A084127 -id:A084127 - cp's OEIS frontend

A096932 Smallest number having exactly s divisors, where s is the n-th semiprime (A001358).

6, 12, 36, 48, 192, 144, 576, 3072, 1296, 12288, 9216, 196608, 5184, 786432, 36864, 12582912, 46656, 589824, 82944, 2359296, 805306368, 3221225472, 331776, 37748736, 206158430208, 746496, 3298534883328, 5308416, 13194139533312, 2415919104, 2985984, 9663676416

Offset: 1

Reinhard Zumkeller, Jul 15 2004

nonn

This is to smallest integer for which the number of divisors is the n-th prime (A061286) as semiprimes (A001358) are to primes (A000040). - Jonathan Vos Post, Feb 03 2011

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000005, A001358, A005179, A061234, A061286, A061299, A084127.

s[n_] := Module[{f = FactorInteger[n], p, q}, If[Total[f[[;;,2]]] == 2, p=f[[1,1]]; q = n/p; 2^(q-1) * 3^(p-1) ,Nothing]]; Array[s, 100] (* Amiram Eldar, Apr 13 2024 *)

A000005(a(n)) = A001358(n) and A000005(m) <> A001358(n) for m < a(n).
a(n) = A005179(A001358(n)).
a(p*q) = 2^(q-1) * 3^(p-1) for primes p <= q.
a(A000040(i)*A000040(j)) = 2^(A084127(j)-1) * 3^(A084127(i)-1) for i <= j.

A070647 Largest prime factor of sequence of numbers of the form p*q (p, q distinct primes).

Original entry on oeis.org

3, 5, 7, 5, 7, 11, 13, 11, 17, 7, 19, 13, 23, 17, 11, 19, 29, 31, 13, 23, 37, 11, 41, 17, 43, 29, 13, 31, 47, 19, 53, 37, 23, 59, 17, 61, 41, 43, 19, 67, 47, 71, 13, 29, 73, 31, 79, 53, 23, 83, 59, 89, 61, 37, 17, 97, 67, 101, 29, 41, 103, 19, 71, 107, 43, 31, 109, 73, 17

Offset: 1

Benoit Cloitre, May 13 2002

			6 = 2*3 is the first number of the form p*q (p, q distinct primes) hence a(1) = 3.

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

Cf. A006530, A006881, A084127, A096916, A195758.

a070647 = a006530 . a006881 -- Reinhard Zumkeller, Sep 23 2011

f[n_]:=Last/@FactorInteger[n]=={1,1};f1[n_]:=Min[First/@FactorInteger[n]];f2[n_]:=Max[First/@FactorInteger[n]];lst={};Do[If[f[n],AppendTo[lst,f2[n]]],{n,0,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 10 2010 *)

go(x)=my(v=List()); forprime(p=2, sqrtint(x\1), forprime(q=p+1, x\p, listput(v, [p*q,q]))); apply(v->v[2], vecsort(Vec(v),1)) \\ Charles R Greathouse IV, Sep 14 2015

a(n) = P(A006881(n)) where P(x) = A006530(x) is the largest prime factor of x.

a(n) = A006881(n)/A096916(n). - Amiram Eldar, Oct 28 2024

A108542 Greater prime factor of n-th golden semiprime.

Original entry on oeis.org

3, 5, 11, 17, 31, 37, 47, 59, 67, 109, 127, 157, 163, 167, 173, 211, 241, 263, 271, 293, 313, 367, 389, 439, 449, 457, 503, 571, 593, 613, 619, 643, 661, 677, 701, 727, 739, 787, 823, 911, 983, 991, 1021, 1069, 1163, 1187, 1231, 1289, 1381, 1429, 1487, 1523

Offset: 1

Reinhard Zumkeller, Jun 09 2005

nonn

abs(phi*A108541(n) - a(n)) < 1, where phi = golden ratio = (1+sqrt(5))/2.

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

Cf. A001622, A084127, A108540, A108541, A108543, A108544.

Subsequence of A108539.

f[p_] := Module[{x = GoldenRatio * p}, p1 = NextPrime[x, -1]; p2 = NextPrime[p1]; q = If[x - p1 < p2 - x, p1, p2]; If[Abs[q - x] < 1, q, 0]]; seq = {}; p=1; Do[p = NextPrime[p]; q = f[p]; If[q > 0, AppendTo[seq, q]], {200}]; seq (* Amiram Eldar, Nov 28 2019 *)

a(n) = A108540(n)/A108541(n).

A089995 Products of pairs of distinct, non-consecutive primes.

Original entry on oeis.org

10, 14, 21, 22, 26, 33, 34, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 201, 202, 203, 205, 206, 209, 213

Offset: 1

Reinhard Zumkeller, Nov 19 2003

nonn

m belongs to the sequence iff m=A001358(k) for some k and A089994(k)>0.

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

Cf. A006881, A001358, A084126, A084127, A089994, A033476.

With[{nn = 213}, TakeWhile[#, # <= nn &] &@ Union@ Flatten@ Table[Function[p, p Prime@ Range[n + 2, Prime@ PrimePi[nn/p]]]@ Prime@ n, {n, Sqrt@ nn}]] (* Michael De Vlieger, Feb 02 2017 *)

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

A061299 Least number whose number of divisors is A007304(n) (the n-th number that is the product of 3 distinct primes).

Original entry on oeis.org

720, 2880, 46080, 25920, 184320, 2949120, 129600, 414720, 11796480, 1658880, 188743680, 3732480, 2073600, 26542080, 12079595520, 14929920, 48318382080, 106168320, 8294400, 3092376453120, 1698693120, 18662400, 238878720

Offset: 1

Labos Elemer, Jun 05 2001

nonn

All terms are divisible by a(1) = 720, the first entry.

All terms [=a(j)], not only arguments [=j] have 3 distinct prime factors at exponents determined by the p,q,r factors of their arguments: a(pqr) = RPQ.

			For n = 5: A007304(5) = 78 = 2*3*13, A005179(78) = 184320 = (2^12)*(3^2)*(5^1) = a(5).

Amiram Eldar, Table of n, a(n) for n = 1..3706 (terms 1..835 from David A. Corneth)

Cf. A000005, A005179, A007304, A061148, A061149.

Cf. A096932, A061234, A061286.

a(n) = A005179(A007304(n)).
Min{x; A000005(x) = pqr} p, q, r are distinct primes. If k = pqr and p > q > r then A005179(k) = 2^(p-1)*3^(q-1)*5^(r-1).
From Reinhard Zumkeller, Jul 15 2004: (Start)
A000005(a(n)) = A007304(n).
A000005(m) != A007304(n) for m < a(n).
a(n) = A005179(A007304(n)).
a(p*m*q) = 2^(q-1) * 3^(m-1) * 5^(p-1) for primes p < m < q.
a(A000040(i)*A000040(j)*A000040(k)) = 2^(A084127(k)-1) * 3^(A084127(j)-1) * 5^(A084127(i)-1) for i < j < k. (End)

Edited by N. J. A. Sloane, Apr 20 2007

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 *)

cp's OEIS Frontend

A096932 Smallest number having exactly s divisors, where s is the n-th semiprime (A001358).

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A070647 Largest prime factor of sequence of numbers of the form p*q (p, q distinct primes).

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A108542 Greater prime factor of n-th golden semiprime.

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A089995 Products of pairs of distinct, non-consecutive primes.

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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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A061299 Least number whose number of divisors is A007304(n) (the n-th number that is the product of 3 distinct primes).

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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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