A054412
Numbers k such that, in the prime factorization of k, the product of exponents equals the product of prime factors.
Original entry on oeis.org
1, 4, 27, 72, 108, 192, 800, 1458, 3125, 5120, 6272, 12500, 21600, 30375, 36000, 48600, 77760, 84375, 114688, 116640, 121500, 138240, 169344, 225000, 247808, 337500, 384000, 395136, 600000, 653184, 750141, 823543, 857304, 979776, 1384448, 1474560, 1500000
Offset: 1
192 is included because 192 =2^6 *3^1 and 2*3 = 6*1.
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peppfQ[n_]:=Module[{f=Transpose[FactorInteger[n]]},Times@@First[f] == Times@@Last[f]]; Select[Range[1.5*10^6],peppfQ] (* Harvey P. Dale, Oct 14 2015 *)
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isok(n) = my(f = factor(n)); prod(i=1, #f~, f[i,2]) == prod(i=1, #f~, f[i,1]); \\ Michel Marcus, May 19 2014
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\\ See Links section.
A008478
Integers of the form Product p_j^k_j = Product k_j^p_j; p_j in A000040.
Original entry on oeis.org
1, 4, 16, 27, 72, 108, 432, 800, 3125, 6272, 12500, 21600, 30375, 50000, 84375, 121500, 169344, 225000, 247808, 337500, 486000, 750141, 823543, 1350000, 1384448, 3000564, 3294172, 6690816, 12002256, 13176688, 19600000, 22235661, 37380096, 37879808, 59295096, 88942644
Offset: 1
16 = 2^4 = 4^2.
27 = 3^3.
108 = 2^2*3^3.
6272 = 2^7*7^2.
121500 = 2^2 * 3^5*5^3.
Some subsequences: p_i^p_i (
A051674), Product_i {p_i^p_i} (
A048102), Product_(j,k)(p_j^p_k * p_k^p_j) with p_j < p_k (
A082949) (see examples).
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f[n_] := Product[{p, e} = pe; e^p, {pe, FactorInteger[n]}];
Reap[For[n = 1, n <= 10^8, n++, If[f[n] == n, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Mar 29 2021 *)
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for(n=2,10^8,if(n==prod(i=1,omega(n), component(component(factor(n),2),i)^component(component(factor(n),1),i)),print1(n,",")))
A098096
Numbers of the form p^2 * 2^p for p prime.
Original entry on oeis.org
16, 72, 800, 6272, 247808, 1384448, 37879808, 189267968, 4437573632, 451508436992, 2063731785728, 188153927303168, 3696558092582912, 16263975998062592, 310889111776919552, 25301222706567446528
Offset: 1
Cf.
A082949, numbers of the form p^q * q^p, with distinct primes p and q.
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a098096 n = a001248 n * a034785 n -- Reinhard Zumkeller, Feb 07 2015
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Table[2^Prime[p]*Prime[p]^2, {p, 16}] (* Alonso del Arte, Oct 28 2005 *)
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forprime(p=2,53,print1(p^2*2^p,","))
A113855
Numbers whose prime factors are raised to the powers of each other.
Original entry on oeis.org
72, 800, 6272, 30375, 247808, 750141, 1384448, 37879808, 189267968, 235782657, 1313046875, 1749600000, 3502727631, 4437573632, 338751673344, 451508436992, 634465620819, 2063731785728, 7863818359375, 7971951402153, 188153927303168
Offset: 1
72 = 8*9 = 2^3*3^2. So primes 2 and 3 are raised to the power of each other.
800 = 2^5*5^2 = 2 to the power 5 times 5 to the power 2.
Cf.
A082949 (numbers of the form p^q * q^p, p, q distinct primes),
A008472 (sum of distinct prime factors of n).
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allpwrfact(n) = { local(x, a, b); a = vector(50); a[1] = 2^3*3^2; a[2] = 2^5*5^2; a[3] = 2^7*7^2; a[4] = 2^11*11^2; a[5] = 2^13*13^2; a[6] = 2^17*17^2; a[7] = 2^19*19^2; a[8] = 2^23*23^2; a[9] = 2^29*29^2; a[10]= 2^31*31^2; a[11]= 2^37*37^2; a[12]= 2^41*41^2; a[13]= 3^5*5^3; a[14]= 3^7*7^3; a[15]= 3^11*11^3; a[16]= 3^13*13^3; a[17]= 3^17*17^3; a[18]= 3^19*19^3; a[19]= 3^23*23^3; a[20]= 3^29*29^3; a[21]= 3^31*31^3; a[22]= 3^37*37^3; a[23]= 2^3*2^5*3^2*3^5*5^2*5^3; a[24]= 2^3*2^7*3^2*3^7*7^2*7^3; a[25]= 2^5*2^7*5^2*5^7*7^2*7^5; a[26]= 2^5*2^11*5^2*5^11*11^2*11^5; a[27]= 3^5*3^7*5^3*5^7*7^3*7^5; a[28]=5^7*7^5; a[29]=5^11*11^5; b= vecsort(a); for(x=1, 42, if(b[x]<>0, print1(b[x]", "))) } (Shepherd)
A276372
Numbers n such that, in the prime factorization of n, the list of the exponents is a rotation of the list of the prime factors.
Original entry on oeis.org
1, 4, 27, 72, 108, 800, 3125, 6272, 12500, 30375, 36000, 48600, 84375, 247808, 337500, 395136, 750141, 823543, 857304, 1384448, 3294172, 22235661, 24532992, 37879808, 53782400, 88942644, 122500000, 161980416, 171478296, 189267968, 235782657, 600112800, 1313046875, 2155524696
Offset: 1
4 is in the sequence because the prime factorization of 4 is 2^2, and the list of exponents (i.e., [2]) is a rotation of the list of prime factors (i.e., [2]).
36000 is in the sequence because the prime factorization of 36000 is 2^5 * 3^2 * 5^3, and the list of exponents (i.e., [5, 2, 3]) is a rotation of the list of prime factors (i.e., [2, 3, 5]).
84 is not in the sequence because the prime factorization of 84 is 2^2 * 3^1 * 7^1, and the list of exponents (i.e., [2, 1, 1]) is not a rotation of the list of prime factors (i.e., [2, 3, 7]).
21600 is not in the sequence because the prime factorization of 21600 is 2^5 * 3^3 * 5^2, and the list of exponents (i.e., [5, 3, 2]) is not a rotation of the list of prime factors (i.e., [2, 3, 5]).
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Select[Range[10^6], Function[w, Total@ Boole@ Map[First@ w == # &, RotateLeft[Last@ w, #] & /@ Range[Length@ Last@ w]] > 0]@ Transpose@ FactorInteger@ # &] (* Michael De Vlieger, Sep 01 2016 *)
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def in_seq( n ):
if n == 1: return True
pf = list( factor( n ) )
primes = [ t[0] for t in pf ]
exponents = [ t[1] for t in pf ]
if primes[0] in exponents:
i = exponents.index(primes[0])
exp_rotated = exponents[i : ] + exponents[0 : i]
return primes == exp_rotated
else:
return False
print([n for n in range(1, 10000000) if in_seq(n)])
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# Much faster program that generates the solutions rather than searching for them.
from sage.misc.misc import powerset
primes = primes_first_n(9)
max_prime = primes[-1]
solutions = set([1])
max_solution = min(2^max_prime * max_prime^2, max_prime^max_prime)
for subset in powerset(primes):
subset_list = list(subset)
for i in range(0, len(subset_list)):
exponents = subset_list[i : ] + subset_list[0 : i]
product = 1
for j in range(0, len(subset_list)):
product = product * subset_list[j]^exponents[j]
if product <= max_solution:
solutions.add(product)
print(sorted(solutions))
A356433
Numbers k such that, in the prime factorization of k, the least common multiple of the exponents equals the least common multiple of the prime factors.
Original entry on oeis.org
1, 4, 27, 72, 108, 192, 576, 800, 1458, 1728, 2916, 3125, 5120, 5832, 6272, 12500, 21600, 25600, 30375, 36000, 46656, 48600, 77760, 84375, 114688, 116640, 121500, 138240, 169344, 225000, 247808, 337500, 384000, 388800, 395136, 583200, 600000, 653184, 691200, 750141, 802816, 823543, 857304, 979776
Offset: 1
576 = 2^6 * 3^2, lcm(2,3) = 6 = lcm(6,2), hence 576 is a term.
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Select[Range[10^6], Equal @@ LCM @@ FactorInteger[#] &] (* Amiram Eldar, Aug 07 2022 *)
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isok(k) = my(f=factor(k)); lcm(f[,1]) == lcm(f[,2]); \\ Michel Marcus, Aug 07 2022
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from math import lcm
from sympy import factorint
def ok(n): f = factorint(n); return lcm(*f.keys()) == lcm(*f.values())
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Aug 07 2022
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