A085721 -id:A085721 - 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

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

A261073 Semiprimes whose prime factors are of equal binary length and which differ from each other in one bit position only.

Original entry on oeis.org

6, 35, 323, 437, 713, 899, 1763, 1961, 2021, 2537, 3233, 4757, 5561, 5609, 6497, 7313, 9797, 10403, 10961, 11009, 18209, 19043, 21353, 22499, 23393, 26969, 27221, 29177, 37001, 38021, 39203, 45113, 71273, 72899, 79523, 87953, 95477, 98201, 99221, 106793, 114857, 114929, 123353

Offset: 1

Antti Karttunen, Sep 22 2015

			6 = 2*3 is present, as 2 in binary is "10" and 3 in binary is "11", so both have two (significant) bits and they differ only in one bit-position from each other.
35 = 5*7 is present, as 5 in binary is "101" and 7 in binary is "111", which both have three bits, differing only in the middle position from each other.

Antti Karttunen, Table of n, a(n) for n = 1..5000

Cf. A000523, A001222, A006530, A020639, A101080.
Cf. also A261074, A261075.
Cf. A071697 (a subsequence).
Intersection of A085721 and A261077.

Select[Range[10^6], And[Length@ # == 2, IntegerLength[#1, 2] == IntegerLength[#2, 2] & @@ #, Total@ BitXor[IntegerDigits[#1, 2], IntegerDigits[#2, 2]] == 1 & @@ #] &@ Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger@ #] &] (* Michael De Vlieger, Oct 08 2016 *)

A000523 = n -> logint(n, 2);
A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
isA261073(n) = { my(a,b); if(bigomega(n)!=2, 0, a=A020639(n); b = (n/a); ((A000523(a) == A000523(b)) && (1 == norml2(binary(bitxor(a,b)))))); };
i=0; n=0; while(i < 5000, n++; if(isA261073(n), i++; write("b261073.txt", i, " ", n)));

;; With Antti Karttunen's IntSeq-library.
(define A261073 (MATCHING-POS 1 1 (lambda (n) (and (= 2 (A001222 n)) (= (A000523 (A020639 n)) (A000523 (A006530 n))) (= 1 (A101080bi (A020639 n) (A006530 n)))))))

A261074 Semiprimes whose prime factors are of equal binary length and which differ from each other in exactly two bit positions.

Original entry on oeis.org

143, 391, 493, 589, 667, 1517, 1739, 1927, 2257, 2419, 2501, 2773, 2867, 3599, 4891, 5293, 5767, 5893, 6499, 6901, 7081, 7169, 7171, 7387, 7811, 7957, 8137, 8453, 8611, 9379, 9991, 10033, 10057, 10379, 10573, 11021, 11227, 11413, 11663, 13081, 13589, 13843, 17947, 19781, 21509, 21877, 22657, 23449, 23701, 23707, 25217, 25283, 26069, 26441, 27029

Offset: 1

Antti Karttunen, Sep 22 2015

			143 = 11*13 is included because 11 ("1011" in binary) and 13 ("1101" in binary) differ from each other in exactly two bit-positions.
56153 = 233 * 241 is included (as term a(119)) because 233 ("11101001" in binary) and 241 ("11110001" in binary) differ from each other in exactly two bit-positions.

Antti Karttunen, Table of n, a(n) for n = 1..10000
N. S. Dattani & N. Bryans, Quantum factorization of 56153 with only 4 qubits, arXiv:1411.6758 [quant-ph], 2014.

Cf. A000523, A001222, A006530, A020639, A101080.
Cf. also A261073, A261075.
Subsequence of A085721.

Select[Range[10^5], And[Length@ # == 2, IntegerLength[#1, 2] == IntegerLength[#2, 2] & @@ #, Total@ BitXor[IntegerDigits[#1, 2], IntegerDigits[#2, 2]] == 2 & @@ #] &@ Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger@ #] &] (* Michael De Vlieger, Oct 08 2016 *)

A000523 = n -> logint(n, 2);
A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
isA261074(n) = { my(a,b); if(bigomega(n)!=2, 0, a = A020639(n); b = (n/a); ((A000523(a) == A000523(b)) && (2 == norml2(binary(bitxor(a,b)))))); };
i=0; n=0; while(i < 10000, n++; if(isA261074(n), i++; write("b261074.txt", i, " ", n)));

;; With Antti Karttunen's IntSeq-library.
(define A261074 (MATCHING-POS 1 1 (lambda (n) (and (= 2 (A001222 n)) (= (A000523 (A020639 n)) (A000523 (A006530 n))) (= 2 (A101080bi (A020639 n) (A006530 n)))))))

A261075 Semiprimes whose prime factors are of equal binary length and which differ from each other in exactly three bit positions.

Original entry on oeis.org

527, 551, 1591, 2173, 2491, 2623, 3127, 5183, 5963, 6059, 6557, 6767, 6887, 7031, 7373, 7571, 7597, 7739, 7979, 8051, 8249, 8549, 8633, 8881, 9017, 9523, 9701, 10541, 10807, 11303, 11639, 12091, 12317, 12827, 14351, 19519, 20413, 20989, 21823, 22331, 23213, 24047, 24613, 24881, 24883, 25777, 25807, 26549, 26671, 26827, 26989, 27661, 28199, 28459, 28757, 29329

Offset: 1

Antti Karttunen, Sep 22 2015

			291311 = 523 * 557 is included (as term a(334)) because 523 ("1000001011" in binary) and 557 ("1000101101" in binary) differ in exactly three bit-positions.

Antti Karttunen, Table of n, a(n) for n = 1..10000
N. S. Dattani & N. Bryans, Quantum factorization of 56153 with only 4 qubits, arXiv:1411.6758 [quant-ph], 2014.

Cf. A000523, A001222, A006530, A020639, A101080.
Cf. also A261073, A261074.
Subsequence of A085721.

Select[Range@ 30000, And[Length@ # == 2, IntegerLength[#1, 2] == IntegerLength[#2, 2] & @@ #, Total@ BitXor[IntegerDigits[#1, 2], IntegerDigits[#2, 2]] == 3 & @@ #] &@ Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger@ #] &] (* Michael De Vlieger, Oct 08 2016 *)

A000523 = n -> logint(n, 2);
A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
isA261075(n) = { my(a,b); if(bigomega(n)!=2, 0, a = A020639(n); b = (n/a); ((A000523(a) == A000523(b)) && (3 == norml2(binary(bitxor(a,b)))))); };
i=0; n=0; while(i < 10000, n++; if(isA261075(n), i++; write("b261075.txt", i, " ", n)));

;; With Antti Karttunen's IntSeq-library.
(define A261075 (MATCHING-POS 1 1 (lambda (n) (and (= 2 (A001222 n)) (= (A000523 (A020639 n)) (A000523 (A006530 n))) (= 3 (A101080bi (A020639 n) (A006530 n)))))))

A266346 Numbers that can be represented as a product of two numbers with an equal number of significant digits (bits) in binary system.

Original entry on oeis.org

0, 1, 4, 6, 9, 16, 20, 24, 25, 28, 30, 35, 36, 42, 49, 64, 72, 80, 81, 88, 90, 96, 99, 100, 104, 108, 110, 112, 117, 120, 121, 126, 130, 132, 135, 140, 143, 144, 150, 154, 156, 165, 168, 169, 180, 182, 195, 196, 210, 225, 256, 272, 288, 289, 304, 306, 320, 323, 324, 336, 340, 342, 352, 357, 360, 361, 368, 374, 378, 380, 384, 391

Offset: 0

Antti Karttunen, Dec 28 2015

Indexing starts from zero as a(0) = 0 is a special case in this sequence.

			1 can be represented as 1*1 (1 being "1" also in base-2 system), thus it is included.
4 can be represented as 2*2, and like any square, is included.
6 can be represented as 2*3, and both "10" and "11" require two bits in binary system, thus 6 is included.

Antti Karttunen, Table of n, a(n) for n = 0..9616

Positions of nonzeros in A266342.
Cf. A266347 (complement).
Cf. A000290, A085721, A261073, A261074, A261075 (subsequences).
Cf. also A266342.

{0}~Join~Flatten[Position[#, k_ /; k > 0] &@ Table[Length@ DeleteCases[Flatten@ Map[Differences@ IntegerLength[#, 2] &, Transpose@ {#, n/#}] &@ TakeWhile[Divisors@ n, # <= Sqrt@ n &], k_ /; k > 0], {n, 400}]] (* Michael De Vlieger, Dec 30 2015 *)

A200878 Composite numbers whose prime factors have equal numbers of bits.

Original entry on oeis.org

4, 6, 8, 9, 12, 16, 18, 24, 25, 27, 32, 35, 36, 48, 49, 54, 64, 72, 81, 96, 108, 121, 125, 128, 143, 144, 162, 169, 175, 192, 216, 243, 245, 256, 288, 289, 323, 324, 343, 361, 384, 391, 432, 437, 486, 493, 512, 527, 529, 551, 576, 589, 625, 648, 667, 713

Offset: 1

Arkadiusz Wesolowski, Nov 23 2011

			7429 = 17*19*23 -> 10001*10011*10111, therefore 7429 is a term.
7430 = 2*5*743 -> 10*101*1011100111, therefore 7430 is not a term.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000

Supersequence of A085721 and of A182302.

lst = {}; Do[b = IntegerDigits[FactorInteger[n], 2]; If[! PrimeQ[n] && Length[b[[-1, 1]]] == Length[b[[1, 1]]], AppendTo[lst, n]], {n, 4, 6!}]; lst (* Arkadiusz Wesolowski, Dec 03 2011 *)
Select[Range[800],CompositeQ[#]&&Length[Union[IntegerLength[ #,2]&/@ FactorInteger[ #][[All,1]]]]==1&] (* Harvey P. Dale, Oct 11 2021 *)

is(n)=my(f=factor(n)[,1]);#binary(f[1])==#binary(f[#f])&&!isprime(n) \\ Charles R Greathouse IV, Dec 23 2011

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Formula

Extensions

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

A261073 Semiprimes whose prime factors are of equal binary length and which differ from each other in one bit position only.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Scheme

A261074 Semiprimes whose prime factors are of equal binary length and which differ from each other in exactly two bit positions.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Scheme

A261075 Semiprimes whose prime factors are of equal binary length and which differ from each other in exactly three bit positions.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Scheme

A266346 Numbers that can be represented as a product of two numbers with an equal number of significant digits (bits) in binary system.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A200878 Composite numbers whose prime factors have equal numbers of bits.

Views

Author

Keywords

Examples

Links

Crossrefs