cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Benoit Cloitre, Jun 10 2002

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

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

Extensions

More terms from Klaus Brockhaus, Jun 12 2002
More terms from Vladeta Jovovic, Jun 13 2002

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

Views

Author

Alex Ratushnyak, Nov 05 2013

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    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

Extensions

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

Views

Author

Alex Ratushnyak, Nov 06 2013

Keywords

Comments

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].

Examples

			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.

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Alex Ratushnyak, Nov 06 2013

Keywords

Comments

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].

Examples

			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.

Crossrefs

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

Programs

  • Mathematica
    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 *)
Showing 1-4 of 4 results.

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