A054411 -id:A054411 - cp's OEIS frontend

A054412 Numbers k such that, in the prime factorization of k, the product of exponents equals the product of prime factors.

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

Leroy Quet, May 09 2000

nonn

For p prime, numbers of the form p^p satisfy the condition, hence A051674 is a subsequence. - Michel Marcus, May 19 2014

Also, numbers of the form p^q * q^p, with distinct primes p and q, satisfy the condition, hence A082949 is a subsequence. - Bernard Schott, Apr 11 2020

			192 is included because 192 =2^6 *3^1 and 2*3 = 6*1.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A054412

Cf. A051674, A054411, A082949.

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 *)

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

PARI
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\\ See Links section.
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More terms from James Sellers, May 23 2000

New name and three more terms from Michel Marcus, May 19 2014

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

A068936 Numbers having the sum of distinct prime factors not greater than the sum of exponents in prime factorization, A008472(k) <= A001222(k).

Original entry on oeis.org

1, 4, 8, 16, 27, 32, 48, 64, 72, 81, 96, 108, 128, 144, 162, 192, 216, 243, 256, 288, 320, 324, 384, 432, 486, 512, 576, 640, 648, 729, 768, 800, 864, 972, 1024, 1152, 1280, 1296, 1458, 1536, 1600, 1728, 1792, 1944, 2000, 2048, 2187, 2304, 2560, 2592, 2916

Offset: 1

Reinhard Zumkeller, Mar 08 2002

nonn

The product of any two terms is also a term. - Amiram Eldar, May 14 2025

			a(5) = 27 = 3^3, 3 = 3.
a(10) = 81 = 3^4, 3 < 4.
a(100) = 16000 = 2^7 * 5^3,  2+5 < 7+3.
a(1000) = 10321920 = 2^15 * 3^2 * 5 * 7, 2+3+5+7 < 15+2+1+1.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)

Cf. A001222, A008472, A068935, A068937, A068938.

Subsequences: A054411, A100716, A257278.

a068936 n = a068936_list !! (n-1)
a068936_list = [x | x <- [1..], a008472 x <= a001222 x]
-- Reinhard Zumkeller, Nov 10 2013

fQ[n_] := Block[{f = FactorInteger@n}, Plus @@ Last /@ f >= Plus @@ First /@ f]; Select[ Range@3000, fQ@ # &] (* Robert G. Wilson v, Jan 16 2006 *)
Select[Range@ 3000, First@ Differences@ Map[Total, Transpose@ FactorInteger@ #] >= 0 &] (* Michael De Vlieger, Dec 08 2016 *)

isok(k) = {my(f = factor(k)); vecsum(f[,1]) <= bigomega(f);} \\ Amiram Eldar, May 14 2025

More terms from Robert G. Wilson v, Jan 16 2006

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

A068938 Numbers having the sum of distinct prime factors greater than the sum of exponents in prime factorization, A008472(n)>A001222(n).

Original entry on oeis.org

2, 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Reinhard Zumkeller, Mar 08 2002

nonn

			12 is included because 12 = 2^2 * 3^1 and 2+3 > 2+1.

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

Cf. A068935, A068936, A054411, A068937.

sdfQ[n_]:=Module[{fi=Transpose[FactorInteger[n]]},Total[fi[[1]]] > Total[ fi[[2]]]]; Select[Range[80],sdfQ] (* Harvey P. Dale, Jul 23 2013 *)

More terms from David Wasserman, Jun 17 2002

A068935 Numbers having the sum of distinct prime factors less than the sum of exponents in prime factorization, A008472(n) < A001222(n).

Original entry on oeis.org

8, 16, 32, 64, 81, 96, 128, 144, 192, 216, 243, 256, 288, 324, 384, 432, 486, 512, 576, 640, 648, 729, 768, 864, 972, 1024, 1152, 1280, 1296, 1458, 1536, 1600, 1728, 1944, 2048, 2187, 2304, 2560, 2592, 2916, 3072, 3200, 3456, 3584, 3888, 4000, 4096, 4374, 4608

Offset: 1

Reinhard Zumkeller, Mar 08 2002

nonn

			144 is included because 144 = 2^4 * 3^2 and 2+3 < 4+2.

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

Cf. A068936, A054411, A068937, A068938.

okQ[n_]:=Module[{tfi=Transpose[FactorInteger[n]]},Total[First[tfi]]Harvey P. Dale, Jan 17 2011 *)

isok(n) = vecsum(factor(n)[,1]) < bigomega(n); \\ Michel Marcus, Apr 25 2016

More terms from Michel Marcus, Apr 25 2016

A068937 Numbers having the sum of distinct prime factors not less than the sum of exponents in prime factorization, A008472(n)>=A001222(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76

Offset: 1

Reinhard Zumkeller, Mar 08 2002

nonn

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

Cf. A068935, A068936, A054411, A068938.

Select[Range[76], Plus @@ (f = FactorInteger[#])[[;;,1]] >= Plus @@ f[[;;,2]] &] (* Amiram Eldar, Nov 14 2019 *)

More terms from David Wasserman, Jun 17 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.

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A054412 Numbers k such that, in the prime factorization of k, the product of exponents equals the product of prime factors.

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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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A068936 Numbers having the sum of distinct prime factors not greater than the sum of exponents in prime factorization, A008472(k) <= A001222(k).

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

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A068938 Numbers having the sum of distinct prime factors greater than the sum of exponents in prime factorization, A008472(n)>A001222(n).

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A068935 Numbers having the sum of distinct prime factors less than the sum of exponents in prime factorization, A008472(n) < A001222(n).

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A068937 Numbers having the sum of distinct prime factors not less than the sum of exponents in prime factorization, A008472(n)>=A001222(n).

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