A376703 -id:A376703 - cp's OEIS frontend

A078972 Brilliant numbers: semiprimes (products of two primes, A001358) whose prime factors have the same number of decimal digits.

4, 6, 9, 10, 14, 15, 21, 25, 35, 49, 121, 143, 169, 187, 209, 221, 247, 253, 289, 299, 319, 323, 341, 361, 377, 391, 403, 407, 437, 451, 473, 481, 493, 517, 527, 529, 533, 551, 559, 583, 589, 611, 629, 649, 667, 671, 689, 697, 703, 713, 731, 737, 767, 779, 781

Offset: 1

Jason Earls, Jan 12 2003

"Brilliant numbers, as defined by Peter Wallrodt, are numbers with two prime factors of the same length (in decimal notation). These numbers are generally used for cryptographic purposes and for testing the performance of prime factoring programs." [Alpern]
Up to 10^8 the approximate sum of reciprocals is ~1.232884485... - Jason Earls, Oct 15 2010
Let f(n) = a(n) - floor(sqrt(a(n)))^2, or how much larger a(n) is than the largest square number <= a(n). Then f(n) is odd for all even squares, and even for all odd squares where n > 5. See "Ulam spiral" in links. - Christian N. K. Anderson, Jun 12 2013

			1711 = 29*59 is in the sequence since both of its factors have two digits.

P. D. James, The Private Patient, Knopf, NY, 2008, p. 192. (from N. J. A. Sloane, Aug 27 2009)

T. D. Noe, Table of n, a(n) for n = 1..10537
Dario Alpern, Brilliant Numbers.
Christian N. K. Anderson, Ulam Spiral of n=1..3000.

Cf. A001358, A085647, A084126, A084127, A055642, A085721, A376703, A376704.

Row 2 of A376738.

import Data.Function (on)
a078972 n = a078972_list !! (n-1)
a078972_list = filter brilliant a001358_list where
   brilliant x = (on (==) a055642) p (x `div` p) where p = a020639 x
-- Reinhard Zumkeller, Nov 10 2013, Mar 22 2014

fQ[n_] := Block[{fi = FactorInteger@n}, Plus @@ Last /@ fi == 2 && Floor[ Log[10, fi[[1, 1]] ]] == Floor[ Log[10, fi[[ -1, 1]] ]]]; Select[ Range@792, fQ@# &] (* Robert G. Wilson v, May 26 2006 *)
Select[Range[800],PrimeOmega[#]==2&&Length[Union[IntegerLength[FactorInteger[#][[;;,1]]]]]==1&] (* Harvey P. Dale, Jan 24 2025 *)
Select[Range@1000, Differences@IntegerLength@Flatten@(ConstantArray@@#&/@FactorInteger[#]) == {0} &] (* Hans Rudolf Widmer, Oct 25 2022 *)
dlist2[d_] := Union[Times @@@ Tuples[Prime[Range[PrimePi[10^(d-1)] + 1, PrimePi[10^d]]], 2]]; (* Generates terms with d-digits prime factors *)
Flatten[Array[dlist2, 2]] (* Paolo Xausa, Oct 05 2024 *)

is(n)=my(f=factor(n));(#f[,1]==1 && f[1,2]==2) || (#f[,1]==2 && f[1,2]==1 && f[2,2]==1 && #Str(f[1,1])==#Str(f[2,1])) \\ Charles R Greathouse IV, Jun 16 2011

from sympy import sieve
A078972 = []
for n in range(3):
    pr = list(sieve.primerange(10**n,10**(n+1)))
    for i,p in enumerate(pr):
        for q in pr[i:]:
            A078972.append(p*q)
A078972 = sorted(A078972)
# Chai Wah Wu, Aug 26 2014

a(n) = A239585(n) * A239586(n). - Reinhard Zumkeller, Mar 22 2014

Edited by N. J. A. Sloane, Aug 27 2009

A376704 4-brilliant numbers: numbers which are the product of four primes having the same number of decimal digits.

Original entry on oeis.org

16, 24, 36, 40, 54, 56, 60, 81, 84, 90, 100, 126, 135, 140, 150, 189, 196, 210, 225, 250, 294, 315, 350, 375, 441, 490, 525, 625, 686, 735, 875, 1029, 1225, 1715, 2401, 14641, 17303, 20449, 22627, 24167, 25289, 26741, 28561, 29887, 30613, 31603, 34969, 35321, 36179

Offset: 1

Paolo Xausa, Oct 02 2024

			35321 is a term because 35321 = 11 * 13 * 13 * 19, and these four prime factors have the same number of digits.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Dario Alpern, 4-Brilliant Numbers.

Subsequence of A014613.

Cf. A078972, A376703, A376705, A376706.

A376704Q[k_] := With[{f = FactorInteger[k]}, Total[f[[All, 2]]] == 4 && Length[Union[IntegerLength[f[[All, 1]]]]] == 1];
Select[Range[40000], A376704Q] (* or *)
dlist4[d_] := Sort[Times @@@ DeleteDuplicates[Map[Sort, Tuples[Prime[Range[PrimePi[10^(d-1)] + 1, PrimePi[10^d]]], 4]]]]; (* Generates terms with d-digits prime factors -- faster but memory intensive *)
Flatten[Array[dlist4, 2]]

A376705 a(n) = smallest 4-brilliant number (A376704) with n decimal digits.

Original entry on oeis.org

16, 100, 1029, 14641, 100529, 1000109, 10005647, 104060401, 1000002847, 10000000861, 100000024219, 1036488922561, 10000000051429, 100000000009729, 1000000000011799, 10028029413722401, 100000000000035241, 1000000000000012861, 10000000000000080101, 100012000540010800081

Offset: 2

Paolo Xausa, Oct 02 2024

See Alpern link for more terms, along with their prime factorization.

			a(6) = 100529 because 100529 = 11 * 13 * 19 * 37 is the smallest 6-digit number with four prime factors having the same number of digits.

Dario Alpern, 4-Brilliant Numbers.

Subsequence of A376704.

Cf. A078972, A083128, A376703, A376706.

A376706 a(n) = largest 4-brilliant number (A376704) with n decimal digits.

Original entry on oeis.org

90, 875, 2401, 99671, 999973, 9991291, 88529281, 999997283, 9999998329, 99999997711, 988053892081, 9999999999809, 99999999988979, 999999999991541, 9892436613211441, 99999999999805729, 999999999999968753, 9999999999999834493, 99964004859708406561, 999999999999999959833

Offset: 2

Paolo Xausa, Oct 02 2024

See Alpern link for more terms, along with their prime factorization.

			a(6) = 999973 because 999973 = 13 * 13 * 61 * 97 is the largest 6-digit number with four prime factors having the same number of digits.

Dario Alpern, 4-Brilliant Numbers.

Subsequence of A376704.

Cf. A078972, A083182, A376703, A376705.

A376738 Array read by ascending antidiagonals: T(n,k) is the k-th number which is the product of n (possibly non-distinct) primes having the same number of decimal digits.

Original entry on oeis.org

2, 4, 3, 8, 6, 5, 16, 12, 9, 7, 32, 24, 18, 10, 11, 64, 48, 36, 20, 14, 13, 128, 96, 72, 40, 27, 15, 17, 256, 192, 144, 80, 54, 28, 21, 19, 512, 384, 288, 160, 108, 56, 30, 25, 23, 1024, 768, 576, 320, 216, 112, 60, 42, 35, 29, 2048, 1536, 1152, 640, 432, 224, 120, 81, 45, 49, 31

Offset: 1

Paolo Xausa, Oct 03 2024

			Array begins:
  n\k|    1     2     3     4     5     6     7      8      9     10  ...
  -----------------------------------------------------------------------
   1 |    2,    3,    5,    7,   11,   13,   17,    19,    23,    29, ... = A000040
   2 |    4,    6,    9,   10,   14,   15,   21,    25,    35,    49, ... = A078972
   3 |    8,   12,   18,   20,   27,   28,   30,    42,    45,    50, ... = A376703
   4 |   16,   24,   36,   40,   54,   56,   60,    81,    84,    90, ... = A376704
   5 |   32,   48,   72,   80,  108,  112,  120,   162,   168,   180, ...
   6 |   64,   96,  144,  160,  216,  224,  240,   324,   336,   360, ...
   7 |  128,  192,  288,  320,  432,  448,  480,   648,   672,   720, ...
   8 |  256,  384,  576,  640,  864,  896,  960,  1296,  1344,  1440, ...
   9 |  512,  768, 1152, 1280, 1728, 1792, 1920,  2592,  2688,  2880, ...
  10 | 1024, 1536, 2304, 2560, 3456, 3584, 3840,  5184,  5376,  5760, ...
  ...    |                                                          \______ A376739 (main diagonal)
      A000079 (from n = 1)
T(9,5) = 1728 because 1728 = 2 * 2 * 2 * 2 * 2 * 2 * 3 * 3 * 3 is the 5th number with nine prime factors all having the same number of digits.

Cf. A000040, A000079, A078972, A376703, A376704, A376739.

Module[{dmax = 15, a, m, f}, a = Table[m = 2^n - 1; Table[While[Total[(f = FactorInteger[++m])[[All, 2]]] != n || Length[Union[IntegerLength[f[[All, 1]]]]] > 1]; m, dmax - n + 1], {n, dmax, 1, -1}]; Array[Diagonal[a, # - dmax] &, dmax]]

T(n,1) = 2^n.

A376800 3-brilliant numbers with distinct prime factors.

Original entry on oeis.org

30, 42, 70, 105, 2431, 2717, 3289, 3553, 4147, 4199, 4301, 4433, 4807, 5083, 5291, 5423, 5681, 5797, 5863, 6061, 6149, 6409, 6479, 6721, 6851, 6919, 7163, 7337, 7429, 7579, 7657, 7667, 7733, 7843, 8041, 8177, 8437, 8569, 8671, 8723, 8789, 8987, 9061, 9139, 9269

Offset: 1

Paul Duckett, Oct 04 2024

			30 = 2*3*5 is a term.
2431 = 11*13*17 is a term.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Intersection of A376703 and A007304.

from sympy import factorint
def ok(n):
    f = factorint(n)
    return len(f) == sum(f.values()) == 3 and len(set([len(str(p)) for p in f])) == 1
print([k for k in range(9300) if ok(k)]) # Michael S. Branicky, Oct 05 2024

from math import prod
from sympy import primerange
from itertools import count, combinations, islice
def bgen(d): # generator of terms that are products of d-digit primes
    primes, out = list(primerange(10**(d-1), 10**d)), set()
    for t in combinations(primes, 3): out.add(prod(t))
    yield from sorted(out)
def agen(): # generator of terms
    for d in count(1): yield from bgen(d)
print(list(islice(agen(), 45))) # Michael S. Branicky, Oct 05 2024

a(6) and beyond from Michael S. Branicky, Oct 05 2024

cp's OEIS Frontend

A078972 Brilliant numbers: semiprimes (products of two primes, A001358) whose prime factors have the same number of decimal digits.

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A376704 4-brilliant numbers: numbers which are the product of four primes having the same number of decimal digits.

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A376705 a(n) = smallest 4-brilliant number (A376704) with n decimal digits.

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A376706 a(n) = largest 4-brilliant number (A376704) with n decimal digits.

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A376738 Array read by ascending antidiagonals: T(n,k) is the k-th number which is the product of n (possibly non-distinct) primes having the same number of decimal digits.

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A376800 3-brilliant numbers with distinct prime factors.

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Python

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