A006881 -id:A006881 - cp's OEIS frontend

A338898 Concatenated sequence of prime indices of semiprimes (A001358).

1, 1, 1, 2, 2, 2, 1, 3, 1, 4, 2, 3, 2, 4, 1, 5, 3, 3, 1, 6, 2, 5, 1, 7, 3, 4, 1, 8, 2, 6, 1, 9, 4, 4, 2, 7, 3, 5, 2, 8, 1, 10, 1, 11, 3, 6, 2, 9, 1, 12, 4, 5, 1, 13, 3, 7, 1, 14, 2, 10, 4, 6, 2, 11, 1, 15, 3, 8, 1, 16, 2, 12, 3, 9, 1, 17, 4, 7, 5, 5, 1, 18, 2

Offset: 1

Gus Wiseman, Nov 15 2020

This is a triangle with two columns and weakly increasing rows, namely {A338912(n), A338913(n)}.

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 sequence of semiprimes together with their prime indices begins:
      4: {1,1}     46: {1,9}      91: {4,6}     141: {2,15}
      6: {1,2}     49: {4,4}      93: {2,11}    142: {1,20}
      9: {2,2}     51: {2,7}      94: {1,15}    143: {5,6}
     10: {1,3}     55: {3,5}      95: {3,8}     145: {3,10}
     14: {1,4}     57: {2,8}     106: {1,16}    146: {1,21}
     15: {2,3}     58: {1,10}    111: {2,12}    155: {3,11}
     21: {2,4}     62: {1,11}    115: {3,9}     158: {1,22}
     22: {1,5}     65: {3,6}     118: {1,17}    159: {2,16}
     25: {3,3}     69: {2,9}     119: {4,7}     161: {4,9}
     26: {1,6}     74: {1,12}    121: {5,5}     166: {1,23}
     33: {2,5}     77: {4,5}     122: {1,18}    169: {6,6}
     34: {1,7}     82: {1,13}    123: {2,13}    177: {2,17}
     35: {3,4}     85: {3,7}     129: {2,14}    178: {1,24}
     38: {1,8}     86: {1,14}    133: {4,8}     183: {2,18}
     39: {2,6}     87: {2,10}    134: {1,19}    185: {3,12}

A112798 restricted to rows of length 2 gives this triangle.
A115392 is the row number for the first appearance of each positive integer.
A176506 gives row differences.
A338899 is the squarefree version.
A338912 is column 1.
A338913 is column 2.
A001221 counts a number's distinct prime indices.
A001222 counts a number's prime indices.
A001358 lists semiprimes.
A004526 counts 2-part partitions.
A006881 lists squarefree semiprimes.
A037143 lists primes and semiprimes.
A046315 and A100484 list odd and even semiprimes.
A046388 and A100484 list odd and even squarefree semiprimes.
A065516 gives first differences of semiprimes.
A084126 and A084127 give the prime factors of semiprimes.
A270650 and A270652 give the prime indices of squarefree semiprimes.
A320655 counts factorizations into semiprimes.
Cf. A056239, A101048, A320892, A320912, A338900, A338901, A338904, A338906, A338907, A338910, A338911.

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

A338912 Lesser prime index of the n-th semiprime.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 4, 2, 3, 2, 1, 1, 3, 2, 1, 4, 1, 3, 1, 2, 4, 2, 1, 3, 1, 2, 3, 1, 4, 5, 1, 2, 2, 4, 1, 2, 1, 5, 3, 1, 3, 1, 2, 4, 1, 6, 2, 1, 2, 3, 5, 1, 2, 1, 4, 3, 1, 5, 2, 1, 3, 4, 1, 2, 6, 1, 3, 2, 6, 2, 5, 1, 4, 1, 3, 2, 1

Offset: 1

Gus Wiseman, Nov 20 2020

nonn

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

			The semiprimes are:
  2*2, 2*3, 3*3, 2*5, 2*7, 3*5, 3*7, 2*11, 5*5, 2*13, ...
so the lesser prime factors are:
  2, 2, 3, 2, 2, 3, 3, 2, 5, 2, ...
with indices:
  1, 1, 2, 1, 1, 2, 2, 1, 3, 1, ...

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

A084126 is the lesser prime factor (not index).
A084127 is the greater factor, with index A338913.
A115392 lists positions of ones.
A128301 lists positions of first appearances of each positive integer.
A270650 is the squarefree case, with greater part A270652.
A338898 has this as first column.
A001221 counts distinct prime indices.
A001222 counts prime indices.
A001358 lists semiprimes, with odds A046315 and evens A100484.
A006881 lists squarefree semiprimes, with odds A046388 and evens A100484.
A087794/A176504/A176506 are product/sum/difference of semiprime indices.
A338910/A338911 list products of pairs of odd/even-indexed primes.
Cf. A037143, A056239, A065516, A112798, A289182, A320732, A320892, A338900, A338904, A338909.

Table[Min[PrimePi/@First/@FactorInteger[n]],{n,Select[Range[100],PrimeOmega[#]==2&]}]

a(n) = A000720(A084126(n)).

A065387 a(n) = sigma(n) + phi(n).

Original entry on oeis.org

2, 4, 6, 9, 10, 14, 14, 19, 19, 22, 22, 32, 26, 30, 32, 39, 34, 45, 38, 50, 44, 46, 46, 68, 51, 54, 58, 68, 58, 80, 62, 79, 68, 70, 72, 103, 74, 78, 80, 106, 82, 108, 86, 104, 102, 94, 94, 140, 99, 113, 104, 122, 106, 138, 112, 144, 116, 118, 118, 184, 122, 126, 140

Offset: 1

Labos Elemer, Nov 05 2001

a(n) = 2n for n listed in A008578, the prime numbers at the beginning of the 20th century. When a(n) = a(n + 1), n is probably listed in A066198, numbers n where phi changes as fast as sigma (the only exceptions below 10000 are 2 and 854). - Alonso del Arte, Nov 16 2005
A. Makowski proved that n is prime if and only if a(n) = n * d(n), where d is A000005. - Charles R Greathouse IV, Mar 19 2012
If n is semiprime, a(n) = 2n+1+ceiling(sqrt(n))-floor(sqrt(n)). - Wesley Ivan Hurt, May 05 2015
Atanassov proves that a(n) >= n + A001414(n). - Charles R Greathouse IV, Dec 06 2016
a(n) = 2*n+1 iff n is square of prime (A001248), a(n) = 2*(n+1) iff n is squarefree semiprime (A006881). - Bernard Schott, Feb 09 2020

			a(10) = 22 because there are 4 coprimes to 10 below 10, the divisors of 10 add up to 18, and 4 + 18 = 22.

K. Atanassov, New integer functions, related to ψ and σ functions. IV., Bull. Number Theory Related Topics 12 (1988), pp. 31-35.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 149.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 162.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000 (First 1000 terms from T. D. Noe.)
A. Makowski, Aufgaben 339, Elemente der Mathematik 15 (1960), pp. 39-40.

Cf. A000010, A000203, A065388, A015702, A051709, A011774.

See A292768 for partial sums, A051612 for sigma - phi.

[DivisorSigma(1,k)+EulerPhi(k):k in [1..65]]; // Marius A. Burtea, Feb 09 2020

with(numtheory); A065387:=n->phi(n) + sigma(n); seq(A065387(n), n=1..100); # Wesley Ivan Hurt, Apr 08 2014

Table[EulerPhi[n] + DivisorSigma[1,n], {n, 65}] (* Alonso del Arte *)
a[n_] := SeriesCoefficient[Sum[(1+MoebiusMu[k])*x^k/(1-x^k)^2, {k, 1, n}], {x, 0, n}]; Array[a, 63] (* Jean-François Alcover, Sep 29 2017, after Ilya Gutkovskiy *)

a(n) = sigma(n) + eulerphi(n) \\ Harry J. Smith, Oct 17 2009

[sigma(n,1)+euler_phi(n) for n in range(1, 64)] # Stefano Spezia, Jul 20 2025

a(n) = A000203(n) + A000010(n).
a(n) = A051709(n) + 2n. - N. J. A. Sloane, Jun 12 2004
G.f.: Sum_{k>=1} (mu(k) + 1)*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Sep 29 2017

A030513 Numbers with 4 divisors.

Original entry on oeis.org

6, 8, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187

Offset: 1

Jeff Burch

Essentially the same as A007422.
Numbers which are either the product of two distinct primes (A006881) or the cube of a prime (A030078).
4*a(n) are the solutions to A048272(x) = Sum_{d|x} (-1)^d = 4. - Benoit Cloitre, Apr 14 2002
Since A119479(4)=3, there are never more than 3 consecutive integers in the sequence. Triples of consecutive integers start at 33, 85, 93, 141, 201, ... (A039833). No such triple contains a term of the form p^3. - Ivan Neretin, Feb 08 2016
Numbers that are equal to the product of their proper divisors (A007956) (proof in Sierpiński). - Bernard Schott, Apr 04 2022

Wacław Sierpiński, Elementary Theory of Numbers, Ex. 2 p. 174, Warsaw, 1964.

Cf. A000005, A007422, A007956, A030515, A035533, A048272, A119479.

Equals the disjoint union of A006881 and A030078.

[n: n in [1..200] | DivisorSigma(0, n) eq 4]; // Vincenzo Librandi, Jul 16 2015

Select[Range[200], DivisorSigma[0,#]==4&] (* Harvey P. Dale, Apr 06 2011 *)

is(n)=numdiv(n)==4 \\ Charles R Greathouse IV, May 18 2015

from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A030513(n):
    def f(x): return int(n+x-primepi(integer_nthroot(x,3)[0])+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 16 2024

{n : A000005(n) = 4}. - Juri-Stepan Gerasimov, Oct 10 2009

Incorrect comments removed by Charles R Greathouse IV, Mar 18 2010

A046387 Products of exactly 5 distinct primes.

Original entry on oeis.org

2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006, 6090, 6270, 6510, 6630, 7410, 7590, 7770, 7854, 8610, 8778, 8970, 9030, 9282, 9570, 9690, 9870, 10010, 10230, 10374, 10626, 11130, 11310, 11730, 12090, 12210, 12390, 12558, 12810, 13090, 13110

Offset: 1

Patrick De Geest, Jun 15 1998

Subsequence of A051270. 4620 = 2^2*3*5*7*11 is in A051270 but not in here, for example. - R. J. Mathar, Nov 10 2014

			a(1) = 2310 = 2 * 3 * 5 * 7 * 11 = A002110(5) = 5#.
a(2) = 2730 = 2 * 3 * 5 * 7 * 13.
a(3) = 3570 = 2 * 3 * 5 * 7 * 17.
a(10) = 6006 = 2 * 3 * 7 * 11 * 13.

T. D. Noe, Table of n, a(n) for n = 1..10000

Products of exactly k distinct primes, for k = 1 to 6: A000040, A006881. A007304, A046386, A046387, A067885.
Cf. A000040, A000961, A001221, A005117, A000977, A002110, A006881, A007304, A007774, A033992, A033993, A046386.
Cf. A014614, A046403, A051270.

A046387 := proc(n)
    option remember;
    local a;
    if n = 1 then
        2*3*5*7*11 ;
    else
        for a from procname(n-1)+1 do
            if A001221(a)= 5 and issqrfree(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Oct 13 2019

f5Q[n_]:=Last/@FactorInteger[n]=={1, 1, 1, 1, 1}; lst={};Do[If[f5Q[n], AppendTo[lst, n]], {n, 8!}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 26 2008 *)

is(n)=factor(n)[,2]==[1,1,1,1,1]~ \\ Charles R Greathouse IV, Sep 17 2015

is(n)= omega(n)==5 && bigomega(n)==5 \\ Hugo Pfoertner, Dec 18 2018

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A046387(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,5)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f) # Chai Wah Wu, Aug 30 2024

Entry revised by N. J. A. Sloane, Apr 10 2006

A067885 Products of exactly 6 distinct primes.

Original entry on oeis.org

30030, 39270, 43890, 46410, 51870, 53130, 62790, 66990, 67830, 71610, 72930, 79170, 81510, 82110, 84630, 85470, 91770, 94710, 98670, 99330, 101010, 102102, 103530, 106590, 108570, 110670, 111930, 114114, 115710, 117390, 122430, 123690, 124410, 125970, 128310

Offset: 1

Benoit Cloitre, Mar 02 2002

nonn

Subsequence of A074969. - R. J. Mathar, Nov 24 2009

Products of exactly k distinct primes, for k = 1 to 6: A000040, A006881. A007304, A046386, A046387, A067885.

Select[Range[125000],PrimeNu[#]==PrimeOmega[#]==6&] (* Harvey P. Dale, May 14 2014 *)

is(n)=factor(n)[,2]==[1,1,1,1,1,1]~ \\ Charles R Greathouse IV, Sep 14 2015

is(n)=omega(n)==6 && bigomega(n)==6 \\ Hugo Pfoertner, Dec 18 2018

list(lim)=lim\=1; my(v=List(), L1,L2,L3,L4,P4,P5); forprime(p=13,lim\2310, L1=lim\p; forprime(q=11,min(L1\210,p-2), L2=L1\q; forprime(r=7, min(L2\30,q-2), L3=L2\r; forprime(s=5,min(L3\6,r-2), L4=L3\s; P4=p*q*r*s; forprime(t=3, min(L4\2,s-2), P5=P4*t; forprime(u=2, min(L4\t,t-1), listput(v,P5*u))))))); Set(v) \\ Charles R Greathouse IV, Aug 27 2021

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A067885(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,6)))
    kmin, kmax = 0,1
    while f(kmax) > kmax:
        kmax <<= 1
    while kmax-kmin > 1:
        kmid = kmax+kmin>>1
        if f(kmid) <= kmid:
            kmax = kmid
        else:
            kmin = kmid
    return kmax # Chai Wah Wu, Aug 29 2024

{k: A001221(k) = A001222(k) = 6}. - R. J. Mathar, Jul 18 2023

A320892 Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into distinct semiprimes.

Original entry on oeis.org

16, 64, 81, 96, 144, 160, 224, 256, 324, 352, 384, 400, 416, 486, 544, 576, 608, 625, 640, 729, 736, 784, 864, 896, 928, 960, 992, 1024, 1184, 1215, 1296, 1312, 1344, 1376, 1408, 1440, 1504, 1536, 1600, 1664, 1696, 1701, 1888, 1936, 1944, 1952, 2016, 2025

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

A semiprime (A001358) is a product of any two not necessarily distinct primes.
If A025487(k) is in the sequence then so is every number with the same prime signature. - David A. Corneth, Oct 23 2018
Numbers for which A001222(n) is even and A322353(n) is zero. - Antti Karttunen, Dec 06 2018

			A complete list of all factorizations of 1296 into semiprimes is:
  1296 = (4*4*9*9)
  1296 = (4*6*6*9)
  1296 = (6*6*6*6)
None of these is strict, so 1296 belongs to the sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A001055, A001358, A005117, A006881, A007717, A025487, A028260, A045778, A318871, A318953, A320462, A320655, A320656, A320891, A320893, A320894, A322353.

strsemfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strsemfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],PrimeOmega[#]==2&]}]];
Select[Range[1000],And[EvenQ[PrimeOmega[#]],strsemfacs[#]=={}]&]

A322353(n, m=n, facs=List([])) = if(1==n, my(u=apply(bigomega,Vec(facs))); (0==length(u)||(2==vecmin(u)&&2==vecmax(u))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A322353(n/d, d-1, newfacs))); (s));
isA300892(n) = if(bigomega(n)%2,0,(0==A322353(n))); \\ Antti Karttunen, Dec 06 2018

A353851 Number of integer compositions of n with all equal run-sums.

Original entry on oeis.org

1, 1, 2, 2, 5, 2, 8, 2, 12, 5, 8, 2, 34, 2, 8, 8, 43, 2, 52, 2, 70, 8, 8, 2, 282, 5, 8, 18, 214, 2, 386, 2, 520, 8, 8, 8, 1957, 2, 8, 8, 2010, 2, 2978, 2, 3094, 94, 8, 2, 16764, 5, 340, 8, 12310, 2, 26514, 8, 27642, 8, 8, 2, 132938, 2, 8, 238, 107411, 8, 236258

Offset: 0

Gus Wiseman, May 31 2022

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

			The a(0) = 1 through a(8) = 12 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
           (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                        (112)            (222)                (224)
                        (211)            (1113)               (422)
                        (1111)           (2112)               (2222)
                                         (3111)               (11114)
                                         (11211)              (41111)
                                         (111111)             (111122)
                                                              (112112)
                                                              (211211)
                                                              (221111)
                                                              (11111111)
For example:
  (1,1,2,1,1) has run-sums (2,2,2) so is counted under a(6).
  (4,1,1,1,1,2,2) has run-sums (4,4,4) so is counted under a(12).
  (3,3,2,2,2) has run-sums (6,6) so is counted under a(12).

David A. Corneth, Table of n, a(n) for n = 0..10000

The version for parts or runs instead of run-sums is A000005.
The version for multiplicities instead of run-sums is A098504.
All parts are divisors of n, see A100346.
The version for partitions is A304442, ranked by A353833.
The version for run-lengths instead of run-sums is A329738, ptns A047966.
These compositions are ranked by A353848.
The distinct instead of equal version is A353850.
A003242 counts anti-run compositions, ranked by A333489.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A353847 represents the composition run-sum transformation.
For distinct instead of equal run-sums: A032020, A098859, A242882, A329739, A351013, A353837, ranked by A353838 (complement A353839), A353852, A354580, ranked by A354581.
Cf. A000005, A006881, A238279, A275870, A333755, A351014, A351016, A351017, A353832, A353834, A353849, A353853-A353859 (run-sum trajectory), A353860, A353863, A353864, A353932.

Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n],SameQ@@Total/@Split[#]&]],{n,0,15}]

a(n) = {if(n <=1, return(1)); my(d = divisors(n), res = 0); for(i = 1, #d, nd = numdiv(d[i]); res+=(nd*(nd-1)^(n/d[i]-1)) ); res } \\ David A. Corneth, Jun 02 2022

From David A. Corneth, Jun 02 2022 (Start)
a(p) = 2 for prime p.
a(p*q) = 8 for distinct primes p and q (Cf. A006881).
a(n) = Sum_{d|n} tau(d)*(tau(d)-1) ^ (n/d - 1) where tau = A000005. (End)

More terms from David A. Corneth, Jun 02 2022

A085987 Product of exactly four primes, three of which are distinct (p^2qr).

Original entry on oeis.org

60, 84, 90, 126, 132, 140, 150, 156, 198, 204, 220, 228, 234, 260, 276, 294, 306, 308, 315, 340, 342, 348, 350, 364, 372, 380, 414, 444, 460, 476, 490, 492, 495, 516, 522, 525, 532, 550, 558, 564, 572, 580, 585, 620, 636, 644, 650, 666, 693, 708, 726

Offset: 1

Alford Arnold, Jul 08 2003

nonn

A014613 is completely determined by A030514, A065036, A085986, A085987 and A046386 since p(4) = 5. (cf. A000041). More generally, the first term of sequences which completely determine the k-almost primes can be found in A036035 (a resorted version of A025487).

A050326(a(n)) = 4. - Reinhard Zumkeller, May 03 2013

			a(1) = 60 since 60 = 2*2*3*5 and has three distinct prime factors.

Cf. A001248, A006881, A030078, A054753, A007304, A050997, A046387, A036035, A086974.

Subsequence of A014613, A307341, A178212.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,2}; Select[Range[2000], f] (* Vladimir Joseph Stephan Orlovsky, May 03 2011 *)
pefp[{a_,b_,c_}]:={a^2 b c,a b^2 c,a b c^2}; Module[{upto=800},Select[ Flatten[ pefp/@Subsets[Prime[Range[PrimePi[upto/6]]],{3}]]//Union,#<= upto&]] (* Harvey P. Dale, Oct 02 2018 *)

list(lim)=my(v=List(),t,x,y,z);forprime(p=2,lim^(1/4),t=lim\p^2;forprime(q=p+1,sqrtint(t),forprime(r=q+1,t\q,x=p^2*q*r;y=p*q^2*r;listput(v,x);if(y<=lim,listput(v,y);z=p*q*r^2;if(z<=lim,listput(v,z))))));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 15 2011

is(n)=vecsort(factor(n)[,2]~)==[1,1,2] \\ Charles R Greathouse IV, Oct 19 2015

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A085987(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((t:=primepi(s:=isqrt(y:=x//r**2)))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(isqrt(x)+1))+sum(primepi(x//p**3) for p in primerange(integer_nthroot(x,3)[0]+1))-primepi(integer_nthroot(x,4)[0])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

More terms from Reinhard Zumkeller, Jul 25 2003

A176506 Difference between the prime indices of the two factors of the n-th semiprime.

Original entry on oeis.org

0, 1, 0, 2, 3, 1, 2, 4, 0, 5, 3, 6, 1, 7, 4, 8, 0, 5, 2, 6, 9, 10, 3, 7, 11, 1, 12, 4, 13, 8, 2, 9, 14, 5, 15, 10, 6, 16, 3, 0, 17, 11, 12, 4, 18, 13, 19, 1, 7, 20, 8, 21, 14, 5, 22, 0, 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, 30, 8, 31, 13, 22

Offset: 1

Juri-Stepan Gerasimov, Apr 19 2010

Are there no adjacent equal terms? I have verified this up to n = 10^6. - Gus Wiseman, Dec 04 2020

			From _Gus Wiseman_, Dec 04 2020: (Start)
The sequence of semiprimes together with the corresponding differences begins:
   4: 1 - 1 = 0
   6: 2 - 1 = 1
   9: 2 - 2 = 0
  10: 3 - 1 = 2
  14: 4 - 1 = 3
  15: 3 - 2 = 1
  21: 4 - 2 = 2
  22: 5 - 1 = 4
  25: 3 - 3 = 0
  26: 6 - 1 = 5
  33: 5 - 2 = 3
(End)

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

Cf. A109313.
Cf. A001358, A049084, A084126, A084127.
A087794 is product of the same indices.
A176504 is the sum of the same indices.
A115392 lists positions of first appearances.
A128301 lists positions of 0's.
A172348 lists positions of 1's.
A338898 has this sequence as row differences.
A338900 is the squarefree case.
A338912/A338913 give the two prime indices of semiprimes.
A006881 lists squarefree semiprimes.
A024697 is the sum of semiprimes of weight n.
A056239 gives sum of prime indices (Heinz weight).
A087112 groups semiprimes by greater factor.
A270650/A270652/A338899 give the prime indices of squarefree semiprimes.
A338904 groups semiprimes by weight.
A338907/A338906 list semiprimes of odd/even weight.
A339114/A339115 give the least/greatest semiprime of weight n.
Cf. A000040, A001222, A062198, A112141, A112798, A325699, A320655, A339005.

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:
A176506 := proc(n) numtheory[pi](A084127(n)) - numtheory[pi](A084126(n)) ; end proc: seq(A176506(n),n=1..120) ; # R. J. Mathar, Apr 22 2010
# Alternative:
N:= 500: # to use the first N semiprimes
Primes:= select(isprime, [2,seq(i,i=3..N/2,2)]):
SP:= NULL:
for i from 1 to nops(Primes) do
  for j from 1 to i do
    sp:= Primes[i]*Primes[j];
    if sp > N then break fi;
    SP:= SP, [sp, i-j]
od od:
SP:= sort([SP],(s,t) -> s[1] t[2], SP); # Robert Israel, Jan 17 2019

M = 500; (* to use the first M semiprimes *)
primes = Select[Join[{2}, Range[3, M/2, 2]], PrimeQ];
SP = {};
For[i = 1, i <= Length[primes], i++,
  For[j = 1, j <= i, j++,
    sp = primes[[i]] primes[[j]];
    If[sp > M, Break []];
    AppendTo[SP, {sp, i - j}]
]];
SortBy[SP, First][[All, 2]] (* Jean-François Alcover, Jul 18 2020, after Robert Israel *)
Table[If[!SquareFreeQ[n],0,-Subtract@@PrimePi/@First/@FactorInteger[n]],{n,Select[Range[100],PrimeOmega[#]==2&]}] (* Gus Wiseman, Dec 04 2020 *)

lista(nn) = {my(vsp = select(x->(bigomega(x)==2), [1..nn])); vector(#vsp, k, my(f=factor(vsp[k])[,1]); primepi(vecmax(f)) - primepi(vecmin(f)));} \\ Michel Marcus, Jul 18 2020

a(n) = A049084(A084127(n)) - A049084(A084126(n)). [corrected by R. J. Mathar, Apr 22 2010]

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

a(51) and a(69) corrected by R. J. Mathar, Apr 22 2010

cp's OEIS Frontend

A338898 Concatenated sequence of prime indices of semiprimes (A001358).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A338912 Lesser prime index of the n-th semiprime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A065387 a(n) = sigma(n) + phi(n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

A030513 Numbers with 4 divisors.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

Extensions

A046387 Products of exactly 5 distinct primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

Extensions

A067885 Products of exactly 6 distinct primes.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Python

Formula

A085987 Product of exactly four primes, three of which are distinct (p^2qr).