001694) and let coreful d | k be such that k/d is also coreful, i.e., rad(d) = rad(d/k) = rad(k), where rad = A007947 is the squarefree kernel. Suppose d < d/k. Then coreful d may either divide k/d or not (indeed, if d/k < d, k/d may either divide d or not).
Then we have either d | k/d (the cardinality of such divisors is A379592(n) for k = A320966(n)) or d does not divide k/d (the cardinality of such divisors is A379552(n) for k = A376936(n)). (The case d = k/d, both certainly coreful, of course pertains to perfect squares k in A000290.)
Coreful divisors are counted by A361430 across natural numbers, and A370329 across powerful numbers A001694. Numbers that set records in A361430 (and A370329) are in A005934 (highly powerful numbers), with records in A036965.

			Let b(n) = A379592(n).
Table showing exponents of prime power factors of a(n) for n = 1..12. Example: a(7) = 20736 = 2^8*3^4, so "8.4" appears in the "exp." column.
   n      a(n)  exp. b(a(n))
  --------------------------
   1        8    3       1   2*4
   2       32    5       2   2*16 = 4*8
   3      128    7       3   2*64 = 4*32 = 8*16
   4      512    9       4   2*256 = 4*128 = 8*64 = 16*32
   5     2048   11       5   2*1024 = 4*512 = 8*256 = 16*128 = 32=64
   6     8192   13       6   2*4096 = 4*2048 = 8*1024 = 16*512 = 32*256 = 64*128
   7    20736    8.4     7
   8    41472    9.4     8
   9    82944   10.4     9
  10   165888   11.4    10
  11   186624    8.6    11
  12   373248    9.6    12

Michael De Vlieger, Table of n, a(n) for n = 1..119
Michael De Vlieger, Prime power decomposition of a(n), n = 1..119.

Cf. A005934, A025487, A320966, A361430, A370329, A379592, A379594.

(* Load function f at A025487 *)
r = 0; s = Union@ Flatten@ f[10]; nn = Length[s];
rad[x_] := Times @@ FactorInteger[x][[All, 1]];
  Transpose@ Reap[Monitor[
    Do[k = s[[i]];
      If[# > r, r = #; Sow[k]] &@
        Count[Transpose@ {#, k/#} &@ #[[2 ;; Ceiling[Length[#]/2]]] &@ Divisors[k],
          _?(And[rad[#1] == rad[#2],
            Xor[Divisible[#2, #1],
                Divisible[#1, #2]]] & @@ # &)], {i, nn}], {i, nn}] ][[-1, 1]]

A085627 Number of divisors of n-th highly powerful number.

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 15, 16, 18, 20, 24, 25, 28, 30, 32, 35, 36, 40, 42, 45, 48, 50, 54, 96, 96, 105, 108, 112, 120, 120, 128, 135, 140, 144, 160, 168, 180, 192, 200, 200, 216, 224, 225, 240, 240, 252, 264, 270, 280, 280, 300, 315, 330, 576, 560, 360, 600, 576, 648, 640, 675, 672

Offset: 1

Jason Earls, Jul 10 2003

nonn

560 is the first term that is less than the preceding term. - David Wasserman, Feb 03 2005

Amiram Eldar, Table of n, a(n) for n = 1..609

Cf. A000005, A005934.

a = {1}; b = {1}; f[n_] := Times @@ Last /@ FactorInteger[n]; Do[If[f@ n > Max[b], And[AppendTo[b, f@ n], AppendTo[a, n]]], {n, 1000000}]; DivisorSigma[0, #] &@ a (* Michael De Vlieger, Aug 28 2015 *)

{prdex(n)=local(s, fac); s=1; fac=factor(n); for(k=1,matsize(fac)[1],s=s*fac[k,2]); return(s)} {dhp(m)=local(rec); rec=0; for(n=1,m,if(prdex(n)>rec,rec=prdex(n); print1(numdiv(n)",")))}

a(n) = A000005(A005934(n)). - Amiram Eldar, Jul 01 2019

More terms from David Wasserman, Feb 03 2005

Two missing terms added by Walter Roscello, Aug 28 2015

A192636 Powerful sums of two powerful numbers.

Original entry on oeis.org

8, 9, 16, 25, 32, 36, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000, 1024, 1089, 1125, 1152, 1156, 1225

Offset: 1

Charles R Greathouse IV, Jul 06 2011

nonn

Browning & Valckenborgh conjecture that a(n) ~ kn^2 with k approximately 0.139485255. See their Conjecture 1 and equation (14). Their Theorems 1 and 2 establish upper and lower asymptotic bounds.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Charles R Greathouse IV)
Tim D. Browning and K. Van Valckenborgh, Sums of three squareful numbers, Experimental Mathematics, Vol. 21, No. 2 (2012), pp. 204-211; arXiv preprint, arXiv:1106.4472 [math.NT], 2011.

Subsequence of A001694 and of A076871.

Cf. A001694, A007532, A005934, A005188, A003321, A014576, A023052, A046074, A013929, A076871, A143813. - Jonathan Vos Post, Jul 10 2011

With[{m = 1225}, pow = Select[Range[m], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]; Intersection[pow, Plus @@@ Tuples[pow, {2}]]] (* Amiram Eldar, Feb 12 2023 *)

isPowerful(n)=if(n>3,vecmin(factor(n)[,2])>1,n==1)
sumset(a,b)={
  my(c=vectorsmall(#a*#b));
  for(i=1,#a,
    for(j=1,#b,
      c[(i-1)*#b+j]=a[i]+b[j]
    )
  );
  vecsort(c,,8)
}; selfsum(a)={
  my(c=vectorsmall(binomial(#a+1,2)),k);
  for(i=1,#a,
    for(j=i,#a,
      c[k++]=a[i]+a[j]
    )
  );
  vecsort(c,,8)
};
list(lim)={
  my(v=select(isPowerful, vector(floor(lim),i,i)));
  select(n->n<=lim && isPowerful(n), Vec(selfsum(v)))
};

Numbers k such that there exists some a, b, c with A001694(a) + A001694(b) = k = A001694(c).

Corrected (on the advice of Donovan Johnson) by Charles R Greathouse IV, Sep 25 2012

A335853 Numbers that are highly powerful in Gaussian integers.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 100, 200, 400, 500, 800, 1000, 2000, 4000, 5000, 8000, 10000, 18000, 20000, 27000, 36000, 40000, 50000, 54000, 80000, 90000, 108000, 135000, 180000, 216000, 270000, 450000, 540000, 810000, 1080000, 1350000, 1620000, 2160000, 2700000

Offset: 1

Amiram Eldar, Jun 26 2020

nonn

Numbers with a record value of the product of the exponents in the prime factorization in Gaussian integers (A335852). Equivalently, numbers with a record number of powerful divisors in Gaussian integers.

The corresponding record values are 1, 2, 4, 6, 8, 10, 12, 16, 24, 32, 36, 40, 54, 72, 90, 96, ... (see the link for more values).

			The factorization of 1, 2, 3 and 4 in Gaussian integers are 1, -i*(1+i)^2, 3 and -(1+i)^4, and the corresponding products of the exponents are 1, 2, 1 and 4. The record values, 1, 2 and 4, occur at 1, 2 and 4 that are the first 3 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..81
Amiram Eldar, Table of n, a(n), A335852(a(n)) for n = 1..81

Cf. A005934, A279254, A335851, A335852.

With[{s = Array[Times @@ FactorInteger[#, GaussianIntegers -> True][[All, -1]] &, 10^5]}, Map[FirstPosition[s, #][[1]] &, Union@FoldList[Max, s]]] (* after Michael De Vlieger at A005934 *)

A349111 Powerful superabundant numbers: numbers m such that psigma(m)/m > psigma(k)/k for all k < m, where psigma(k) is the sum of powerful divisors of k (A183097).

Original entry on oeis.org

1, 4, 8, 16, 32, 64, 128, 144, 216, 432, 864, 1296, 1728, 2592, 5184, 10368, 15552, 31104, 54000, 108000, 162000, 216000, 324000, 648000, 1296000, 1944000, 3240000, 3888000, 6480000, 9720000, 19440000, 38880000, 58320000, 74088000, 111132000, 222264000, 444528000, 666792000

Offset: 1

Amiram Eldar, Nov 08 2021

nonn

The corresponding record values are 1, 5/4, 13/8, 29/16, 61/32, 125/64, ...

The least term k with psigma(k)/k > m, for m = 2, 3, ..., is 144, 54000, 666792000, ...

Subsequence of A349112.
Cf. A005934, A183097.
Similar sequences: A002110 (unitary), A037992 (infinitary), A061742, A292984, A329882, A348273.

f[p_,e_] := (p^(e+1)-1)/(p-1) - p; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; seq = {}; rm = 0; Do[r1 = s[n]/n; If[r1 > rm, rm = r1; AppendTo[seq, n]], {n, 1, 10^6}]; seq

A363330 Numbers with a record number of divisors that are both coreful and infinitary.

Original entry on oeis.org

1, 8, 128, 216, 3456, 27000, 279936, 432000, 9261000, 34992000, 148176000, 8957952000, 12002256000, 197222256000, 3072577536000, 7501410000000, 15975002736000, 433297296432000, 1920360960000000, 4089600700416000, 9984376710000000, 35097081010992000, 2128789617370416000

Offset: 1

Amiram Eldar, May 28 2023

nonn

Indices of records in A363329.

The corresponding record values are 1, 3, 7, 9, 21, 27, 49, 63, 81, 147, 189, 315, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..64
Amiram Eldar, Table of n, a(n), A363329(a(n)) for n = 1..64

Cf. A363329.
Subsequence of A025487.
Similar sequences: A005934, A037992.

f[p_, e_] := 2^DigitCount[e, 2, 1] - 1; d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n];
v = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]];
seq = {}; dm = 0; Do[If[(dk = d[v[[k]]]) > dm, dm = dk; AppendTo[seq, v[[k]]]], {k, 1, Length[v]}]; seq

A363333 Numbers with a record number of divisors that are both coreful and bi-unitary.

Original entry on oeis.org

1, 8, 32, 128, 216, 864, 3456, 7776, 13824, 31104, 108000, 279936, 432000, 972000, 1728000, 3888000, 15552000, 34992000, 62208000, 97200000, 139968000, 248832000, 333396000, 559872000, 592704000, 874800000, 1333584000, 5334336000, 12002256000, 21337344000, 33339600000

Offset: 1

Amiram Eldar, May 28 2023

nonn

Indices of records in A363332.

The corresponding record values are 1, 3, 5, 7, 9, 15, 21, 25, 27, 35, 45, 49, 63, 75, 81, ... (see the link for more values).

Amiram Eldar, Table of n, a(n) for n = 1..261
Amiram Eldar, Table of n, a(n), A363332(a(n)) for n = 1..261

Cf. A363332.
Subsequence of A025487.
Similar sequences: A005934, A293185.

f[p_, e_] := If[OddQ[e], e, e - 1]; d[1] = 1; d[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 120]
v = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]];
seq = {}; dm = 0; Do[If[(dk = d[v[[k]]]) > dm, dm = dk; AppendTo[seq, v[[k]]]], {k, 1, Length[v]}]; seq

A141422 Positions of highly powerful numbers in the EKG sequence.

Original entry on oeis.org

1, 3, 8, 17, 31, 64, 122, 136, 199, 270, 411, 817, 1234, 1649, 2484, 3321, 4981, 7480, 9993

Offset: 1

Parthasarathy Nambi, Aug 05 2008

nonn

			10368 is a highly powerful number located at position 9993 in the EKG sequence.

Cf. A005934, A064413.

cp's OEIS Frontend

A364990 Coreful triperfect numbers: numbers k such that csigma(k) = 3*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

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A379593 Numbers that set records in A379592.

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A085627 Number of divisors of n-th highly powerful number.

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A192636 Powerful sums of two powerful numbers.

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A335853 Numbers that are highly powerful in Gaussian integers.

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A349111 Powerful superabundant numbers: numbers m such that psigma(m)/m > psigma(k)/k for all k < m, where psigma(k) is the sum of powerful divisors of k (A183097).

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A363330 Numbers with a record number of divisors that are both coreful and infinitary.

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A363333 Numbers with a record number of divisors that are both coreful and bi-unitary.

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