A100484 -id:A100484 - cp's OEIS frontend

A099777 Number of divisors of 2n.

2, 3, 4, 4, 4, 6, 4, 5, 6, 6, 4, 8, 4, 6, 8, 6, 4, 9, 4, 8, 8, 6, 4, 10, 6, 6, 8, 8, 4, 12, 4, 7, 8, 6, 8, 12, 4, 6, 8, 10, 4, 12, 4, 8, 12, 6, 4, 12, 6, 9, 8, 8, 4, 12, 8, 10, 8, 6, 4, 16, 4, 6, 12, 8, 8, 12, 4, 8, 8, 12, 4, 15, 4, 6, 12, 8, 8, 12, 4, 12, 10, 6, 4, 16, 8, 6, 8, 10, 4, 18, 8, 8, 8, 6, 8

Offset: 1

N. J. A. Sloane, Nov 19 2004

			a(7) = 4 because the divisors of 14 are: 1, 2, 7 and 14.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Bisection of A000005.

Cf. A099774, A001248, A065091, A100484, A372713.

with(numtheory): seq(tau(2*n),n=1..100);

DivisorSigma[0, 2*Range[100]] (* Vladimir Joseph Stephan Orlovsky, Jul 20 2011 *)

a(n)=if(n<1, 0, numdiv(2*n)) /* Michael Somos, Sep 20 2005 */

Moebius transform is period 2 sequence [2, 1, ...]. - Michael Somos, Sep 20 2005
G.f.: Sum_{k>0} x^k(2+x^k)/(1-x^(2k)) = Sum_{k>0} 2*x^(2k-1)/(1-x^(2k-1))+x^(2k)/(1-x^(2k)). - Michael Somos, Sep 20 2005
a(n) = A000005(n) + A001227(n). - Matthew Vandermast, Sep 30 2014
Sum_{k=1..n} a(k) ~ n/2 * (3*log(n) + log(2) + 6*gamma - 3), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 13 2019
From Bernard Schott, Sep 14 2020: (Start)
a(n) = 2 iff n = 1;
a(n) = prime(m) iff n = 2^(prime(m)-2);
a(n) = 4 iff n = 4 or n is an odd prime (A065091);
a(n) = 6 iff n = 16, or n = 2p for p an odd prime (A100484 \ {4}), or n = p^2 for p an odd prime (A001248 \ {4});
a(n) = 2*A000005(n) iff n is odd. (End)
Dirichlet g.f.: zeta(s)^2 * (2 - 1/2^s). - Amiram Eldar, Jun 08 2025

More terms from Emeric Deutsch, Dec 03 2004

A087112 Triangle in which the n-th row contains n distinct semiprimes not listed previously with all prime factors from among the first n primes.

Original entry on oeis.org

4, 6, 9, 10, 15, 25, 14, 21, 35, 49, 22, 33, 55, 77, 121, 26, 39, 65, 91, 143, 169, 34, 51, 85, 119, 187, 221, 289, 38, 57, 95, 133, 209, 247, 323, 361, 46, 69, 115, 161, 253, 299, 391, 437, 529, 58, 87, 145, 203, 319, 377, 493, 551, 667, 841, 62, 93, 155, 217, 341, 403, 527, 589, 713, 899, 961

Offset: 1

Ray Chandler, Aug 21 2003

Terms through row n, sorted, will provide terms for A077553 through row n*(n+1)/2.

			Triangle begins:
   4;
   6,   9;
  10,  15,  25;
  14,  21,  35,  49;
  22,  33,  55,  77, 121;
  26,  39,  65,  91, 143, 169;

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A100484 (left edge), A001248 (right edge), A143215 (row sums), A219603 (central terms of odd-indexed rows); A000040, A065342.

Cf. A276086, A370121.

a087112 n k = a087112_tabl !! (n-1) !! (k-1)
a087112_row n = map (* last ps) ps where ps = take n a000040_list
a087112_tabl = map a087112_row [1..]
-- Reinhard Zumkeller, Nov 25 2012

T := (n, k) -> ithprime(n) * ithprime(k):
seq(print(seq(T(n, k), k = 1..n)), n = 1..11);  # Peter Luschny, Jun 25 2024

Table[ Prime[j]*Prime[k], {j, 11}, {k, j}] // Flatten (* Robert G. Wilson v, Feb 06 2017 *)

A087112(n) = { n--; my(c = (sqrtint(8*n + 1) - 1) \ 2); (prime(1+c) * prime(1+(n-binomial(1+c, 2)))); }; \\ Antti Karttunen, Feb 29 2024

The n-th row consists of n terms, prime(n)*prime(i), i=1..n.
T(n, k) = A000040(n) * A000040(k).
For n >= 2, a(n) = A276086(A370121(n-1)). - Antti Karttunen, Feb 29 2024

A176504 a(n) = m + k where prime(m)*prime(k) = semiprime(n).

Original entry on oeis.org

2, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 7, 9, 8, 10, 8, 9, 8, 10, 11, 12, 9, 11, 13, 9, 14, 10, 15, 12, 10, 13, 16, 11, 17, 14, 12, 18, 11, 10, 19, 15, 16, 12, 20, 17, 21, 11, 13, 22, 14, 23, 18, 13, 24, 12, 19, 25, 20, 15, 12, 26, 21, 27, 14, 16, 28, 13, 22, 29, 17, 15, 30, 23, 13, 31

Offset: 1

Juri-Stepan Gerasimov, Apr 19 2010

nonn

			From _Gus Wiseman_, Dec 04 2020: (Start)
A semiprime (A001358) is a product of any two prime numbers. The sequence of all semiprimes together with their prime indices and weights begins:
   4: 1 + 1 = 2
   6: 1 + 2 = 3
   9: 2 + 2 = 4
  10: 1 + 3 = 4
  14: 1 + 4 = 5
  15: 2 + 3 = 5
  21: 2 + 4 = 6
  22: 1 + 5 = 6
  25: 3 + 3 = 6
  26: 1 + 6 = 7
(End)

A056239 is the version for not just semiprimes.
A087794 gives the product of the same two indices.
A176506 gives the difference of the same two indices.
A338904 puts the n-th semiprime in row a(n).
A001358 lists semiprimes.
A006881 lists squarefree semiprimes.
A338898/A338912/A338913 give the prime indices of semiprimes.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes, with product/sum/difference A339361/A339362/A338900.
Cf. A001222, A046315, A065516, A084126, A084127, A100484, A112798, A115392, A128301, A338906/A338907.

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:
A176504 := proc(n) numtheory[pi](A084126(n)) + numtheory[pi](A084127(n)) ; end proc: seq(A176504(n),n=1..80) ; (End)

Table[If[SquareFreeQ[n],Total[PrimePi/@First/@FactorInteger[n]],2*PrimePi[Sqrt[n]]],{n,Select[Range[100],PrimeOmega[#]==2&]}] (* Gus Wiseman, Dec 04 2020 *)

a(n) = A056239(A001358(n)) = A338912(n) + A338913(n). - Gus Wiseman, Dec 04 2020

sqrt(n/(log n log log n)) << a(n) << n/log log n. - Charles R Greathouse IV, Apr 17 2024

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

A239929 Numbers n with the property that the symmetric representation of sigma(n) has two parts.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 41, 43, 44, 46, 47, 52, 53, 58, 59, 61, 62, 67, 68, 71, 73, 74, 76, 78, 79, 82, 83, 86, 89, 92, 94, 97, 101, 102, 103, 106, 107, 109, 113, 114, 116, 118, 122, 124, 127, 131, 134, 136, 137, 138

Offset: 1

Omar E. Pol, Apr 06 2014

nonn

All odd primes are in the sequence because the parts of the symmetric representation of sigma(prime(i)) are [m, m], where m = (1 + prime(i))/2, for i >= 2.
There are no odd composite numbers in this sequence.
First differs from A173708 at a(13).
Since sigma(p*q) >= 1 + p + q + p*q for odd p and q, the symmetric representation of sigma(p*q) has more parts than the two extremal ones of size (p*q + 1)/2; therefore, the above comments are true. - Hartmut F. W. Hoft, Jul 16 2014
From Hartmut F. W. Hoft, Sep 16 2015: (Start)
The following two statements are equivalent:
  (1) The symmetric representation of sigma(n) has two parts, and
  (2) n = q * p where q is in A174973, p is prime, and 2 * q < p.
For a proof see the link and also the link in A071561.
This characterization allows for much faster computation of numbers in the sequence - function a239929F[] in the Mathematica section - than computations based on Dyck paths. The function a239929Stalk[] gives rise to the associated irregular triangle whose columns are indexed by A174973 and whose rows are indexed by A065091, the odd primes. (End)
From Hartmut F. W. Hoft, Dec 06 2016: (Start)
For the respective columns of the irregular triangle with fixed m: k = 2^m * p, m >= 1, 2^(m+1) < p and p prime:
(a) each number k is representable as the sum of 2^(m+1) but no fewer consecutive positive integers [since 2^(m+1) < p].
(b) each number k has 2^m as largest divisor <= sqrt(k) [since 2^m < sqrt(k) < p].
(c) each number k is of the form 2^m * p with p prime [by definition].
m = 1: (a) A100484 even semiprimes (except 4 and 6)
       (b) A161344 (except 4, 6 and 8)
       (c) A001747 (except 2, 4 and 6)
m = 2: (a) A270298
       (b) A161424 (except 16, 20, 24, 28 and 32)
       (c) A001749 (except 8, 12, 20 and 28)
m = 3: (a) A270301
       (b) A162528 (except 64, 72, 80, 88, 96, 104, 112 and 128)
       (c) sequence not in OEIS
b(i,j) = A174973(j) * {1,5) mod 6 * A174973(j), for all i,j >= 1; see A091999 for j=2. (End)

			From _Hartmut F. W. Hoft_, Sep 16 2015: (Start)
a(23) = 52 = 2^2 * 13 = q * p with q = 4 in A174973 and 8 < 13 = p.
a(59) = 136 = 2^3 * 17 = q * p with q = 8 in A174973 and 16 < 17 = p.
The first six columns of the irregular triangle through prime 37:
   1    2    4    6    8   12 ...
  -------------------------------
   3
   5   10
   7   14
  11   22   44
  13   26   52   78
  17   34   68  102  136
  19   38   76  114  152
  23   46   92  138  184
  29   58  116  174  232  348
  31   62  124  186  248  372
  37   74  148  222  296  444
  ...
(End)

Hartmut F. W. Hoft, Proof of Characterization Theorem

Column 2 of A240062.
Cf. A000203, A006254, A065091, A071561, A174973, A196020, A236104, A235791, A237591, A237593, A237270, A237271, A238443, A239660, A239663, A239665, A239931-A239934, A244050, A245062, A262626.
Cf. A000040, A001747, A001749, A091999, A100484, A161344, A161424, A162528, A270298, A270301. - Hartmut F. W. Hoft, Dec 06 2016

isA174973 := proc(n)
    option remember;
    local k,dvs;
    dvs := sort(convert(numtheory[divisors](n),list)) ;
    for k from 2 to nops(dvs) do
        if op(k,dvs) > 2*op(k-1,dvs) then
            return false;
        end if;
    end do:
    true ;
end proc:
A174973 := proc(n)
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA174973(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
isA239929 := proc(n)
    local i,p,j,a73;
    for i from 1 do
        p := ithprime(i+1) ;
        if p > n then
            return false;
        end if;
        for j from 1 do
            a73 := A174973(j) ;
            if a73 > n then
                break;
            end if;
            if p > 2*a73 and n = p*a73 then
                return true;
            end if;
        end do:
    end do:
end proc:
for n from 1 to 200 do
    if isA239929(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Oct 04 2018

(* sequence of numbers k for m <= k <= n having exactly two parts *)
(* Function a237270[] is defined in A237270 *)
a239929[m_, n_]:=Select[Range[m, n], Length[a237270[#]]==2&]
a239929[1, 260] (* data *)
(* Hartmut F. W. Hoft, Jul 07 2014 *)
(* test for membership in A174973 *)
a174973Q[n_]:=Module[{d=Divisors[n]}, Select[Rest[d] - 2 Most[d], #>0&]=={}]
a174973[n_]:=Select[Range[n], a174973Q]
(* compute numbers satisfying the condition *)
a239929Stalk[start_, bound_]:=Module[{p=NextPrime[2 start], list={}}, While[start p<=bound, AppendTo[list, start p]; p=NextPrime[p]]; list]
a239929F[n_]:=Sort[Flatten[Map[a239929Stalk[#, n]&, a174973[n]]]]
a239929F[138] (* data *)(* Hartmut F. W. Hoft, Sep 16 2015 *)

Entries b(i, j) in the irregular triangle with rows indexed by i>=1 and columns indexed by j>=1 (alternate indexing of the example):

b(i,j) = A000040(i+1) * A174973(j) where A000040(i+1) > 2 * A174973(j). - Hartmut F. W. Hoft, Dec 06 2016

Extended beyond a(56) by Michel Marcus, Apr 07 2014

A338900 Difference between the two prime indices of the n-th squarefree semiprime.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 16 2020

nonn

A squarefree semiprime is a product of any two distinct prime numbers. A prime index of n is a number m such that the m-th prime number divides n. The multiset of prime indices of n is row n of A112798.

Is this sequence an anti-run, i.e., are there no adjacent equal parts? I have verified this conjecture up to n = 10^6. - Gus Wiseman, Nov 18 2020

A176506 is the not necessarily squarefree version.
A338899 has row-differences equal to this sequence.
A338901 gives positions of first appearances.
A001221 counts distinct prime indices.
A001222 counts prime indices.
A001358 lists semiprimes.
A002100 and A338903 count partitions using squarefree semiprimes.
A004526 counts 2-part partitions, with strict case A140106 (shifted left).
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odds A046388 and evens A100484.
A065516 gives first differences of semiprimes.
A166237 gives first differences of squarefree semiprimes.
A270650 and A270652 give the prime indices of squarefree semiprimes.
A338912 and A338913 give the prime indices of semiprimes.
Cf. A000040, A056239, A112798, A167171, A320656, A320891, A320894, A320911, A338898, A338905, A338908.

-Subtract@@PrimePi/@First/@FactorInteger[#]&/@Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&]

If the n-th squarefree semiprime is prime(x) * prime(y) with x < y, then a(n) = y - x.

a(n) = A270652(n) - A270650(n).

A338904 Irregular triangle read by rows where row n lists all semiprimes whose prime indices sum to n.

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 35, 34, 39, 49, 55, 38, 51, 65, 77, 46, 57, 85, 91, 121, 58, 69, 95, 119, 143, 62, 87, 115, 133, 169, 187, 74, 93, 145, 161, 209, 221, 82, 111, 155, 203, 247, 253, 289, 86, 123, 185, 217, 299, 319, 323, 94, 129, 205

Offset: 2

Gus Wiseman, Nov 28 2020

A semiprime is a product of any two prime numbers. A prime index of n is a number m such that the m-th prime number divides n. The multiset of prime indices of n is row n of A112798.

			Triangle begins:
   4
   6
   9  10
  14  15
  21  22  25
  26  33  35
  34  39  49  55
  38  51  65  77
  46  57  85  91 121
  58  69  95 119 143
  62  87 115 133 169 187
  74  93 145 161 209 221
  82 111 155 203 247 253 289
  86 123 185 217 299 319 323
  94 129 205 259 341 361 377 391

A004526 gives row lengths.
A024697 gives row sums.
A087112 is a different triangle of semiprimes.
A098350 has antidiagonals with the same distinct terms as these rows.
A338905 is the squarefree case, with row sums A025129.
A338907/A338906 are the union of odd/even rows.
A339114/A339115 are the row minima/maxima.
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A014342 is the self-convolution of primes.
A037143 lists primes and semiprimes.
A056239 gives sum of prime indices (Heinz weight).
A062198 gives partial sums of semiprimes.
A084126 and A084127 give the prime factors of semiprimes.
A289182/A115392 list the positions of odd/even terms in A001358.
A332765 gives the greatest squarefree semiprime of weight n.
A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
A338899, A270650, and A270652 give the prime indices of squarefree semiprimes, with difference A338900.
Cf. A000040, A001221, A001222, A005117, A112798, A320732, A332877, A338908, A338910, A338911, A339116.

Table[Sort[Table[Prime[k]*Prime[n-k],{k,n/2}]],{n,2,10}]

A338907 Semiprimes whose prime indices sum to an odd number.

Original entry on oeis.org

6, 14, 15, 26, 33, 35, 38, 51, 58, 65, 69, 74, 77, 86, 93, 95, 106, 119, 122, 123, 141, 142, 143, 145, 158, 161, 177, 178, 185, 201, 202, 209, 214, 215, 217, 219, 221, 226, 249, 262, 265, 278, 287, 291, 299, 302, 305, 309, 319, 323, 326, 327, 329, 346, 355

Offset: 1

Gus Wiseman, Nov 28 2020

nonn

All terms are squarefree (A005117).
A semiprime is a product of any two prime numbers. A prime index of n is a number m such that the m-th prime number divides n. The multiset of prime indices of n is row n of A112798.
The semiprimes in A300063; the semiprimes in A332820. - Peter Munn, Dec 25 2020

			The sequence of terms together with their prime indices begins:
      6: {1,2}      95: {3,8}     202: {1,26}
     14: {1,4}     106: {1,16}    209: {5,8}
     15: {2,3}     119: {4,7}     214: {1,28}
     26: {1,6}     122: {1,18}    215: {3,14}
     33: {2,5}     123: {2,13}    217: {4,11}
     35: {3,4}     141: {2,15}    219: {2,21}
     38: {1,8}     142: {1,20}    221: {6,7}
     51: {2,7}     143: {5,6}     226: {1,30}
     58: {1,10}    145: {3,10}    249: {2,23}
     65: {3,6}     158: {1,22}    262: {1,32}
     69: {2,9}     161: {4,9}     265: {3,16}
     74: {1,12}    177: {2,17}    278: {1,34}
     77: {4,5}     178: {1,24}    287: {4,13}
     86: {1,14}    185: {3,12}    291: {2,25}
     93: {2,11}    201: {2,19}    299: {6,9}

A031368 looks at primes instead of semiprimes.
A098350 has this as union of odd-indexed antidiagonals.
A300063 looks at all numbers (not just semiprimes).
A338904 has this as union of odd-indexed rows.
A338906 is the even version.
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A056239 gives the sum of prime indices (Heinz weight).
A084126 and A084127 give the prime factors of semiprimes.
A087112 groups semiprimes by greater factor.
A289182/A115392 list the positions of odd/even terms in A001358.
A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
A338899, A270650, and A270652 give the prime indices of squarefree semiprimes, with difference A338900.
A338908 lists squarefree semiprimes of even weight.
A339114/A339115 give the least/greatest semiprime of weight n.
Cf. A000040, A001222, A014342, A024697, A062198, A112798, A300061, A319242, A320655, A338910, A339003.
Subsequence of A332820.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],PrimeOmega[#]==2&&OddQ[Total[primeMS[#]]]&]

from math import isqrt
from sympy import primepi, primerange
def A338907(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum((primepi(x//p)-a>>1) for a,p in enumerate(primerange(isqrt(x)+1)))
    return bisection(f,n,n) # Chai Wah Wu, Apr 03 2025

Complement of A338906 in A001358.

A356867 For n >= 1, write n = 3^m + k, where m >= 0 is the greatest power of 3 <= n, and k is in the range 0 <= k < 3^(m+1) - 3^m, then for n such that k=0, a(n)=n, and for n such that k > 0, a(n) is the smallest prime multiple p*a(k), p != 3, that is not already a term.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 10, 8, 9, 7, 14, 15, 25, 20, 12, 50, 16, 18, 35, 28, 30, 125, 40, 24, 100, 32, 27, 11, 22, 21, 55, 44, 42, 70, 56, 45, 49, 98, 75, 175, 140, 60, 250, 80, 36, 245, 196, 150, 625, 200, 48, 500, 64, 54, 77, 110, 105, 275, 88, 84, 350, 112, 90, 343

Offset: 1

David James Sycamore, Sep 01 2022

Any prime p may be used to generate a sequence D(p) of this kind. The present sequence is D(3), and D(2) is the Doudna sequence, A005940.
Conjectured to be a permutation of the positive integers in which the primes appear in order.
From Antti Karttunen, Sep 16 2023: (Start)
The conjecture is true: Sequence is a permutation of natural numbers. By definition it is injective, and the surjectivity is guaranteed by the fact that there are infinitely many such n > k encountered by the greedy algorithm that a(n) will be a multiple of a(k), and "the smallest prime multiple" condition guarantees that all multiples of a(k) will eventually appear. That the primes and A100484 appear in order follows from the formulas a(3^m + 1) = prime(m+2), and a(3^m + 2) = 2*prime(m+2).
If the base-3 representation of n-1 has the base-3 representation of k-1 as its suffix, then a(n) is a multiple of a(k). For example, A007089(16-1) = 120, and A007089(43-1) = 1120, thus the former is the suffix of the latter, and a(16) = 50 indeed divides a(43) = 250.
(End)

			n=1=3^0+0 so a(1)=1. n=2=3^0+1 so k=1 and a(2)=2. Similarly a(3)=3 and a(9)=9.
n=10=3^2+1, therefore k=1 and a(1)=1 so a(10)=1*7=7 (since 2 and 5 have already occurred).

Michael De Vlieger, Table of n, a(n) for n = 1..19683 (19683 = 3^9)
Michael De Vlieger, Fan style ternary tree showing a(n) for n = 1..3^9, with a heat map color function for level m where 3^m is blue, smaller values are bluer, and larger are yellow-green. The smallest value in level m is shown in purple and largest is shown in red.
Index entries for sequences that are permutations of the natural numbers

Cf. A007089, A007949, A011655, A048473, A100484, A053735, A364958 (fixed points), A365390 (inverse permutation), A365424, A365459, A365462 [= a(n)-n], A365463 [= gcd(a(n),n)], A365464, A365465, A365717 [= A348717(a(1+n))], A365719 [= A046523(a(1+n))], A365721 [= omega(a(1+n))], A365722 [= bigomega(a(1+n))].

Cf. also A005940, A364611, A364628 for variants D(2), D(5) and D(7).

Block[{a, c, i, j, k, m, t, nn}, nn = 64; m = 1; i = 2; p = Prime[i]; c[] = False; Do[Set[{m, k}, {1, n - p^Floor[Log[p, n]]}]; If[k == 0, Set[{a[n], c[n]}, {n, True}], While[Set[t, Prime[m] a[k]]; Or[m == i, c[t]], m++]; Set[{a[n], c[t]}, {t, True}]], {n, nn}]; Array[a, nn] ] (* _Michael De Vlieger, Sep 01 2022 *)

up_to = 19683;
A356867list(up_to) = { my(v=vector(up_to),met=Map(),h=0,ak); for(i=1,#v,if(1==vecsum(digits(i,3)), v[i] = i; h = i, ak = v[i-h]; forprime(p=2,,if(3!=p && !mapisdefined(met,p*ak), v[i] = p*ak; break))); mapput(met,v[i],i)); (v); };
v356867 = A356867list(up_to);
A356867(n) = v356867[n]; \\ Antti Karttunen, Sep 15 2023

from sympy import nextprime
from sympy.ntheory import digits
from itertools import count, islice
def b(n): return n - 3**(len(digits(n,3)) - 2)
def agen():
    aset, alst = set(), [None]
    for n in count(1):
        k = b(n)
        if k == 0: an = n
        else:
            ak, p = alst[k], 2
            while p == 3 or p*ak in aset: p = nextprime(p)
            an = p*ak
        yield an; aset.add(an); alst.append(an)
print(list(islice(agen(), 64))) # Michael S. Branicky, Sep 02 2022

a(3^m + 1) = prime(m+2) for m >= 1.
Conjectures from Jianing Song, Nov 23 2022: (Start)
(1) a(3^m+2) = 2*prime(m+2) for m >= 2. - [The conjecture is true because a(2) = 2 and 3^m + 2 < 3^(1+m) + (3^m) + 1 for all m - Antti Karttunen, Sep 16 2023]
(2) For n > m >= 1, a(3^n+3^m+1) = prime(m+2)^2 for n = m+1; prime(n+2)*prime(m+2)^2 for n >= m+2.
(3) For n > m >= 1, a(3^n+3^m+2) = 4*prime(n+2) for n >= 3, m = 1; 2*prime(m+2)^2 for n = m+1, m >= 2; 2*prime(m+2)*prime(m+3) for n = m+2, m >= 2; 2*prime(n+2)*prime(m+2)^2 for n >= m+3, m >= 2. (End)
From Antti Karttunen, Sep 17 2023: (Start)
If A053735(n) = 1, then a(n) = n, otherwise a(n) = A365424(n) * a(A365459(n)).
For all n >= 1, A007949(a(n)) = A007949(n) and a(3*n) = 3*a(n).
For n >= 1, a(3^n - 1) = 2^(2n - 1), a(A048473(n)) = 2^(2*(n-1)).
These are conjectures so far:
For n >= 1, a(3^n - 2) = 10^(n-1).
For n >= 2, a(3^n - 3) = A002023(n-2) = 6*4^(n-2).
(End)

More terms from Michael De Vlieger, Sep 01 2022

A077065 Semiprimes of form prime - 1.

Original entry on oeis.org

4, 6, 10, 22, 46, 58, 82, 106, 166, 178, 226, 262, 346, 358, 382, 466, 478, 502, 562, 586, 718, 838, 862, 886, 982, 1018, 1186, 1282, 1306, 1318, 1366, 1438, 1486, 1522, 1618, 1822, 1906, 2026, 2038, 2062, 2098, 2206, 2446, 2458, 2578, 2818, 2878, 2902

Offset: 1

Reinhard Zumkeller, Oct 23 2002

nonn

There are 670 semiprimes of form prime-1 below 10^5.

			A001358(16) = 46 = 2*23 is a term as 46 = A000040(15) - 1 = 47 - 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Intersection of A006093 and A001358.
Intersection of A006093 and A100484.
Cf. A077068, A232342, A064911.
Cf. A005384, A005385.
Cf. A000010, A023900, A194593.

a077065 n = a077065_list !! (n-1)
a077065_list = filter ((== 1) . a010051' . (`div` 2)) a006093_list
-- Reinhard Zumkeller, Nov 22 2013, Oct 27 2012

IsSemiprime:=func; [s: n in [2..500] | IsSemiprime(s) where s is NthPrime(n)-1]; // Vincenzo Librandi, Oct 17 2012

q:= n-> (n::even) and andmap(isprime, [n+1, n/2]):
select(q, [$1..5000])[];  # Alois P. Heinz, Jul 19 2023

Select[Range[6000],Plus@@Last/@FactorInteger[#]==2&&PrimeQ[#+1]&] (* Vladimir Joseph Stephan Orlovsky, May 08 2011 *)
Select[Range[3000],PrimeOmega[#]==2&&PrimeQ[#+1]&] (* Harvey P. Dale, Oct 16 2012 *)
Select[ Prime@ Range@ 430 - 1, PrimeOmega@# == 2 &] (* Robert G. Wilson v, Feb 18 2014 *)

[x-1|x<-primes(10^4),bigomega(x-1)==2] \\ Charles R Greathouse IV, Nov 22 2013

a(n) = A005385(n) - 1 = 2*A005384(n).
A010051(A006093(a(n))/2) = A064911(A006093(a(n))) = 1. - Reinhard Zumkeller, Nov 22 2013
a(n) = A077068(n) - A232342(n). - Reinhard Zumkeller, Dec 16 2013
a(n) = A000010(A194593(n+1)). - Torlach Rush, Aug 23 2018
A000010((a(n)*2)+2) = A023900((a(n)*2)+2). - Torlach Rush, Aug 23 2018

A079704 a(n) = 2*prime(n)^2.

Original entry on oeis.org

8, 18, 50, 98, 242, 338, 578, 722, 1058, 1682, 1922, 2738, 3362, 3698, 4418, 5618, 6962, 7442, 8978, 10082, 10658, 12482, 13778, 15842, 18818, 20402, 21218, 22898, 23762, 25538, 32258, 34322, 37538, 38642, 44402, 45602, 49298, 53138, 55778

Offset: 1

Jon Perry, Jan 31 2003

Numbers of the form 2*p^2 where p runs through the primes.

For these numbers m, there are precisely 5 groups of order m, hence this is a subsequence of A054397. If p = 2, these 5 groups of order 8 are described in example section of A054397, and when p is odd prime, the five corresponding groups are described in a comment of A143928. - Bernard Schott, Dec 11 2021

			a(2) = prime(2)^2*2 = 3^2*2 = 9*2 = 18.

Pascal Ortiz, Exercices d'Algèbre, Collection CAPES / Agrégation, Ellipses, problème 1.35, pp. 70-74, 2004.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000040, A001248, A054397, A100484.

A143928 is a subsequence.

a079704 = (* 2) . a001248  -- Reinhard Zumkeller, Nov 19 2013

[2*p^2: p in PrimesUpTo(200)]; // Vincenzo Librandi, Mar 27 2014

2 Prime[Range[40]]^2 (* Vincenzo Librandi, Mar 27 2014 *)

PARI
```
forprime (p=2,100,print1(p^2*2","))
```

from sympy import primerange
print([2*p**2 for p in primerange(1, 200)]) # Michael S. Branicky, Dec 11 2021

a(n) = 2*A001248(n) = A100484(n)*A000040(n). - Reinhard Zumkeller, Nov 19 2013

More terms from Vincenzo Librandi, Jan 29 2010

Offset corrected by Reinhard Zumkeller, Nov 19 2013

cp's OEIS Frontend

A099777 Number of divisors of 2n.

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A087112 Triangle in which the n-th row contains n distinct semiprimes not listed previously with all prime factors from among the first n primes.

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A176504 a(n) = m + k where prime(m)*prime(k) = semiprime(n).

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A239929 Numbers n with the property that the symmetric representation of sigma(n) has two parts.

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A338900 Difference between the two prime indices of the n-th squarefree semiprime.

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A338904 Irregular triangle read by rows where row n lists all semiprimes whose prime indices sum to n.

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A338907 Semiprimes whose prime indices sum to an odd number.

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