A054411 -id:A054411 - cp's OEIS frontend

A272859 Numbers m such that sigma(Product(p_j)) = sigma(Product(e_j)), where m = Product((p_i)^e_i) and sigma = A000203.

1, 4, 27, 72, 108, 192, 800, 1458, 3125, 5120, 6144, 6272, 10976, 12500, 21600, 30375, 36000, 48600, 54675, 77760, 84375, 114688, 116640, 121500, 134456, 138240, 169344, 173056, 225000, 229376, 247808, 337500, 354294, 384000, 395136, 600000, 653184, 655360, 703125, 750141, 823543, 857304, 913952, 979776

Offset: 1

Giuseppe Coppoletta, May 08 2016

nonn

A048102 is clearly a subsequence, as for any prime p, p^p satisfy the herein condition. Moreover, due to the multiplicativity of the arithmetic function sigma, A122406 is also a subsequence. More generally, if a number is a term, then any permutation of the exponents in its prime factorization (i.e., any permutation of its prime signature) gives also a term.

The condition defining this sequence coincides with the condition in A272858 at least for the terms of A114129.

			173056 is included because 173056 = 2^10 * 13^2 and sigma(2*13) = sigma(10*2).
653184 is included because 653184 = 2^7 * 3^6 * 7 and sigma(2*3*7) = sigma(7*6*1).

Giuseppe Coppoletta and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 100 terms from G. Coppoletta)

Cf. A048102, A054411, A054412, A071174, A114129, A122406, A272818, A272858.

Select[Range[10^6], First@ # == Last@ # &@ Map[DivisorSigma[1, Times @@ #] &, Transpose@ FactorInteger@ #] &] (* Michael De Vlieger, May 12 2016 *)

A272859 = []
for n in (1..10000):
    v = factor(n)
    if prod(1 + w[0] for w in v) == sigma(prod(w[1] for w in v)): A272859.append(n)
print(A272859)

A055977 Numbers k such that Product_{q|k} p(q) divides p(k), where p(k) is number of unrestricted partitions of k and the product is over all distinct primes q that divide k.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 10, 11, 13, 17, 19, 23, 29, 31, 37, 40, 41, 43, 47, 53, 59, 61, 64, 67, 71, 73, 75, 79, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 180, 181, 189, 191, 193, 197, 199, 211, 223, 225, 227

Offset: 1

Leroy Quet, Jul 20 2000

			10 is included because p(10) = 42 is divisible by p(2)*p(5) = 2*7 and 2 and 5 are the distinct prime divisors of 10.

John Tyler Rascoe, Table of n, a(n) for n = 1..2000

Cf. A000041, A054411.

isok(k) = my(f=factor(k)); numbpart(k) % prod(i=1, #f~, numbpart(f[i,1])) == 0; \\ Michel Marcus, Jul 25 2024

from itertools import count, islice
from math import prod
from sympy.ntheory import npartitions, factorint
def a_gen():
    for n in count(1):
        if npartitions(n)%prod([npartitions(i) for i in factorint(n)]) < 1:
            yield n
A055977_list = list(islice(a_gen(), 61)) # John Tyler Rascoe, Jul 24 2024

Name and offset edited by John Tyler Rascoe, Jul 24 2024

A071837 Numbers k with the property that in the prime factorization of k all prime exponents are prime, their sum is also prime and equals the sum of distinct prime factors of k.

Original entry on oeis.org

4, 27, 72, 108, 800, 3125, 12500, 247808, 823543, 22579200, 37879808, 190512000, 266716800, 428652000, 529200000, 600112800, 868020300, 1190700000, 1234800000, 1452124800, 2420208000, 2679075000, 3267280800, 3307500000, 4984012800, 6994132992, 7351381800, 7441875000, 7717500000, 9376762500

Offset: 1

Reinhard Zumkeller, Jun 08 2002

			800 is a term as 800 = 2^5 * 5^2, 2+5 = 5+2 = 7, and 7,5,2 are primes.

Cf. A054411, A008472, A001222, A056166, A070215.

A240983 and A051674 are subsequences. - Zak Seidov, Aug 21 2014

terms = 24; fromFactors[s_List] := (Times @@ (s^#)&) /@ Permutations[s]; Clear[f]; f[n_] := f[n] = (ssp = Select[Subsets[Prime[Range[n]]] // Rest, PrimeQ[Total[#]]&]; fromFactors /@ ssp // Flatten // Union // PadRight[#, terms]& ); f[2]; f[n = 4]; While[Print["n = ", n]; f[n] != f[n-2], n = n+2]; f[n] (* Jean-François Alcover, Jul 20 2015 *)

isok(n) = {f = factor(n); for (i=1, #f~, if (! isprime(f[i, 2]), return (0));); isprime(se = sum(i=1, #f~, f[i, 2])) && (se == sum(i=1, #f~, f[i, 1]));} \\ Michel Marcus, Aug 21 2014

from sympy import factorint, isprime
A071837 = []
for n in range(1,10**5):
    f = factorint(n)
    fp, fe = list(f.keys()),list(f.values())
    if sum(fp) == sum(fe) and isprime(sum(fe)) and all([isprime(e) for e in fe]):
        A071837.append(n)
# Chai Wah Wu, Aug 27 2014

Missing terms inserted by Sean A. Irvine, Aug 17 2024

A231230 Numbers k such that, in the prime factorization of k, the sum of the primes equals the squared sum of exponents.

Original entry on oeis.org

1, 28, 98, 132, 198, 351, 368, 726, 1092, 1375, 1488, 1521, 1540, 1638, 2232, 2295, 2320, 3008, 3025, 3348, 3822, 3825, 3850, 4048, 4232, 5022, 5390, 5800, 6375, 6591, 6655, 7098, 7980, 8470, 11328, 11375, 11970, 12012, 12432, 13005, 14500, 15925, 16992, 18018

Offset: 1

Alex Ratushnyak, Nov 05 2013

nonn

Numbers k such that sopf(k) = Omega(k)^2, i.e., A008472(k) = A001222(k)^2. - Amiram Eldar, Jun 13 2025

			98 = 7^2 * 2, sum of primes is 9, sum of exponents is 3, so 98 is in the sequence.

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

Cf. A000040, A001222, A008472, A054411, A054412, A060205, A071174, A071175, A122406.

t = {1}; n = 2; While[Length[t] < 50, n++; {p, e} = Transpose[FactorInteger[n]]; If[Total[p] == Total[e]^2, AppendTo[t, n]]]; t (* T. D. Noe, Nov 08 2013 *)

isok(k) = my(f = factor(k)); sum(i=1, #f~, f[i, 1]) == sum(i=1, #f~, f[i, 2])^2; \\ Michel Marcus, Nov 07 2013

a(1) = 1 inserted by Amiram Eldar, Jun 13 2025

A231231 Numbers m such that, in the prime factorization of m, the product of the exponents equals the sum of prime factors and exponents.

Original entry on oeis.org

432, 648, 1152, 4000, 5400, 8748, 9000, 12800, 12960, 13500, 17280, 19440, 21952, 25000, 48000, 48384, 50625, 60000, 78400, 87480, 100352, 114048, 150000, 189000, 202176, 263424, 303264, 303750, 304128, 340736, 356400, 367416, 368640, 370440, 374544, 384912

Offset: 1

Alex Ratushnyak, Nov 06 2013

nonn

If m = p_1^c_1 * p_2^c_2 * p_3^c_3 * ... * p_k^c_k, where c's are positive integers and p's are distinct primes, then product{j=1 to k}[c_j] = sum{j=1 to k}[p_j+c_j].

			9000 = 3^2 * 2^3 * 5^3. Product of exponents is 2*3*3=18, sum of prime factors and exponents is 3+2+2+3+5+3=18, hence 9000 is in the sequence.

Cf. A054411, A054412, A071174, A071175, A122406, A231293.

t = {}; n = 1; While[Length[t] < 38, n++; f = FactorInteger[n]; sm = Total[Flatten[f]]; pr = Times @@ Transpose[f][[2]]; If[sm == pr, AppendTo[t, n]]]; t (* T. D. Noe, Nov 08 2013 *)
peQ[n_]:=Module[{fi=FactorInteger[n]},Times@@fi[[All,2]]==Total[ Flatten[ fi]]]; Select[Range[400000],peQ] (* Harvey P. Dale, May 21 2019 *)

A231293 Numbers m such that, in the prime factorization of m, the product of the prime factors equals the sum of prime factors and exponents.

Original entry on oeis.org

20, 50, 112, 392, 1372, 2816, 3645, 4802, 6075, 10125, 13312, 15488, 16875, 28125, 46875, 85184, 86528, 278528, 413343, 468512, 562432, 964467, 1245184, 2250423, 2367488, 2576816, 3655808, 3932160, 5250987, 5898240, 8847360, 9830400, 11829248, 12252303

Offset: 1

Alex Ratushnyak, Nov 06 2013

nonn

If m = p_1^c_1 * p_2^c_2 * p_3^c_3 * ... * p_k^c_k, where c's are positive integers and p's are distinct primes, then product{j=1 to k}[p_j] = sum{j=1 to k}[p_j+c_j].

			50 = 2 * 5^2; the product of the prime factors is 2 * 5 = 10, the sum of the prime factors and exponents is 2 + 1 + 5 + 2 = 10, hence 50 is in the sequence.
112 = 2^4 * 7; the product of the prime factors is 2 * 7 = 14, the sum of the prime factors and exponents is 2 + 4 + 7 + 1 = 14, hence 112 is in the sequence.
14172488 = 2^3 * 11^6, product of prime factors is 2*11 = 22, sum of prime factors and exponents is 2 + 3 + 11 + 6 = 22, hence 14172488 is in the sequence.

Cf. A054411, A054412, A071174, A071175, A122406, A231231.

t = {}; n = 1; While[Length[t] < 30, n++; f = FactorInteger[n]; sm = Total[Flatten[f]]; pr = Times @@ Transpose[f][[1]]; If[sm == pr, AppendTo[t, n]]]; t
ppfQ[n_]:=Module[{f=FactorInteger[n]},Times@@[f][[All,1]] == Total[ Flatten[f]]]; Select[Range[13*10^6],ppfQ] (* Harvey P. Dale, Aug 17 2016 *)

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

Robert C. Lyons, Aug 31 2016

nonn

			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]).

Subsequence of A122406 and of A056166. A048102 is a subsequence.

Cf. A008475, A008478, A054411, A054412, A082949, A113855.

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 *)

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)])

# 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

Jean-Marc Rebert, Aug 07 2022

nonn

Numbers k such that A072411(k) = A007947(k). - Michel Marcus, Aug 29 2022
Terms p^p, p prime, form the subsequence A051674. - Bernard Schott, Sep 21 2022
Terms p^q * q^p with distinct primes p and q form the subsequence A082949. - Bernard Schott, Feb 01 2023

			576 = 2^6 * 3^2, lcm(2,3) = 6 = lcm(6,2), hence 576 is a term.

Cf. A054411, A054412, A068935, A068936, A068937, A068938.

Cf. A072411, A007947, A051674, A082949.

Select[Range[10^6], Equal @@ LCM @@ FactorInteger[#] &] (* Amiram Eldar, Aug 07 2022 *)

isok(k) = my(f=factor(k)); lcm(f[,1]) == lcm(f[,2]); \\ Michel Marcus, Aug 07 2022

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

cp's OEIS Frontend

A272859 Numbers m such that sigma(Product(p_j)) = sigma(Product(e_j)), where m = Product((p_i)^e_i) and sigma = A000203.

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A055977 Numbers k such that Product_{q|k} p(q) divides p(k), where p(k) is number of unrestricted partitions of k and the product is over all distinct primes q that divide k.

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A071837 Numbers k with the property that in the prime factorization of k all prime exponents are prime, their sum is also prime and equals the sum of distinct prime factors of k.

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A231230 Numbers k such that, in the prime factorization of k, the sum of the primes equals the squared sum of exponents.

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A231231 Numbers m such that, in the prime factorization of m, the product of the exponents equals the sum of prime factors and exponents.

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A231293 Numbers m such that, in the prime factorization of m, the product of the prime factors equals the sum of prime factors and exponents.

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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.

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Sage

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.

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