A054412 -id:A054412 - cp's OEIS frontend

A333936 a(n) is the product of the distinct prime factors of A054412(n).

1, 2, 3, 6, 6, 6, 10, 6, 5, 10, 14, 10, 30, 15, 30, 30, 30, 15, 14, 30, 30, 30, 42, 30, 22, 30, 30, 42, 30, 42, 21, 7, 42, 42, 26, 30, 30, 30, 30, 42, 42, 14, 30, 66, 30, 30, 66, 42, 10, 70, 21, 66, 78, 34, 42, 22, 70, 42, 15, 78, 42, 70, 66, 42, 42, 78, 66

Offset: 1

Rémy Sigrist, Apr 11 2020

nonn

Equivalently, a(n) is the product of the exponents of the prime factorization of A054412(n).

Every squarefree number s appears A001221(s)^A001221(s) times.

			For n = 36:
- A054412(36) = 1474560 = 2^15 * 3^2 * 5,
- hence a(36) = 2 * 3 * 5 = 30.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A333936

Cf. A001221, A005361, A007947, A054412.

PARI
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See Links section.
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a(n) = A005361(A054412(n)) = A007947(A054412(n)).

A054411 Numbers k such that Sum_{j} p_j = Sum_{j} e_j where Product_{j} p_j^(e_j) is the prime factorization of k.

Original entry on oeis.org

1, 4, 27, 48, 72, 108, 162, 320, 800, 1792, 2000, 3125, 3840, 5000, 5760, 6272, 8640, 9600, 10935, 12500, 12960, 14400, 18225, 19440, 21504, 21600, 21952, 24000, 29160, 30375, 31250, 32256, 32400, 36000, 43740, 45056, 48384, 48600, 50625, 54000, 60000, 65610

Offset: 1

Leroy Quet, May 09 2000

nonn

Numbers for which the sum of distinct prime factors equals the sum of exponents in the prime factorization, A008472(n)=A001222(n). - Reinhard Zumkeller, Mar 08 2002

			320 is included because 320 = 2^6 * 5^1 and 2+5 = 6+1.

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

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

f[n_]:=Plus@@First/@FactorInteger[n]==Plus@@Last/@FactorInteger[n]; lst={};Do[If[f[n],AppendTo[lst,n]],{n,0,3*8!}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 19 2010 *)
max = 10^12; Sort@Reap[Sow@1; Do[p = Select[IntegerPartitions[se, All, Prime@ Range@ PrimePi@ se], Sort[#] == Union[#] &]; Do[ np = Length[f]; va = IntegerPartitions[se, {np}, Range[se]]; Do[pe = Permutations[v]; Do[z = Times @@ (f^e); If[z <= max, Sow@z], {e, pe}], {v, va}], {f, p}], {se, 2, Log2[max]}]][[2, 1]] (* Giovanni Resta, May 07 2016 *)

for(n=1,10^6,if(bigomega(n)==sumdiv(n,d,isprime(d)*d),print1(n,",")))

is(n)=my(f=factor(n)); sum(i=1,#f~, f[i,1]-f[i,2])==0 \\ Charles R Greathouse IV, Sep 08 2016

def d(n):
    v=factor(n)[:]; L=len(v); s0=sum(v[j][0] for j in range(L)); s1=sum(v[j][1] for j in range(L))
    return s0-s1
[k for k in (1..100000) if d(k)==0] # Giuseppe Coppoletta, May 07 2016

A122406 Numbers of the form Product_i p_i^e_i, where the p_i are distinct primes and the e_i are a permutation of the p_i.

Original entry on oeis.org

1, 4, 27, 72, 108, 800, 3125, 6272, 12500, 21600, 30375, 36000, 48600, 84375, 121500, 169344, 225000, 247808, 337500, 395136, 750141, 823543, 857304, 1384448, 3000564, 3294172, 6690816, 19600000, 22235661, 24532992, 37380096, 37879808, 53782400, 59295096, 88942644

Offset: 1

Franklin T. Adams-Watters, Sep 01 2006

nonn

Numbers m such that if m = Product_i [p_i^e_i] then m = Product_i [e_i * (p_i^(e_i - 1))]. Example: 21600 = 2^5 * 3^3 * 5^2 = 5*2^4 * 3*3^2 * 2*5^1. - Jaroslav Krizek, Jun 23 2011
From Rémy Sigrist, Oct 29 2017: (Start)
If gcd(a(i), a(j)) = 1, then a(i)*a(j) belongs to the sequence.
This sequence has similarities with A109297, where the prime exponents are a permutation of the prime indices. (End)

			2^5 * 3^3 * 5^2 = 21600, so 21600 is in the sequence. - corrected by _Jaroslav Krizek_, Jun 23 2011

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Subsequence of A054411, A054412, and A122405.

Cf. A109297.

Clear[f, seq]; f[sub_] := f[sub] = (Times @@ (sub^#) & ) /@ Permutations[sub]; seq[0] = {1}; seq[k_] := seq[k] = Union[seq[k - 1], f /@ Subsets[Prime /@ Range[17], {k}] // Flatten // Union // Select[#, # <= 6836638277409177600000 &] &]; seq[k = 1]; While[nterms = Length[seq[k]]; nterms < 1000, k++; Print["nterms = ", nterms]]; seq[k] (* Jean-François Alcover, Dec 09 2013, using Alois P. Heinz's data *)

is(n)=n=factor(n);vecsort(n[,1])==vecsort(n[,2]) \\ Charles R Greathouse IV, Jun 24 2011

A071174 Numbers whose sum of exponents is equal to the product of prime factors.

Original entry on oeis.org

4, 27, 96, 144, 216, 324, 486, 2560, 3125, 6400, 16000, 40000, 57344, 100000, 200704, 250000, 625000, 702464, 823543, 1562500, 2458624, 3906250, 8605184, 23068672, 23914845, 30118144, 39858075, 66430125, 105413504, 110716875, 126877696, 184528125, 307546875, 368947264, 436207616

Offset: 1

Benoit Cloitre, Jun 10 2002

Number k such that A001222(k) = A007947(k). - Amiram Eldar, Jun 24 2022

			57344 = 2^13 * 7^1 and 2*7 = 13+1 hence 57344 is in the sequence.
16000 = 2^7 * 5^3 and 2*5 = 7+3 hence 16000 is in the sequence.

David A. Corneth, Table of n, a(n) for n = 1..10760 (terms <= 10^52)

Cf. A001222, A007947, A054411, A054412, A071175.

q[n_] := Times @@(f = FactorInteger[n])[[;; , 1]] == Total[f[[;; , 2]]]; Select[Range[2, 10^5], q] (* Amiram Eldar, Jun 24 2022 *)

for(n=1,200000,o=omega(n); if(prod(i=1,o, component(component(factor(n),1),i))==sum(i=1,o, component(component(factor(n),2),i)),print1(n,",")))

from math import prod
from sympy import factorint
def ok(n): f = factorint(n); return sum(f[p] for p in f)==prod(p for p in f)
print(list(filter(ok, range(10**6)))) # Michael S. Branicky, Apr 27 2021

More terms from Klaus Brockhaus, Jun 12 2002

More terms from Vladeta Jovovic, Jun 13 2002

A071175 Numbers whose product of exponents is equal to the sum of prime factors.

Original entry on oeis.org

4, 27, 96, 486, 640, 1440, 2025, 2400, 2744, 3024, 3125, 3528, 3584, 4032, 4536, 4860, 5292, 5625, 9408, 11907, 12150, 12348, 14256, 15360, 16464, 17424, 20412, 22400, 22464, 25344, 31360, 32805, 36504, 37500, 39204, 55566, 56250, 57624, 59904, 70304, 71442

Offset: 1

Benoit Cloitre, Jun 10 2002

Number k such that A005361(k) = A008472(k). - Amiram Eldar, Jun 24 2022

			55566 = 2^1 * 3^4 * 7^3 and 1*4*3 = 2+3+7 hence 55566 is in the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A005361, A008472, A054411, A054412.

q[n_] := Total[(f = FactorInteger[n])[[;; , 1]]] == Times @@ f[[;; , 2]]; Select[Range[2, 10^5], q] (* Amiram Eldar, Jun 24 2022 *)

for(n=1,200000,o=omega(n); if(prod(i=1,o, component(component(factor(n),2),i))==sum(i=1,o, component(component(factor(n),1),i)),print1(n,",")))

from math import prod
from sympy import factorint
def ok(n): f = factorint(n); return prod(f[p] for p in f)==sum(p for p in f)
print(list(filter(ok, range(10**5)))) # Michael S. Branicky, Apr 27 2021

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

Original entry on oeis.org

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)

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 *)

cp's OEIS Frontend

A333936 a(n) is the product of the distinct prime factors of A054412(n).

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A054411 Numbers k such that Sum_{j} p_j = Sum_{j} e_j where Product_{j} p_j^(e_j) is the prime factorization of k.

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A122406 Numbers of the form Product_i p_i^e_i, where the p_i are distinct primes and the e_i are a permutation of the p_i.

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A071174 Numbers whose sum of exponents is equal to the product of prime factors.

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A071175 Numbers whose product of exponents is equal to the sum of prime factors.

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

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

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