A084126 -id:A084126 - cp's OEIS frontend

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

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

A087718 Semiprimes with greater factor less than twice the smaller factor.

Original entry on oeis.org

4, 6, 9, 15, 25, 35, 49, 77, 91, 121, 143, 169, 187, 209, 221, 247, 289, 299, 323, 361, 391, 437, 493, 527, 529, 551, 589, 667, 703, 713, 841, 851, 899, 943, 961, 989, 1073, 1147, 1189, 1247, 1271, 1333, 1363, 1369, 1457, 1517, 1537, 1591, 1643, 1681

Offset: 1

Reinhard Zumkeller, Sep 29 2003

nonn

A084127(a(n)) < A084126(a(n))*2; subsequence of A001358; A001248 is a subsequence.
Odd composite integers which do not have a representation as the sum of an even number of consecutive integers. For instance, 27 is not in the sequence because it has a representation as the sum of an even number of consecutive integers (2+3+4+5+6+7). 35 is in the sequence because it does not have such a representation. - Andrew S. Plewe, May 14 2007
Decker & Moree prove that this sequence has (x log 4)/(log x)^2 + O(x/(log x)^3) members up to x. - Charles R Greathouse IV, Jul 07 2016

			35=5*7 is a term, as 7<5*2=10;
21=3*7 is not a term, as 7>3*2=6.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Andreas Decker and Pieter Moree, Counting RSA-integers, Results in Mathematics 52 (2008), pp. 35-39.
Eric Weisstein's World of Mathematics, Semiprime

Cf. A001358.

Select[Range[1700],PrimeOmega[#]==2&&(IntegerQ[Sqrt[#]]|| FactorInteger[ #] [[-1,1]] < 2*FactorInteger[#][[1,1]])&] (* Harvey P. Dale, Sep 12 2017 *)

list(lim)=my(v=List()); forprime(p=2, sqrtint(lim\2), forprime(q=2, min(lim\p,2*p), listput(v,p*q))); Set(v) \\ Charles R Greathouse IV, Jul 07 2016

a(n) ~ kx log^2 x with k = 1/log 4 = 0.7213..., see Decker & Moree. - Charles R Greathouse IV, Jul 07 2016

A338902 Number of integer partitions of the n-th semiprime into semiprimes.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 4, 7, 7, 10, 17, 25, 21, 34, 34, 73, 87, 103, 149, 176, 206, 281, 344, 479, 725, 881, 1311, 1597, 1742, 1841, 2445, 2808, 3052, 3222, 6784, 9298, 11989, 14533, 15384, 17414, 18581, 19680, 28284, 35862, 38125, 57095, 60582, 64010, 71730, 76016

Offset: 1

Gus Wiseman, Nov 24 2020

nonn

A semiprime (A001358) is a product of any two prime numbers.

			The a(1) = 1 through a(33) = 17 partitions of 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, where A-Z = 10-35:
  4  6  9  A   E    F   L     M      P      Q       X
           64  A4   96  F6    994    FA     M4      EA9
               644      966   A66    L4     AA6     F99
                        9444  E44    A96    E66     FE4
                              6664   F64    9944    L66
                              A444   9664   A664    P44
                              64444  94444  E444    9996
                                            66644   AA94
                                            A4444   E964
                                            644444  F666
                                                    FA44
                                                    L444
                                                    96666
                                                    A9644
                                                    F6444
                                                    966444
                                                    9444444

A002100 counts partitions into squarefree semiprimes.
A056768 uses primes instead of semiprimes.
A101048 counts partitions into semiprimes.
A338903 is the squarefree version.
A339112 includes the Heinz numbers of these partitions.
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A037143 lists primes and semiprimes.
A084126 and A084127 give the prime factors of semiprimes.
A320655 counts factorizations into semiprimes.
A338898/A338912/A338913 give prime indices of semiprimes, with sum/difference/product A176504/A176506/A087794.
A338899/A270650/A270652 give prime indices of squarefree semiprimes.
Cf. A000041, A000607, A006881, A065516, A115392, A128301, A320656, A320732, A320892, A320912, A338915, A338916, A339113.

nn=100;Table[Length[IntegerPartitions[n,All,Select[Range[nn],PrimeOmega[#]==2&]]],{n,Select[Range[nn],PrimeOmega[#]==2&]}]

a(n) = A101048(A001358(n)).

A089994 Number of primes between factors of n-th semiprime.

Original entry on oeis.org

0, 0, 0, 1, 2, 0, 1, 3, 0, 4, 2, 5, 0, 6, 3, 7, 0, 4, 1, 5, 8, 9, 2, 6, 10, 0, 11, 3, 12, 7, 1, 8, 13, 4, 14, 9, 5, 15, 2, 0, 16, 10, 11, 3, 17, 12, 18, 0, 6, 19, 7, 20, 13, 4, 21, 0, 14, 22, 15, 8, 1, 23, 16, 24, 5, 9, 25, 2, 17, 26, 10, 6, 27, 18, 0, 28, 11, 19, 1, 20, 3, 29, 7, 30

Offset: 1

Reinhard Zumkeller, Nov 19 2003

nonn

a(m)=0 iff m in A033476; a(m)>0 iff m in A089995.

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

Cf. A001358, A033476, A089995, A084126, A084127, A000720.

pbfs[n_]:=Module[{f=PrimePi/@Transpose[FactorInteger[n]][[1]]}, Max[ 0,Last[f]-First[f]-1]]; pbfs/@Select[Range[300],PrimeOmega[#]==2&] (* Harvey P. Dale, Apr 09 2012 *)

A106554 Concatenation of the two prime divisors of a semiprime, smallest divisor first.

Original entry on oeis.org

22, 23, 33, 25, 27, 35, 37, 211, 55, 213, 311, 217, 57, 219, 313, 223, 77, 317, 511, 319, 229, 231, 513, 323, 237, 711, 241, 517, 243, 329, 713, 331, 247, 519, 253, 337, 523, 259, 717, 1111, 261, 341, 343, 719, 267, 347, 271, 1113

Offset: 1

Eric Angelini, May 09 2005

Concatenation of the divisors starting with the largest one leads to another sequence.

			First semiprime is 4; 4 is 2*2 -> 22.
Second semiprime is 6; 6 is 2*3 -> 23.
Third semiprime is 9; 9 is 3*3 -> 33.
Fourth semiprime is 10; 10 is 2*5 -> 25.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A106550, A084126, A084127, A001358.

read("transforms") :
A106554 := proc(n)
    A037276(A001358(n)) ;
end proc: # R. J. Mathar, Oct 29 2012

FromDigits@ Flatten[IntegerDigits@ Table[#1, {#2}] & @@@ FactorInteger@ #] & /@ Select[Range@ 144, PrimeOmega@ # == 2 &] (* Michael De Vlieger, Oct 01 2015 *)

do(x)=my(v=List()); forprime(p=2, sqrt(x), forprime(q=p, x\p, listput(v, [p*q, eval(Str(p, q))]))); Vec(apply(u->u[2], vecsort(v, 1))) \\ Charles R Greathouse IV, Sep 30 2015

a(n) = A037276(A001358(n)). - R. J. Mathar, Oct 29 2012

More terms from Reinhard Zumkeller, May 19 2005

Data corrected by Giovanni Teofilatto and Altug Alkan, Oct 01 2015

A106550 a(n) = n-th semiprime + (concatenation of its two prime factors, smallest factor first).

Original entry on oeis.org

26, 29, 42, 35, 41, 50, 58, 233, 80, 239, 344, 251, 92, 257, 352, 269, 126, 368, 566, 376, 287, 293, 578, 392, 311, 788, 323, 602, 329, 416, 804, 424, 341, 614, 359, 448, 638, 377, 836, 1232, 383, 464, 472, 852, 401, 488, 413, 1256, 674, 419, 686, 437, 512, 884

Offset: 1

Eric Angelini, May 09 2005

			First semiprime is 4; 4 is 2*2; 26=4+22.
Second semiprime is 6; 6 is 2*3; 29=6+23.
Third semiprime is 9; 9 is 3*3; 41=9+33.
Fourth semiprime is 10; 10 is 2*5; 35=10+25.

Cf. A084126, A084127, A001358, A106554.

R=165;SP=Select[Range[R], PrimeOmega[#]==2&]; FromDigits@ Flatten[IntegerDigits@ Table[#1, {#2}] & @@@ FactorInteger@ #] & /@ SP +SP (* James C. McMahon, Feb 18 2024 *)

a(n) = A001358(n) + A106554(n). - Reinhard Zumkeller, May 19 2005

More terms from Reinhard Zumkeller, May 19 2005

Typo corrected by James C. McMahon, Feb 06 2024

A131284 Numbers n such that difference between prime factors of n-th semiprime is n.

Original entry on oeis.org

5, 80, 86, 613668, 6384425704

Offset: 1

Zak Seidov, Sep 25 2007

The 6384425704th semiprime is 44690979977 = 7*6384425711. 6384425711 - 7 = 6384425704. - Donovan Johnson, Jul 11 2010

			sp(5) = 14 = 2*7 and 7 - 2 = 5, sp(80) = 249 = 3*83 and 83 - 3 = 80, sp(86) = 267 = 3*89 and 89 - 3 = 86; sp(n) = n-th semiprime.

Cf. A064910, A084126, A084127, A109313.

{ n=0; j=1; /* n=3068365-1; j=613668;*/
while( l=(j\10^4+1)*10^4, until( l < j++, until(bigomega(n+=1)==2,);
if(2!=#f=factor(n)[,1],next); if(j==f[2]-f[1],print("\n",[j,n,f])));
print1(j-1,":"n", "))} \\ M. F. Hasler, Sep 28 2007

a(4) = 613668 (p=5, q=613673) from M. F. Hasler, Sep 28 2007

a(5) from Donovan Johnson, Jul 11 2010

A239585 Prime factor <= other prime factor of n-th brilliant number, cf. A078972.

Original entry on oeis.org

2, 2, 3, 2, 2, 3, 3, 5, 5, 7, 11, 11, 13, 11, 11, 13, 13, 11, 17, 13, 11, 17, 11, 19, 13, 17, 13, 11, 19, 11, 11, 13, 17, 11, 17, 23, 13, 19, 13, 11, 19, 13, 17, 11, 23, 11, 13, 17, 19, 23, 17, 11, 13, 19, 11, 13, 17, 11, 19, 29, 23, 11, 13, 19, 29, 17, 11

Offset: 1

Reinhard Zumkeller, Mar 22 2014

a(n) = A020639(A078972(n)) = A078972(n) / A239586(n).

A055642(a(n)) = A055642(A239586(n)).

			      n  | A239585(n) | A239586(n) | A078972(n)   Lengths of factors
  -------+------------+------------+-----------   ------------------
      1  |        2   |        2   |        4          1
      5  |        2   |        7   |       14
     10  |        7   |        7   |       49
         |.........................|              ..................
     11  |       11   |       11   |      121          2
     78  |       11   |       97   |     1067
    100  |       37   |       37   |     1369
    241  |       97   |       97   |     9409
         |.........................|              ..................
    242  |      101   |      101   |    10201          3
   1000  |      193   |      263   |    50759
   2530  |      101   |      997   |   100697
  10000  |      743   |      937   |   696191
  10537  |      997   |      997   |   994009
         |.........................|              ..................
  10538  |     1009   |     1009   |  1018081          4

Reinhard Zumkeller, Table of n, a(n) for n = 1..20000
Dario Alpern, Brilliant Numbers

Subsequence of A084126.

Cf. A020639, A055642, A078972, A239586.

Haskell
```
a239585 = a020639 . a078972
```

Table[With[{f = FactorInteger[k]}, If[Total[f[[All, 2]]] == 2 && Length[Union[IntegerLength[f[[All, 1]]]]] == 1, f[[1, 1]], Nothing]], {k, 1000}] (* Paolo Xausa, Oct 02 2024 *)
dlist2[d_] := Union[Times @@@ Tuples[Prime[Range[PrimePi[10^(d-1)] + 1, PrimePi[10^d]]], 2]]; (* Generates terms with d-digits prime factors -- faster but memory intensive *)
Map[FactorInteger[#][[1, 1]]&, Flatten[Array[dlist2, 2]]] (* Paolo Xausa, Oct 09 2024 *)

A176347 n-th semiprime minus sum of its prime factors.

Original entry on oeis.org

0, 1, 3, 3, 5, 7, 11, 9, 15, 11, 19, 15, 23, 17, 23, 21, 35, 31, 39, 35, 27, 29, 47, 43, 35, 59, 39, 63, 41, 55, 71, 59, 45, 71, 51, 71, 87, 57, 95, 99, 59, 79, 83, 107, 65, 91, 69, 119, 111, 71, 119, 77, 103, 131, 81, 143, 115, 87, 119, 143, 159, 95, 131, 99, 167, 159, 101

Offset: 1

Juri-Stepan Gerasimov, Apr 15 2010

nonn

From R. J. Mathar, Apr 20 2010: (Start)
isA001358 := proc(n) numtheory[bigomega](n) = 2 ; end proc:
A001358 := proc(n) option remember ; if n = 1 then return 4 ; else for a from procname(n-1)+1 do if isA001358(a) then return a; end if; end do; end if; end proc:
A084126 := proc(n) min(op(numtheory[factorset](A001358(n)))) ; end proc:
A084127 := proc(n) max(op(numtheory[factorset](A001358(n)))) ; end proc:
A068318 := proc(n) A084126(n)+A084127(n) ; end proc:
A176347 := proc(n) A001358(n)-A068318(n) ; end proc: seq(A176347(n),n=1..80) ; (End)

a(n) = A001358(n) - A068318(n).

Entries checked by R. J. Mathar, Apr 20 2010

cp's OEIS Frontend

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

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

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A087718 Semiprimes with greater factor less than twice the smaller factor.

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A338902 Number of integer partitions of the n-th semiprime into semiprimes.

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A089994 Number of primes between factors of n-th semiprime.

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A106554 Concatenation of the two prime divisors of a semiprime, smallest divisor first.

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A106550 a(n) = n-th semiprime + (concatenation of its two prime factors, smallest factor first).

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