A122406 -id:A122406 - cp's OEIS frontend

A272818 Numbers such that (sum + product) of all their prime factors equals (sum + product) of all exponents in their prime factorization.

1, 4, 27, 72, 96, 108, 486, 800, 1280, 3125, 6272, 10976, 12500, 14336, 21600, 30375, 36000, 48600, 51840, 54675, 69120, 84375, 121500, 134456, 169344, 174960, 192000, 225000, 240000, 247808, 337500, 340736, 395136, 435456, 451584, 703125, 750141, 781250, 787320, 823543, 857304, 885735

Offset: 1

Giuseppe Coppoletta, May 08 2016

nonn

For p prime, p^p satisfy the condition, hence A051674 (and also A048102) is a subsequence. Moreover, if p and q are primes and i and j are positive integers, if p^i * q^j verify the condition, then the same is true for p^j * q^i. So 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. In addition, any number having no more than two distinct prime factors (apart their multiplicity) is a term iff it belongs also to A272858.

			885735 = 3^11 * 5 is included because (3+5) + 3*5 = (11+1) + 11*1.
2^10 * 3^6 * 19^2 is included because (2+3+19)+ 2*3*19 = (10+6+2)+ 10*6*2.

Giuseppe Coppoletta and Giovanni Resta, Table of n, a(n) for n = 1..1413 (terms < 10^19, first 100 terms from G. Coppoletta)

Cf. A048102, A054411, A054412, A071174, A122406, A272858, A272859.

Select[Range[10^6], Total@ First@ # + Times @@ First@ # == Total@ Last@ # + Times @@ Last@ # &@ Transpose@ FactorInteger@ # &] (* Michael De Vlieger, May 08 2016 *)

spp(v) = vecsum(v) + prod(k=1, #v, v[k]);
isok(n) = my(f = factor(n)); spp(f[,1]) == spp(f[,2]); \\ Michel Marcus, May 08 2016

def d(n):
    v = factor(n)
    d1 = sum(w[0] for w in v) + prod(w[0] for w in v)
    d2 = sum(w[1] for w in v) + prod(w[1] for w in v)
    return d1 == d2
[k for k in (1..10000) if d(k)]

A272858 Numbers m such that Product(1 + p_i) = Product(1 + e_i), where m = Product((p_i)^e_i).

Original entry on oeis.org

1, 4, 27, 72, 96, 108, 486, 800, 1280, 3125, 6272, 10976, 12500, 14336, 21600, 28800, 30375, 34560, 36000, 38880, 48600, 54675, 84375, 92160, 96000, 121500, 134456, 153600, 169344, 217728, 218700, 225000, 247808, 262440, 296352, 300000, 337500, 340736, 387072, 395136, 489888, 666792, 703125, 750141, 781250, 823543, 857304, 885735

Offset: 1

Giuseppe Coppoletta, May 08 2016

nonn

A048102 is clearly a subsequence, as for any prime p, p^p satisfy the herein condition. Similarly, 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 A272859 at least for the terms of A114129.

			92160 is included because 92160 = 2^11 * 3^2 * 5 and (2+1)*(3+1)*(5+1) = (11+1)*(2+1)*(1+1).

Giuseppe Coppoletta and Giovanni Resta, Table of n, a(n) for n = 1..6058 (terms < 10^18, first 100 terms from G. Coppoletta)

Cf. A048102, A054411, A054412, A071174, A114129, A122406, A272818, A272859.

ok[n_] := Block[{p,e}, {p,e} = Transpose@ FactorInteger@ n; Times @@ (1+p) == Times @@ (1+e)]; Select[Range[10^6], ok] (* Giovanni Resta, May 08 2016 *)

is(n)=my(f=factor(n)); prod(i=1,#f~, f[i,1]+1)==prod(i=1,#f~,f[i,2]) \\ Charles R Greathouse IV, Sep 08 2016

def d(n):
    v = factor(n)
    d1 = prod(1 + w[0] for w in v)
    d2 = prod(1 + w[1] for w in v)
    return d1 == d2
[k for k in (1..10000) if d(k)]

If N is a positive integer and N = Product_{i=1..k} (p_i)^e_i is its prime factorization, then N is in A272858 iff Product_{i=1..k} (1 + p_i) = Product_{i=1..k} (1 + e_i).

For a number with three different prime factors N = p1^e1 * p2^e2 * p3^e3, the defining condition can be expressed as: p1 + p2 + p3 + p1*p2 + p1*p3 + p2*p3 + p1*p2*p3 = e1 + e2 + e3 + e1*e2 + e1*e3 + e2*e3 + e1*e2*e3.

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

Original entry on oeis.org

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)

A122405 Numbers of the form Product_i b_i^e_i, where the b_i are all distinct values > 1 and the e_i are a permutation of the b_i.

Original entry on oeis.org

1, 4, 27, 72, 108, 256, 800, 1024, 2304, 3125, 5184, 6272, 6912, 9216, 10368, 12500, 16384, 18432, 20736, 21600, 27648, 30375, 36000, 41472, 46656, 48600, 62208, 84375, 102400, 121500, 124416, 157464, 169344, 186624, 204800, 209952, 225000

Offset: 1

Franklin T. Adams-Watters, Sep 01 2006

nonn

			2^2 * 3^4 * 4^3 = 20736, so 20736 is in the sequence.

Cf. A122406.

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))

cp's OEIS Frontend

A272818 Numbers such that (sum + product) of all their prime factors equals (sum + product) of all exponents in their prime factorization.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Sage

A272858 Numbers m such that Product(1 + p_i) = Product(1 + e_i), where m = Product((p_i)^e_i).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Sage

A122405 Numbers of the form Product_i b_i^e_i, where the b_i are all distinct values > 1 and the e_i are a permutation of the b_i.

Views

Author

Keywords

Examples

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Examples

Crossrefs

Programs