A349109
Powerful numbers (A001694) whose sum of powerful divisors (including 1) is also powerful.
Original entry on oeis.org
1, 64, 243, 441, 1764, 9800, 15552, 28224, 41616, 60516, 82369, 88200, 189728, 226576, 329476, 336200, 648675, 741321, 968256, 1317904, 1428025, 1707552, 1943236, 2039184, 2056356, 2381400, 2446227, 2798929, 2965284, 2986568, 4372281, 5189400, 5271616, 6508832
Offset: 1
64 = 2^6 is a term since it is powerful and the sum of its powerful divisors, A183097(64) = 1 + 4 + 8 + 16 + 32 + 64 = 125 = 5^3 is also powerful.
-
powQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;;,2]], # > 1 &]; f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - p; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := powQ[n] && powQ[s[n]]; Select[Range[7*10^6], q]
-
isok(n) = ispowerful(n) && ispowerful(sumdiv(n, d, d*ispowerful(d))); \\ Michel Marcus, Nov 08 2021
-
is(k) = {my(f = factor(k)); ispowerful(f) && ispowerful(prod(i = 1, #f~, (f[i,1]^(f[i,2]+1) - 1)/(f[i,1] - 1) - f[i,1]));} \\ Amiram Eldar, Sep 14 2024
A363175
Primitive abundant numbers (A071395) that are powerful numbers (A001694).
Original entry on oeis.org
342225, 570375, 3172468, 4636684, 63126063, 99198099, 117234117, 171991125, 280495504, 319600125, 327921075, 404529741, 581549787, 635689593, 762155163, 1029447225, 1148667664, 1356949503, 1435045924, 1501500375, 1558495125, 1596961444, 1757705625, 1778362047
Offset: 1
-
f1[p_, e_] := (p^(e + 1) - 1)/(p^(e + 1) - p^e); f2[p_, e_] := (p^(e + 1) - p)/(p^(e + 1) - 1);
primAbQ[n_] := (r = Times @@ f1 @@@ (f = FactorInteger[n])) > 2 && r * Max @@ f2 @@@ f < 2;
seq[max_] := Module[{pow = Union[Flatten[Table[i^2*j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}]]]}, Select[Rest[pow], primAbQ]]; seq[10^10]
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isPrimAb(n) = {my(f = factor(n), r, p, e); r = sigma(f, -1); r > 2 && vecmax(vector(#f~, i, p = f[i, 1]; e = f[i, 2]; (p^(e + 1) - p)/(p^(e + 1) - 1))) * r < 2; }
lista(lim) = {my(pow = List(), t); for(j=1, sqrtnint(lim\1, 3), for(i=1, sqrtint(lim\j^3), listput(pow, i^2*j^3))); select(x->isPrimAb(x), Set(pow)); }
A380254
Number of powerful numbers (in A001694) that do not exceed primorial A002110(n).
Original entry on oeis.org
1, 1, 2, 7, 22, 85, 330, 1433, 6450, 31555, 172023, 964560, 5891154, 37807505, 248226019, 1702890101, 12401685616, 95277158949, 744210074157, 6091922351106, 51332717836692, 438592279944173, 3898316990125822, 35515462315592564, 335052677538616216, 3299888425002527366
Offset: 0
Let P = A002110 and let s = A001694.
a(0) = 1 since P(0) = 1, and the set s(1) = {1} contains k that do not exceed 1.
a(1) = 1 since P(1) = 2, and the set s(1) = {1} contains k <= 2.
a(2) = 2 since P(2) = 6, and the set s(1..2) = {1, 4} contains k <= 6.
a(3) = 7 since P(3) = 30, and the set s(1..7) = {1, 4, 8, 9, 16, 25, 27} contains k <= 30.
a(4) = 22 since P(4) = 210, and the set s(1..19) = {1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200} contains k <= 210, etc.
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f[x_] := Sum[If[SquareFreeQ[ii], Floor[Sqrt[x/ii^3]], 0], {ii, x^(1/3)}];
Table[f[#[[k + 1]]], {k, 0, Length[#] - 1}] &[
FoldList[Times, 1, Prime[Range[12] ] ] ] (* function f after Robert G. Wilson v at A118896 *)
-
from math import isqrt
from sympy import primorial, integer_nthroot, mobius
def A380254(n):
def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
if n == 0: return 1
m = primorial(n)
c, l, j = squarefreepi(integer_nthroot(m, 3)[0]), 0, isqrt(m)
while j>1:
k2 = integer_nthroot(m//j**2,3)[0]+1
w = squarefreepi(k2-1)
c += j*(w-l)
l, j = w, isqrt(m//k2**3)
return c-l # Chai Wah Wu, Jan 24 2025
A115675
Brilliant numbers (A078972) whose digit reversal is a powerful(1) number (A001694).
Original entry on oeis.org
4, 9, 10, 121, 169, 869, 961, 1273, 1843, 10201, 10609, 12769, 16171, 44521, 48361, 48613, 70597, 76121, 94249, 96721, 106009, 108853, 121879, 129307, 211591, 255953, 276491, 278699, 291149, 445051, 526567, 613927, 616571, 1026169
Offset: 1
869 = 11*79 is brilliant and 968 = 2^3*11^2 is powerful.
A115676
Powerful(1) numbers (A001694) whose digit reversal is a brilliant number (A078972).
Original entry on oeis.org
4, 9, 121, 169, 400, 900, 961, 968, 3481, 3721, 4000, 9000, 10201, 12100, 12167, 12544, 12769, 16384, 16900, 17161, 31684, 40000, 79507, 90000, 90601, 94249, 96100, 96721, 96800, 121000, 150544, 169000, 175616, 192200, 194672, 195112
Offset: 1
968=2^3*11^2 is powerful and 869=11*79 is brilliant.
A115687
Powerful(1) numbers (A001694) whose digit reversal is a semiprime (A001358).
Original entry on oeis.org
4, 9, 49, 64, 121, 169, 289, 400, 512, 625, 900, 961, 968, 1156, 1225, 1568, 1849, 2048, 2401, 2888, 3125, 3136, 3364, 3481, 3721, 4000, 4900, 4913, 5041, 5329, 5408, 6400, 6859, 6889, 7396, 8192, 8575, 9000, 9604, 10201, 10648, 10816, 10952
Offset: 1
968=2^3*11^2 is powerful and 869=11*79 is semiprime.
A115688
Semiprimes (A001358) whose digit reversal is a powerful(1) number (A001694).
Original entry on oeis.org
4, 9, 10, 46, 94, 121, 169, 215, 526, 869, 961, 982, 1042, 1273, 1405, 1843, 2918, 3194, 4069, 4633, 5213, 5221, 5758, 6313, 6511, 6937, 8045, 8402, 8651, 8882, 9235, 9481, 9586, 9886, 10201, 10609, 12538, 12769, 14023, 16171, 16327, 16582
Offset: 1
869=11*79 is semiprime and 968=2^3*11^2 is powerful.
-
N:= 99999:
S:= {1}:
p:= 1:
do
p:= nextprime(p);
if p^2 > N then break fi;
S:= S union map(t -> seq(t*p^j,j=2..floor(log[p](N/t))), S);
od:
digrev:= proc(x) local L;
L:= convert(x,base,10);
add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
sort(convert({10} union select(t -> numtheory:-bigomega(t)=2, map(digrev, select(t -> t mod 10 <> 0, S))),list)); # Robert Israel, Dec 03 2019
A115691
Powerful(1) numbers (A001694) whose digit reversal is a triangular number.
Original entry on oeis.org
1, 100, 1000, 5400, 10000, 54000, 100000, 540000, 1000000, 1306449, 1728243, 5400000, 10000000, 50086125, 54000000, 100000000, 130644900, 172824300, 540000000, 1000000000, 1044000721, 1306449000, 1728243000, 5008612500
Offset: 1
5400=2^3*3^3*5^2 is powerful and 45=T(9).
A115694
Cubes whose digit reversal is a powerful(1) number (A001694).
Original entry on oeis.org
1, 8, 27, 343, 1000, 1331, 8000, 27000, 343000, 1000000, 1030301, 1331000, 1367631, 8000000, 27000000, 343000000, 1000000000, 1003003001, 1030301000, 1033364331, 1331000000, 1334633301, 1367631000, 8000000000, 10662526601, 27000000000, 343000000000, 1000000000000
Offset: 1
27 = 3^3 and 72 = 2^3*3^2 is powerful.
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Select[Range[2300]^3,Min[FactorInteger[IntegerReverse[#]][[All,2]]]>1 || IntegerReverse[#]==1&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 08 2021 *)
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lista(kmax) = {my(c); for(k = 1, kmax, c = k^3; if(ispowerful(fromdigits( Vecrev(digits(c)))), print1(c, ", ")));} \\ Amiram Eldar, Feb 24 2024
A115695
Powerful(1) numbers (A001694) whose digit reversal is a pentagonal number (A000326).
Original entry on oeis.org
1, 100, 500, 529, 1000, 1089, 2116, 5000, 6241, 10000, 50000, 52900, 100000, 103041, 108900, 151875, 211600, 500000, 529000, 549152, 624100, 1000000, 1089000, 2116000, 5000000, 5290000, 6241000, 10000000
Offset: 1
529=23^2 is powerful and 925 is the 25th pentagonal number.
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