A001358 -id:A001358 - cp's OEIS frontend

A122824 Prime(semiprime(n)) - semiprime(prime(n)). Commutator [A000040,A001358] at n.

1, 4, 9, 8, 10, 12, 24, 24, 32, 15, 46, 24, 27, 34, 25, 40, 44, 46, 51, 54, 53, 46, 54, 60, 70, 70, 98, 105, 104, 91, 64, 72, 45, 48, 95, 118, 120, 120, 116, 108, 100, 96, 101, 118, 102, 144, 123, 86, 76, 81, 136, 138, 143, 112, 132, 131, 153, 160, 171, 169

Offset: 1

Jonathan Vos Post, Oct 23 2006

			a(1) = prime(semiprime(1)) - semiprime(prime(1)) = prime(4) - semiprime(2) = 7 - 6 = 1.
a(2) = prime(semiprime(2)) - semiprime(prime(2)) = prime(6) - semiprime(3) = 13 - 9 = 4.
a(3) = prime(semiprime(3)) - semiprime(prime(3)) = prime(9) - semiprime(5) = 23 - 14 = 9.
a(4) = prime(semiprime(4)) - semiprime(prime(4)) = prime(10) - semiprime(7) = 29 - 21 = 8.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000040, A001358, A106349, A106350.

sp = Select[Range[1000], PrimeOmega[#] == 2 &]; Table[ Prime[ sp[[i]]] - sp[[Prime[i]]], {i, PrimePi@ Length@ sp}] (* Giovanni Resta, Jun 13 2016 *)

a(n) = A106349(n) - A106350(n).

a(33)-a(54) corrected by and a(55)-a(60) from Giovanni Resta, Jun 13 2016

A105282 Positive integers n such that n^20 + 1 is semiprime (A001358).

Original entry on oeis.org

2, 4, 46, 154, 266, 472, 748, 1434, 1738, 2058, 2204, 2222, 2428, 2478, 2510, 2866, 3132, 3288, 3576, 3688, 3756, 4142, 4506, 4940, 5164, 6252, 6330, 6786, 7180, 7300, 7338, 7416, 7628, 7806, 9270, 9312, 10044, 10722, 10860, 12126, 12422, 12668, 12998, 13350

Offset: 1

Jonathan Vos Post, Apr 25 2005

We have the polynomial factorization: n^20 + 1 = (n^4 + 1) * (n^16 - n^12 + n^8 - n^4 + 1). Hence after the initial n=1 prime, the binomial can never be prime. It can be semiprime iff n^4+1 is prime and (n^16 - n^12 + n^8 - n^4 + 1) is prime.

			2^20 + 1 = 1048577 = 17 * 61681,
4^20 + 1 = 1099511627777 = 257 * 4278255361,
46^20 + 1 = 1799519816997495209117766334283777 = 4477457 * 401906666439788301510827761,
1434^20 + 1 =
1352019721694375552250489804528860551814233886722212960509362177 =
4228599998737 * 319732233386510278346888399489424537759394853595121.

Robert Price, Table of n, a(n) for n = 1..1405

Cf. A000040, A001358, A006313, A103854, A104238, A104335, A105041, A105066, A105078, A105122, A105142, A105237, A104479, A104494, A104657.

IsSemiprime:=func< n | &+[ k[2]: k in Factorization(n) ] eq 2 >; [n: n in [1..1000] | IsSemiprime(n^20+1)] // Vincenzo Librandi, Dec 21 2010

Select[Range[1000000], PrimeQ[#^4 + 1] && PrimeQ[(#^20 + 1)/(#^4 + 1)] &] (* Robert Price, Mar 09 2015 *)

a(n)^20 + 1 is semiprime (A001358).

a(9)-a(44) from Robert Price, Mar 09 2015

A074985 Squares of semiprimes (A001358).

Original entry on oeis.org

16, 36, 81, 100, 196, 225, 441, 484, 625, 676, 1089, 1156, 1225, 1444, 1521, 2116, 2401, 2601, 3025, 3249, 3364, 3844, 4225, 4761, 5476, 5929, 6724, 7225, 7396, 7569, 8281, 8649, 8836, 9025, 11236, 12321, 13225, 13924, 14161, 14641, 14884, 15129

Offset: 1

Jani Melik, Oct 07 2002

Disjoint union of 4th powers of primes, A030514, and squares of squarefree semiprimes, A085986. - M. F. Hasler, Nov 12 2021

			4 is divisible by 2 (twice) and 4*4 = 16.
6 is divisible by exactly 2 and 3 and 6*6 = 36.

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

Cf. A001358, A085548, A085964.

Cf. A030514 (4th powers of primes), A085986 (squares of squarefree semiprimes).

a074985 = a000290 . a001358  -- Reinhard Zumkeller, Aug 02 2012

readlib(issqr): ts_kv_sp := proc(n); if (numtheory[bigomega](n)=4 and issqr(n)='true') then RETURN(n); fi; end: seq(ts_kv_sp(i), i=1..50000);

Select[Range[200],PrimeOmega[#]==2&]^2 (* Harvey P. Dale, Oct 03 2011 *)

is(n)=if(issquare(n,&n), isprimepower(n)==2 || factor(n)[,2]==[1,1]~, 0) \\ Charles R Greathouse IV, Oct 16 2015

list(lim)=lim=sqrtint(lim\1); my(v=List()); forprime(p=2, sqrtint(lim), forprime(q=p, lim\p, listput(v, (p*q)^2))); Set(v) \\ Charles R Greathouse IV, Nov 13 2021

a(n) ~ (n log n/log log n)^2. - Charles R Greathouse IV, Oct 16 2015

Sum_{n>=1} 1/a(n) = (P(2)^2 + P(4))/2 = (A085548^2 + A085964)/2 = 0.1407604343..., where P is the prime zeta function. - Amiram Eldar, Oct 30 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

A255609 a(1)=2; a(n) = the smallest prime p such that a(n)-a(n-1) is semiprime (A001358).

Original entry on oeis.org

2, 11, 17, 23, 29, 43, 47, 53, 59, 73, 79, 83, 89, 103, 107, 113, 127, 131, 137, 151, 157, 163, 167, 173, 179, 193, 197, 211, 233, 239, 277, 281, 307, 311, 317, 331, 337, 347, 353, 359, 373, 379, 383, 389, 463, 467, 541, 547, 557, 563, 569, 607

Offset: 1

Zak Seidov, Feb 28 2015

nonn

Sequence with any initial prime term a(1) eventually merges with this sequence: 3,7,11; 5,11; 13,17; 19,23; 31,37,41,47.

For n > 1, a(n) = A289750(n+1). - Jon E. Schoenfield, Nov 26 2017

			a(2) - a(1) = 11 - 2 = 9 = 3*3;
a(3) - a(2) = 17 - 11 = 6 = 2*3;
a(81) - a(80) = 1009 - 887 = 122 = 2*61.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001358, A156829, A289750.

A:= Vector(100): A[1]:= 2:
for n from 2 to 100 do
  p:= A[n-1];
  do
    p:= nextprime(p);
    until numtheory:-bigomega(p-A[n-1]) = 2;
  A[n]:= p;
od:
convert(A,list); # Robert Israel, Dec 28 2022

s = {2}; p = 2; Do[q = NextPrime[p]; While[2 != PrimeOmega[q - p], q = NextPrime[q]]; AppendTo[s, q]; p = q, {100}]; s
sp[n_]:=Module[{p=NextPrime[n]},While[PrimeOmega[p-n]!=2,p= NextPrime[ p]];p]; NestList[sp,2,60] (* Harvey P. Dale, Oct 10 2015 *)

v=[2];forprime(p=3,300,if(bigomega(p-v[#v])==2,v=concat(v,p)));v \\ Derek Orr, Feb 28 2015

A048623 Binary encoding of semiprimes (A001358).

Original entry on oeis.org

2, 3, 4, 5, 9, 6, 10, 17, 8, 33, 18, 65, 12, 129, 34, 257, 16, 66, 20, 130, 513, 1025, 36, 258, 2049, 24, 4097, 68, 8193, 514, 40, 1026, 16385, 132, 32769, 2050, 260, 65537, 72, 32, 131073, 4098, 8194, 136, 262145, 16386, 524289, 48, 516, 1048577, 1028

Offset: 1

Antti Karttunen, Jul 14 1999

Permutation of A048645 (without the term 1).

			Squares p_i^2 are encoded with a single bit in position i (e.g. 25=ithprime(3)*ithprime(3) => 2^3 = 8) and other terms p_i*p_j are encoded with two bits, as sum 2^(i-1)+2^(j-1)

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

Cf. A001358, A048639, A048672, A048645.

nthprime := proc(n) local i; if(isprime(n)) then for i from 1 to 1000000 do if(ithprime(i) = n) then RETURN(i); fi; od; else RETURN(0); fi; end; # nthprime(2) = 1, nthprime(3) = 2, nthprime(5) = 3, etc.
bef := proc(n) local s,d; s := 0; for d in ifactors(n)[ 2 ] do s := s + d[ 2 ]*(2^(nthprime(d[ 1 ])-1)); od; RETURN(s); end; # bef = Binary Encode Factorization.
encode_semiprimes := proc(upto_n) local b,i; b := [ ]; for i from 1 to upto_n do if((3 = tau(i)) or ((0 <> mobius(i)) and (4 = tau(i)))) then b := [ op(b), bef(i) ]; fi; od: RETURN(b); end;

f[n_] := Block[{p = FactorInteger@ n}, Total[2^PrimePi@ # &@ Map[First, p - If[Length@ p == 2, 1, 0]]]]; f /@ Select[Range@ 156, PrimeOmega@ # == 2 &] (* Michael De Vlieger, Oct 01 2015 *)

lista(nn) = {for (n=1, nn, if (bigomega(n)==2, if (issquare(n), x = 2^primepi(sqrtint(n)), f = factor(n); x = sum(k=1, #f~, 2^(primepi(f[k,1]) - 1))); print1(x, ", ");););} \\ Michel Marcus, Oct 02 2015

from math import isqrt
from sympy import primepi, primerange, factorint
def A048623(n):
    def f(x): return int(n+x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//p) for p in primerange(s+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return sum(e<Chai Wah Wu, Feb 22 2025

A118713 a(n) = determinant of n X n circulant matrix whose first row is A001358(1), A001358(2), ..., A001358(n) where A001358(n) = n-th semiprime.

Original entry on oeis.org

4, -20, 361, -3567, 218053, -3455872, 736439027, -16245418225, 1519211613654, -37662452460912, 20199655476042865, -643524421698841536, 46513669467992431114, -3754367220494585505280, 277686193779526116536293, -123973821931125256333959105, 20103033234038999233385180658

Offset: 1

Jonathan Vos Post, May 20 2006

Semiprime analog of A066933 Circulant of prime numbers. a(n) alternates in sign. A048954 Wendt determinant of n-th circulant matrix C(n). A052182 Circulant of natural numbers. A086459 Circulant of powers of 2.

			a(2) = -20 = determinant
|4,6|
|6,4|.
a(3) = 361 = 19^2 = determinant
|4,6,9|
|9,4,6|
|6,9,4|.

Eric Weisstein's World of Mathematics, Circulant Matrix.

Cf. A001358, A048954, A052182, A066933, A086459, A086569.

A118713 := proc(n)
    local C,r,c ;
    C := Matrix(1..n,1..n) ;
    for r from 1 to n do
    for c from 1 to n do
        C[r,c] := A001358(1+((c-r) mod n)) ;
    end do:
    end do:
    LinearAlgebra[Determinant](C) ;
end proc:
seq(A118713(n),n=1..13) ;

nmax = 13;
sp = Select[Range[3 nmax], PrimeOmega[#] == 2&];
a[n_] := Module[{M}, M[1] = sp[[1 ;; n]];
   M[k_] := M[k] = RotateRight[M[k - 1]];
   Det[Table[M[k], {k, 1, n}]]];
Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Feb 16 2023 *)

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Aug 23 2007

A096003 Smallest semiprime (A001358) which is at the end of an arithmetic progression of n semiprimes.

Original entry on oeis.org

4, 6, 14, 46, 58, 221, 445, 497, 1211, 1561, 4195, 4393, 6347, 10717, 14233, 28213, 31451, 72965, 119029, 121603, 124177, 611261, 632171, 2003171, 2012771, 7466993, 7619243, 7771493, 7923743, 68029097, 142247113, 168901739

Offset: 1

N. J. A. Sloane, Jul 28 2004

nonn

a(33) <= 1320196303 (gap of 39029760). a(34) <= 1359226063 (gap of 39029760). a(35) <= 1398255823 (gap of 39029760). - Donovan Johnson, Jun 03 2012

			a(3)=14 since (4,9,14) [or (6,10,14)] is an arithmetic progression of 3 semiprimes ending in 14 and 14 is the smallest semiprime with this property.

Cf. A001358 (semiprimes), A005115 (analog for primes).

For the associated gaps see A097824.

More terms from T. D. Noe and Ray Chandler, Jul 29 2004
a(18)-a(20) from Ray Chandler, Aug 01 2004
More terms from Hugo Pfoertner, Aug 27 2004
a(26)-a(30) from Hugo Pfoertner, Sep 07 2004
a(31)-a(32) from Donovan Johnson, Jun 03 2012

A130533 a(n) = smallest k such that A001358(n+1) = A001358(n) + (A001358(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 2, 6, 13, 9, 2, 19, 2, 19, 2, 3, 4, 37, 8, 43, 47, 47, 53, 2, 6, 59, 61, 8, 71, 6, 79, 2, 5, 83, 89, 2, 3, 12, 101, 107, 4, 3, 3, 2, 11

Offset: 1

Rémi Eismann, Aug 16 2007 - Jan 20 2011

nonn

a(n) is the "weight" of semiprimes.

The decomposition of semiprimes into weight * level + gap is A001358(n) = a(n) * A184729(n) + A065516(n) if a(n) > 0.

			For n = 1 we have A001358(n) = 4, A001358(n+1) = 6; there is no k such that 6 - 4 = 2 = (4 mod k), hence a(1) = 0.
For n = 3 we have A001358(n) = 9, A001358(n+1) = 10; 2 is the smallest k such that 10 - 9 = 1 = (9 mod k), hence a(3) = 2.
For n = 19 we have A001358(n) = 55, A001358(n+1) = 57; 53 is the smallest k such that 57 - 55 = 2 = (55 mod k), hence a(19) = 53.

Remi Eismann, Table of n, a(n) for n=1..9999

Cf. A001358, A065516, A184729, A184728, A117078, A117563, A001223, A118534.

A105997 Semiprime function n -> A001358(n) applied three times to n.

Original entry on oeis.org

26, 39, 74, 77, 118, 119, 178, 194, 219, 235, 299, 301, 329, 377, 381, 454, 471, 502, 535, 565, 566, 634, 679, 703, 721, 779, 842, 886, 893, 914, 973, 995, 998, 1006, 1126, 1174, 1227, 1282, 1294, 1317, 1337, 1343, 1389, 1418, 1457, 1563, 1577, 1623, 1642

Offset: 1

Jonathan Vos Post, Apr 29 2005

			a(1) = semiprime(semiprime(semiprime(1))) = semiprime(semiprime(4)) = semiprime(10) = 26.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A001358, A007097, A091022, A105998, A105999.

issp:= n-> not isprime(n) and numtheory[bigomega](n)=2:
sp:= proc(n) option remember; local k; if n=1 then 4 else
       for k from 1+sp(n-1) while not issp(k) do od; k fi end:
a:= n-> (sp@@3)(n):
seq(a(n), n=1..49);  # Alois P. Heinz, Aug 16 2024

f[n_] := Plus @@ Flatten[ Table[ # [[2]], {1}] & /@ FactorInteger[ n]]; t = Select[ Range[ 1700], f[ # ] == 2 &]; Table[ Nest[ t[[ # ]] &, n, 3], {n, 50}] (* Robert G. Wilson v, Apr 30 2005 *)

from math import isqrt
from sympy import primepi, primerange
def A105997(n):
    def f(x,n): return int(n+x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    def A001358(n):
        m, k = n, f(n,n)
        while m != k:
            m, k = k, f(k,n)
        return m
    return A001358(A001358(A001358(n))) # Chai Wah Wu, Aug 16 2024

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

Corrected and extended by Robert G. Wilson v, Apr 30 2005

cp's OEIS Frontend

A122824 Prime(semiprime(n)) - semiprime(prime(n)). Commutator [A000040,A001358] at n.

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A105282 Positive integers n such that n^20 + 1 is semiprime (A001358).

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A074985 Squares of semiprimes (A001358).

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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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A255609 a(1)=2; a(n) = the smallest prime p such that a(n)-a(n-1) is semiprime (A001358).

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Maple

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A048623 Binary encoding of semiprimes (A001358).

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A118713 a(n) = determinant of n X n circulant matrix whose first row is A001358(1), A001358(2), ..., A001358(n) where A001358(n) = n-th semiprime.

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