A078972 -id:A078972 - cp's OEIS frontend

A116047 n+phi(n)+phi(phi(n)) is a brilliant number (A078972).

2, 3, 6, 7, 8, 11, 73, 91, 93, 105, 109, 119, 127, 151, 157, 185, 195, 209, 223, 225, 229, 249, 289, 303, 325, 355, 371, 385, 395, 397, 401, 417, 473, 495, 545, 559, 589, 641, 665, 673, 681, 699, 709, 779, 795, 815, 821, 901, 905, 1007, 1041, 1061, 1065

Offset: 1

Giovanni Resta, Feb 13 2006

			779+phi(779)+phi(phi(779)) = 1691 = 19*89.

Cf. A078972, A116046.

A116052 Numbers k such that k + sigma(k) + sigma(sigma(k)) is a brilliant number (A078972).

Original entry on oeis.org

2, 3, 57, 63, 64, 83, 85, 91, 98, 133, 141, 145, 149, 154, 156, 187, 189, 232, 247, 249, 255, 263, 279, 281, 293, 310, 319, 415, 419, 425, 467, 512, 575, 589, 593, 653, 791, 799, 813, 837, 839, 970, 1019, 1103, 1113, 1145, 1247, 1259, 1427, 2183, 2223

Offset: 1

Giovanni Resta, Feb 13 2006

			1103 is in the sequence since 1103 + sigma(1103) + sigma(sigma(1103)) = 71*73 (a brilliant number).

Cf. A078972 (brilliant numbers), A116051.

A338378 a(1) = 4. a(n) is the smallest semiprime number, which is not an earlier term, for which a(n - 1) + a(n) is a brilliant semiprime number (A078972).

Original entry on oeis.org

4, 6, 9, 26, 95, 74, 69, 118, 25, 10, 15, 34, 87, 82, 39, 214, 33, 358, 49, 94, 93, 206, 155, 14, 21, 122, 65, 254, 35, 86, 57, 262, 115, 106, 141, 46, 123, 166, 55, 382, 91, 346, 183, 38, 209, 194, 129, 58, 85, 466, 51, 158, 161, 398, 119, 134, 185, 62, 159

Offset: 1

Marius A. Burtea, Oct 26 2020

The brilliant semiprime numbers in order of appearance are: 10, 15, 35, 121, 169, 143, 187, 143, 35, 25, 49, 121, 169, 121, 253, 247, 391, 407, 143, 187, 299, 361, 169, 35, 143, 187, 319, 289, 121, 143, ... It is observed that some numbers repeat: 35 = 9 + 26 = 25 + 10 = 14 + 21 or 143 = 74 + 69 = 118 + 25 = 49 + 94 = 21 + 122 = 86 + 57.

			a(1) + a(2) = 4 + 6 = A001358(1) + A001358(2) = 10 = A078972(4).
a(2) + a(3) = 6 + 9 = A001358(2) + A001358(3) = 15 = A078972(6).
a(3) + a(4) = 9 + 26 = A001358(3) + A001358(10) = 35 = A078972(9).
a(4) + a(5) = 26 + 95 = A001358(10) + A001358(34) = 121 = A078972(11).

Cf. A001358, A078972.

bs:=func; s:=func; a:=[ 4 ]; for n in [2..60] do  k:=2; while k in a or  not s(k) or not bs(k+a[n-1]) do k:=k+1; end while; Append(~a,k); end for; a;

Block[{a = {4}}, Do[Block[{k = 6}, While[Nand[FreeQ[a, k], PrimeOmega[k] == 2, If[PrimeOmega[#] == 2, SameQ @@ Map[IntegerLength, FactorInteger[#][[All, 1]] ], False] &[a[[-1]] + k]], k++]; AppendTo[a, k]], {i, 58}]; a] (* Michael De Vlieger, Nov 06 2020 *)

A138510 Smallest number b such that in base b the prime factors of the n-th semiprime (A001358) have equal lengths.

Original entry on oeis.org

1, 2, 1, 6, 8, 3, 3, 12, 1, 14, 12, 18, 2, 20, 14, 24, 1, 18, 4, 20, 30, 32, 4, 24, 38, 4, 42, 5, 44, 30, 4, 32, 48, 5, 54, 38, 5, 60, 5, 1, 62, 42, 44, 5, 68, 48, 72, 2, 30, 74, 32, 80, 54, 5, 84, 1, 60, 90, 62, 38, 3, 98, 68, 102, 6, 42, 104, 3, 72, 108, 44, 6, 110, 74, 3, 114, 48, 80

Offset: 1

Reinhard Zumkeller, Mar 21 2008

a(n) = 1 iff A001358(n) is the square of a prime (A001248);
a(A174956(A138511(n))) = A084127(A174956(A138511(n))) + 1.
Equally, 1 if A001358(n) = p^2, otherwise, if A001358(n) = p*q (p, q primes, p < q), then a(n) = A252375(n) = the least r such that r^k <= p < q < r^(k+1), for some k >= 0. - Antti Karttunen, Dec 16 2014
a(A174956(A085721(n))) <= 2. - Reinhard Zumkeller, Dec 19 2014

			For n=31, the n-th semiprime is A001358(31) = 91 = 7*13;
     7 =  111_2 =  21_3 = 13_4
and 13 = 1101_2 = 111_3 = 31_4, so a(31) = 4. [corrected by _Jon E. Schoenfield_, Sep 23 2018]
.
Illustration of initial terms, n <= 25:
.   n | A001358(n) =  p * q |  b = a(n) | p and q in base b
. ----+---------------------+-----------+-------------------
.   1 |       4       2   2 |      1    |     [1]        [1]
.   2 |       6       2   3 |      2    |   [1,0]      [1,1]
.   3 |       9       3   3 |      1    | [1,1,1]    [1,1,1]
.   4 |  **  10       2   5 |      6    |     [2]        [5]
.   5 |  **  14       2   7 |      8    |     [2]        [7]
.   6 |      15       3   5 |      3    |   [1,0]      [1,2]
.   7 |      21       3   7 |      3    |   [1,0]      [2,1]
.   8 |  **  22       2  11 |     12    |     [2]       [11]
.   9 |      25       5   5 |      1    |   [1]^5      [1]^5
.  10 |  **  26       2  13 |     14    |     [2]       [13]
.  11 |  **  33       3  11 |     12    |     [3]       [11]
.  12 |  **  34       2  17 |     18    |     [2]       [17]
.  13 |      35       5   7 |      2    | [1,0,1]    [1,1,1]
.  14 |  **  38       2  19 |     20    |     [2]       [19]
.  15 |  **  39       3  13 |     14    |     [3]       [13]
.  16 |  **  46       2  23 |     24    |     [2]       [23]
.  17 |      49       7   7 |      1    |   [1]^7      [1]^7
.  18 |  **  51       3  17 |     18    |     [3]       [17]
.  19 |      55       5  11 |      4    |   [1,1]      [2,3]
.  20 |  **  57       3  19 |     20    |     [3]       [19]
.  21 |  **  58       2  29 |     30    |     [2]       [29]
.  22 |  **  62       2  31 |     32    |     [2]       [31]
.  23 |      65       5  13 |      4    |   [1,1]      [3,1]
.  24 |  **  69       3  23 |     24    |     [3]       [23]
.  25 |  **  74       2  37 |     38    |     [2]       [37]
where p = A084126(n) and q = A084127(n),
semiprimes marked with ** indicate terms of A138511, i.e. b = q + 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A078972, A085721.

Cf. A138511, A251728, A252375, A084127, A020639, A162319, A174956, A001358.

import Data.List (genericIndex, unfoldr); import Data.Tuple (swap)
import Data.Maybe (mapMaybe)
a138510 n = genericIndex a138510_list (n - 1)
a138510_list = mapMaybe f [1..] where
  f x | a010051' q == 0 = Nothing
      | q == p          = Just 1
      | otherwise       = Just $
        head [b | b <- [2..], length (d b p) == length (d b q)]
      where q = div x p; p = a020639 x
  d b = unfoldr (\z -> if z == 0 then Nothing else Just $ swap $ divMod z b)
-- Reinhard Zumkeller, Dec 16 2014

(define (A138510 n) (A251725 (A001358 n))) ;; Antti Karttunen, Dec 16 2014

a(n) = A251725(A001358(n)). - Antti Karttunen, Dec 16 2014

Wrong comment corrected by Reinhard Zumkeller, Dec 16 2014

A085721 Semiprimes whose prime factors have an equal number of digits in binary representation.

Original entry on oeis.org

4, 6, 9, 25, 35, 49, 121, 143, 169, 289, 323, 361, 391, 437, 493, 527, 529, 551, 589, 667, 713, 841, 899, 961, 1369, 1517, 1591, 1681, 1739, 1763, 1849, 1927, 1961, 2021, 2173, 2183, 2209, 2257, 2279, 2419, 2491, 2501, 2537, 2623, 2773, 2809

Offset: 1

Reinhard Zumkeller, Jul 20 2003

A138510(A174956(a(n))) <= 2. - Reinhard Zumkeller, Dec 19 2014

			A078972(35) = 527 = 17*31 -> 10001*11111, therefore 527 is a term;
A078972(37) = 533 = 13*41 -> 1101*101001, therefore 533 is not a term;
A001358(1920) = 7169 = 67*107 -> 1000011*1101011: therefore 7169 a term, but not of A078972.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Dario A. Alpern, Brilliant Numbers.

Cf. A078972, A007088, A070939, A055642.
Cf. A084126, A084127, A070939.
Cf. A138510, A174956.
Cf. A261073, A261074, A261075 (subsequences).
Intersection of A001358 and A266346.

a085721 n = a085721_list !! (n-1)
a085721_list = [p*q | (p,q) <- zip a084126_list a084127_list,
                      a070939 p == a070939 q]
-- Reinhard Zumkeller, Nov 10 2013

fQ[n_] := Block[{fi = FactorInteger@ n}, Plus @@ Last /@ fi == 2 && IntegerLength[ fi[[1, 1]], 2] == IntegerLength[ fi[[-1, 1]], 2]]; Select[ Range@ 2866, fQ] (* Robert G. Wilson v, Oct 29 2011 *)
Select[Range@ 3000, And[Length@ # == 2, IntegerLength[#1, 2] == IntegerLength[#2, 2] & @@ #] &@ Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger@ #] &] (* Michael De Vlieger, Oct 08 2016 *)

is(n)=bigomega(n)==2&&#binary(factor(n)[1,1])==#binary(n/factor(n)[1,1]) \\ Charles R Greathouse IV, Nov 08 2011

Edited by Charles R Greathouse IV, Aug 02 2010

A376703 3-brilliant numbers: numbers which are the product of three primes having the same number of decimal digits.

Original entry on oeis.org

8, 12, 18, 20, 27, 28, 30, 42, 45, 50, 63, 70, 75, 98, 105, 125, 147, 175, 245, 343, 1331, 1573, 1859, 2057, 2197, 2299, 2431, 2717, 2783, 2873, 3179, 3211, 3289, 3509, 3553, 3751, 3757, 3887, 3971, 4147, 4199, 4301, 4433, 4477, 4693, 4807, 4901, 4913, 4961, 5083

Offset: 1

Paolo Xausa, Oct 02 2024

			4961 is a term because 4961 = 11 * 11 * 41, and these three prime factors have the same number of digits.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Dario Alpern, 3-Brilliant Numbers.

Subsequence of A014612.

Cf. A078972, A083128, A083182, A376704.

A376703Q[k_] := With[{f = FactorInteger[k]}, Total[f[[All, 2]]] == 3 && Length[Union[IntegerLength[f[[All, 1]]]]] == 1];
Select[Range[6000], A376703Q] (* or *)
dlist3[d_] := Sort[Times @@@ DeleteDuplicates[Map[Sort, Tuples[Prime[Range[PrimePi[10^(d-1)] + 1, PrimePi[10^d]]], 3]]]]; (* Generates terms with d-digits prime factors -- faster but memory intensive *)
Flatten[Array[dlist3, 2]]

from sympy import factorint
def ok(n):
    f = factorint(n)
    return sum(f.values()) == 3 and len(set([len(str(p)) for p in f])) == 1
print([k for k in range(5100) if ok(k)]) # Michael S. Branicky, Oct 05 2024

from math import prod
from sympy import primerange
from itertools import count, combinations_with_replacement as cwr, 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 cwr(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(), 50))) # Michael S. Branicky, Oct 05 2024

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]]

A084475 a(n) defines the first brilliant number, b_n, greater than 10^n. If n is odd or zero, then b_n is 10^n+a(n); and if n is a positive even number, then b_n is {10^(n/2)+a(n)}^2.

Original entry on oeis.org

3, 0, 1, 3, 1, 13, 9, 43, 7, 81, 3, 147, 3, 73, 19, 3, 7, 831, 7, 49, 19, 987, 3, 691, 39, 183, 37, 4153, 31, 279, 37, 667, 61, 709, 3, 277, 3, 1687, 51, 997, 39, 1207, 117, 91, 9, 1411, 117, 393, 7, 951, 13, 9793, 67, 2217, 103, 6229, 331, 2317, 319, 213, 57, 399, 33, 19

Offset: 0

Jason Earls, Jun 03 2003

			a(5)=13 because 10^5+13 = 100013 = 103*971 and a(6)=9 because (10^3+9)^2 = 1009^2. For n>0, a(2n) = A033873(n).

Harry Metrebian, Table of n, a(n) for n = 0..130
Dario Alejandro Alpern, Brilliant numbers

Cf. A078972, A084476.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; LengthBase10[n_] := Floor[ Log[10, n] + 1]; f[n_] := Block[{k = 0}, If[ EvenQ[n] && n > 1, NextPrim[ 10^(n/2)]^2 - 10^(n/2), While[fi = FactorInteger[10^n + k]; Plus @@ Flatten[ Table[ # [[2]], {1}] & /@ fi] != 2 || Length[ Union[ LengthBase10 /@ Flatten[ Table[ # [[1]], {1}] & /@ fi]]] != 1, k++ ]; k]]; Table[ f[n], {n, 0, 63}]

Edited and extended by Robert G. Wilson v, Jun 27 2003

A083128 Least 3-brilliant number of size n.

Original entry on oeis.org

8, 12, 105, 1331, 10013, 100181, 1030301, 10000127, 100000727, 1027243729, 10000002797, 100000000757, 1002101470343, 10000000000493, 100000000005643, 1000090002700027, 10000000000001251, 100000000000000649

Offset: 1

Robert G. Wilson v, May 11 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.
a(3n+1) will always be the cube of the least prime greater than 10^n.
2-brilliant numbers are A078972. 3-brilliant numbers addressed in A083128 and A083182. The sum of all 1, 2 and 3-digit 2-brilliant numbers is a 3-brilliant number. 37789 = 23 * 31 * 53 = 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 + 793 + 799 + 803 + 817 + 841 + 851 + 869 + 871 + 893 + 899 + 901 + 913 + 923 + 943 + 949 + 961 + 979 + 989 - Jonathan Vos Post, Jun 17 2007

			a(5) = 10013 = 17 * 19 * 31 and there is no lesser number of five digits which has three prime factors, not necessarily different, of the same size in decimal notation.

Dario Alpern, Brilliant numbers

Cf. A083182.

Cf. A078972, A083128, A083182.

A114125 a(n) is the 10^n-th semiprime.

Original entry on oeis.org

4, 26, 314, 3595, 40882, 459577, 5109839, 56168169, 611720495, 6609454805, 70937808071, 757060825018, 8040423200947, 85037651263063, 896113850117314, 9413000361625346, 98597629032410971, 1030179406403917981, 10739422018595513973, 111729397883168684917, 1160260967837159869621

Offset: 0

Robert G. Wilson v, Feb 11 2006

nonn

Eric Weisstein's World of Mathematics, Semiprime.

Cf. A001358, A006880, A006988, A066265, A078972.

fQ[n_] := Plus @@ Last /@ FactorInteger@n == 2; c = 0; k = 2; Do[While[c < 10^n, If[fQ@k, c++ ]; k++ ]; Print[k - 1], {n, 0, 8}]
(* checked by *) SemiPrimePi[n_] := Sum[ PrimePi[n/Prime@i] - i + 1, {i, PrimePi@ Sqrt@n}]

use ntheory ":all"; print "$ ",nth_semiprime(10**$),"\n" for 0..15; # Dana Jacobsen, Oct 08 2018

a(14) from Donovan Johnson, Sep 27 2010
Corrected a(14), added a(15)-a(18) from Dana Jacobsen, Oct 10 2018
a(19)-a(20) from Henri Lifchitz, Nov 08 2024

cp's OEIS Frontend

A116047 n+phi(n)+phi(phi(n)) is a brilliant number (A078972).

Views

Author

Keywords

Examples

Crossrefs

A116052 Numbers k such that k + sigma(k) + sigma(sigma(k)) is a brilliant number (A078972).

Views

Author

Keywords

Examples

Crossrefs

A338378 a(1) = 4. a(n) is the smallest semiprime number, which is not an earlier term, for which a(n - 1) + a(n) is a brilliant semiprime number (A078972).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

A138510 Smallest number b such that in base b the prime factors of the n-th semiprime (A001358) have equal lengths.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Scheme

Formula

Extensions

A085721 Semiprimes whose prime factors have an equal number of digits in binary representation.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Extensions

A376703 3-brilliant numbers: numbers which are the product of three primes having the same number of decimal digits.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Python

Python

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A084475 a(n) defines the first brilliant number, b_n, greater than 10^n. If n is odd or zero, then b_n is 10^n+a(n); and if n is a positive even number, then b_n is {10^(n/2)+a(n)}^2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A083128 Least 3-brilliant number of size n.