A006881 -id:A006881 - cp's OEIS frontend

A338899 Concatenated sequence of prime indices of squarefree semiprimes (A006881).

1, 2, 1, 3, 1, 4, 2, 3, 2, 4, 1, 5, 1, 6, 2, 5, 1, 7, 3, 4, 1, 8, 2, 6, 1, 9, 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, 1, 18, 2, 13, 2, 14, 4, 8, 1, 19, 2, 15

Offset: 1

Gus Wiseman, Nov 16 2020

This is a triangle with two columns and strictly increasing rows, namely {A270650(n), A270652(n)}.

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.

			The sequence of terms together with their prime indices begins:
      6: {1,2}     57: {2,8}     106: {1,16}    155: {3,11}
     10: {1,3}     58: {1,10}    111: {2,12}    158: {1,22}
     14: {1,4}     62: {1,11}    115: {3,9}     159: {2,16}
     15: {2,3}     65: {3,6}     118: {1,17}    161: {4,9}
     21: {2,4}     69: {2,9}     119: {4,7}     166: {1,23}
     22: {1,5}     74: {1,12}    122: {1,18}    177: {2,17}
     26: {1,6}     77: {4,5}     123: {2,13}    178: {1,24}
     33: {2,5}     82: {1,13}    129: {2,14}    183: {2,18}
     34: {1,7}     85: {3,7}     133: {4,8}     185: {3,12}
     35: {3,4}     86: {1,14}    134: {1,19}    187: {5,7}
     38: {1,8}     87: {2,10}    141: {2,15}    194: {1,25}
     39: {2,6}     91: {4,6}     142: {1,20}    201: {2,19}
     46: {1,9}     93: {2,11}    143: {5,6}     202: {1,26}
     51: {2,7}     94: {1,15}    145: {3,10}    203: {4,10}
     55: {3,5}     95: {3,8}     146: {1,21}    205: {3,13}

A270650 is the first column.
A270652 is the second column.
A320656 counts multiset partitions using these rows, or factorizations into squarefree semiprimes.
A338898 is the version including squares, with columns A338912 and A338913.
A338900 gives row differences.
A338901 gives the row numbers for first appearances.
A001221 and A001222 count distinct/all prime indices.
A001358 lists semiprimes.
A004526 counts 2-part partitions, with strict case shifted right once.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes.
A046315 and A100484 list odd and even semiprimes.
A046388 lists odd squarefree semiprimes.
A166237 gives first differences of squarefree semiprimes.
Cf. A030229, A056239, A065516, A112798, A115392, A167171, A176506, A320893, A338904, A338906, A338907, A338910, A338911.

Join@@Cases[Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&],k_:>PrimePi/@First/@FactorInteger[k]]

A270650 Min(i, j), where p(i)*p(j) is the n-th term of A006881.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 25 2016

			A006881 = (6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, ... ), the increasing sequence of all products of distinct primes.  The first 4 factorizations are 2*3, 2*5, 2*7, 3*5, so that (a(1), a(2), a(3), a(4)) = (1,1,1,2).

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000040, A006881, A096916, A070647, A270652, A270003.

mx = 350; t = Sort@Flatten@Table[Prime[n]*Prime[m], {n, Log[2, mx/3]}, {m, n + 1, PrimePi[mx/Prime[n]]}]; (* A006881, Robert G. Wilson v, Feb 07 2012 *)
u = Table[FactorInteger[t[[k]]][[1]], {k, 1, Length[t]}];
u1 = Table[u[[k]][[1]], {k, 1, Length[t]}]  (* A096916 *)
PrimePi[u1]  (* A270650 *)
v = Table[FactorInteger[t[[k]]][[2]], {k, 1, Length[t]}];
v1 = Table[v[[k]][[1]], {k, 1, Length[t]}]  (* A070647 *)
PrimePi[v1]  (* A270652 *)
d = v1 - u1  (* A176881 *)
Map[PrimePi[FactorInteger[#][[1, 1]]] &, Select[Range@ 240, And[SquareFreeQ@ #, PrimeOmega@ # == 2] &]] (* Michael De Vlieger, Apr 25 2016 *)

A270652 Max(i,j), where p(i)*p(j) is the n-th term of A006881.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 25 2016

			A006881 = (6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, ... ), the increasing sequence of all products of distinct primes.  The first 4 factorizations are 2*3, 2*5, 2*7, 3*5, so that (a(1), a(2), a(3), a(4)) = (2,3,4,3).

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000040, A006881, A096916, A270650, A070647, A270003.

mx = 350; t = Sort@Flatten@Table[Prime[n]*Prime[m], {n, Log[2, mx/3]}, {m, n + 1, PrimePi[mx/Prime[n]]}]; (* A006881, Robert G. Wilson v, Feb 07 2012 *)
u = Table[FactorInteger[t[[k]]][[1]], {k, 1, Length[t]}];
u1 = Table[u[[k]][[1]], {k, 1, Length[t]}]  (* A096916 *)
PrimePi[u1]  (* A270650 *)
v = Table[FactorInteger[t[[k]]][[2]], {k, 1, Length[t]}];
v1 = Table[v[[k]][[1]], {k, 1, Length[t]}]  (* A070647 *)
PrimePi[v1]  (* A270652 *)
d = v1 - u1  (* A176881 *)
Map[PrimePi[FactorInteger[#][[-1, 1]]] &, Select[Range@ 240, And[SquareFreeQ@ #, PrimeOmega@ # == 2] &]] (* Michael De Vlieger, Apr 25 2016 *)

from math import isqrt
from sympy import primepi, primerange, primefactors
def A270652(n):
    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
    def f(x): return int(n+x+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    return primepi(max(primefactors(bisection(f,n,n)))) # Chai Wah Wu, Oct 23 2024

A168472 Partial sums of products of two distinct primes (A006881).

Original entry on oeis.org

6, 16, 30, 45, 66, 88, 114, 147, 181, 216, 254, 293, 339, 390, 445, 502, 560, 622, 687, 756, 830, 907, 989, 1074, 1160, 1247, 1338, 1431, 1525, 1620, 1726, 1837, 1952, 2070, 2189, 2311, 2434, 2563, 2696, 2830, 2971, 3113, 3256, 3401, 3547, 3702, 3860, 4019

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 26 2009

nonn

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

N:= 1001: # to get all a(n) where A006881(n) < N
Primes:= select(isprime, [2, seq(2*k+1, k=1..floor(N/2))]):
L:= sort(convert({seq(seq(p*q, q=Primes[1..ListTools:-BinaryPlace(Primes, N/p)]), p=Primes)} minus {seq(p^2, p=Primes)},list)):
ListTools:-PartialSums(L); # Robert Israel, Apr 29 2018

f[n_]:=Last/@FactorInteger[n]=={1,1}; s=0;lst={};Do[If[f[n],AppendTo[lst,s+=n]],{n,6!}];lst
With[{nn=50},Take[Accumulate[Union[Times@@@Subsets[Prime[Range[nn]],{2}]]],nn]] (* Harvey P. Dale, Aug 08 2013 *)

A166237 Differences between consecutive products of two distinct primes: a(n) = A006881(n+1) - A006881(n).

Original entry on oeis.org

4, 4, 1, 6, 1, 4, 7, 1, 1, 3, 1, 7, 5, 4, 2, 1, 4, 3, 4, 5, 3, 5, 3, 1, 1, 4, 2, 1, 1, 11, 5, 4, 3, 1, 3, 1, 6, 4, 1, 7, 1, 1, 2, 1, 9, 3, 1, 2, 5, 11, 1, 5, 2, 2, 7, 7, 1, 1, 2, 1, 3, 4, 1, 1, 2, 1, 1, 2, 5, 9, 2, 10, 2, 4, 1, 5, 3, 3, 2, 7, 4, 9, 4, 4, 3, 1, 2, 1, 1, 2, 4, 5, 5, 2, 2, 3, 1, 2, 5, 1, 4, 2, 5, 9, 3

Offset: 1

Vladimir Joseph Stephan Orlovsky, Oct 09 2009

nonn

Goldston, Graham, Pintz & Yıldırım (2005) prove that a(n+1) - a(n) <= 26 infinitely often. - Charles R Greathouse IV, Dec 26 2020

Michael De Vlieger, Table of n, a(n) for n = 1..10000
D. A. Goldston, S. W. Graham, J. Pintz and C. Y. Yıldırım, Small gaps between primes and almost primes, arXiv:math/0506067 [math.NT], 2005; Proceedings of the London Mathematical Society 98:3 (May 2009), pp. 741-774.
Yang Liu, Peter S. Park, and Zhuo Qun Song, Bounded gaps between products of distinct primes, arXiv:1607.03887 [math.NT], 2016-2017; Research in Number Theory 3:26 (2017).
Keiju Sono, Small gaps between the set of products of at most two primes, arXiv:1605.02920 [math.NT], 2016-2018; Journal of the Mathematical Society of Japan 72:1 (2020), pp. 81-118.

Cf. A006881 (products of two distinct primes), A001358 (semiprimes: products of two primes), A065516 (differences between products of two primes), A001223 (differences between consecutive primes).

T:=[ n: n in [1..360] | #PrimeDivisors(n) eq 2 and &*[ d[2]: d in Factorization(n) ] eq 1 ]; [ T[j+1]-T[j]: j in [1..#T-1] ]; // Klaus Brockhaus, Oct 13 2009

f[n_]:=Last/@FactorInteger[n]=={1,1}; a=6;lst={};Do[If[f[n],AppendTo[lst,n-a];a=n],{n,9,6!}];lst

{m=106; v=vector(m); n=0; c=0; while(cKlaus Brockhaus, Oct 13 2009

Edited by Klaus Brockhaus, Oct 13 2009

Added formula to clarify the definition. - N. J. A. Sloane, Jul 19 2022

A048639 Binary encoding of A006881, numbers with two distinct prime divisors.

Original entry on oeis.org

3, 5, 9, 6, 10, 17, 33, 18, 65, 12, 129, 34, 257, 66, 20, 130, 513, 1025, 36, 258, 2049, 24, 4097, 68, 8193, 514, 40, 1026, 16385, 132, 32769, 2050, 260, 65537, 72, 131073, 4098, 8194, 136, 262145, 16386, 524289, 48, 516, 1048577, 1028, 2097153, 32770

Offset: 1

Antti Karttunen, Jul 14 1999

nonn

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

Permutation of A018900. Cf. A048640, A048623.

encode_A006881 := proc(upto_n) local b,i; b := [ ]; for i from 1 to upto_n do if((0 <> mobius(i)) and (4 = tau(i))) then b := [ op(b), bef(i) ]; fi; od: RETURN(b); end; # see A048623 for bef

Total[2^PrimePi@ # &@ (Map[First, FactorInteger@ #] - 1)] & /@ Select[Range@ 160, SquareFreeQ@ # && PrimeOmega@ # == 2 &] (* Michael De Vlieger, Oct 01 2015 *)

lista(nn) = {for (n=1, nn, if (issquarefree(n) && bigomega(n)==2, f = factor(n); x = sum(k=1, #f~, 2^(primepi(f[k,1])-1)); print1(x, ", ");););} \\ Michel Marcus, Oct 01 2015

from math import isqrt
from sympy import primepi, primerange, primefactors
def A048639(n):
    def f(x): return int(n+x+(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 sum(1<Chai Wah Wu, Feb 22 2025

a(n) = 2^(i-1) + 2^(j-1), where A006881(n) = p_i*p_j (p_i and p_j stand for the i-th and j-th primes respectively, where the first prime is 2).

A228578 Sum of the distinct prime factors of the squarefree semiprimes (A006881).

Original entry on oeis.org

5, 7, 9, 8, 10, 13, 15, 14, 19, 12, 21, 16, 25, 20, 16, 22, 31, 33, 18, 26, 39, 18, 43, 22, 45, 32, 20, 34, 49, 24, 55, 40, 28, 61, 24, 63, 44, 46, 26, 69, 50, 73, 24, 34, 75, 36, 81, 56, 30, 85, 62, 91, 64, 42, 28, 99, 70, 103, 36, 46, 105, 30, 74, 109, 48, 38, 111

Offset: 1

Wesley Ivan Hurt, Aug 28 2013

Sum of the distinct prime factors of A006881(n). If A006881(n) is even then a(n) = A006881(n)/2 + 2. If A006881(n) is odd then a(n) is even.

			a(1) = 5, since 6 is the first squarefree semiprime and the sum of the distinct prime factors of 6 is 2 + 3 = 5. a(2) = 7 since 10 is the second squarefree semiprime and the sum of the distinct prime factors of 10 is 2 + 5 = 7.

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

Cf. A006881, A001414, A008472.

Total[First /@ FactorInteger@ #] & /@ Select[Range@ 240, PrimeNu@ # == 2 && SquareFreeQ@ # &] (* Michael De Vlieger, Oct 28 2015 *)

do(x)=my(v=List()); forprime(p=3,x\2, forprime(q=2,min(x\p,p-1), listput(v,[p*q,p+q]))); v=vecsort(Vec(v),1); apply(u->u[2],v) \\ Charles R Greathouse IV, Nov 05 2017

from math import isqrt
from sympy import primepi, primerange, primefactors
def A228578(n):
    def f(x): return int(n+x+(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 sum(primefactors(m)) # Chai Wah Wu, Aug 16 2024

a(n) = sopf(A006881(n)) = A008472(A006881(n)).

Also, a(n) = sopfr(A006881(n)) = A001414(A006881(n)) because A006881 are squarefree. - Zak Seidov, Oct 28 2015

a(61)-a(67) corrected by Michael De Vlieger, Oct 28 2015

A324333 Numbers d such that A324331(x) = d^2 only when x is a squarefree semiprime (A006881).

Original entry on oeis.org

1, 2, 4, 6, 11, 12, 14, 16, 18, 20, 21, 22, 24, 25, 27, 29, 30, 32, 34, 35, 36, 38, 39, 41, 42, 43, 44, 45, 47, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79, 81, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 100

Offset: 1

Michel Marcus, Feb 23 2019

nonn

Problem posed by Albert A. Mullin, who asks if this sequence is infinite.

B. Alspach, Research problems, Problem 18, Discrete Math 40 (1982), page 126.

Cf. A000010, A000203, A006881.

Cf. A324331, A324334 (complement).

A324334 Numbers d such that A324331(x)=d^2 not only for squarefree semiprimes x (A006881) but also for x in A324333.

Original entry on oeis.org

3, 5, 7, 8, 9, 10, 13, 15, 17, 19, 23, 26, 28, 31, 33, 37, 40, 46, 48, 49, 55, 61, 62, 63, 65, 78, 80, 82, 96, 99, 118, 122, 126, 127, 129, 142, 144, 145, 148, 157, 159, 163, 166, 172, 176, 179, 185, 226, 228, 230, 240, 242, 244, 246, 249, 255, 257, 258, 288, 296, 303, 320, 321, 328, 342, 354, 357, 358, 360

Offset: 1

Michel Marcus, Feb 23 2019

nonn

			A324331(45) = 64, a square, even though 45 is not squarefree semiprime, so 8 is a term, and 45 is in A324332.

B. Alspach, Research problems, Problem 18, Discrete Math 40 (1982), page 126.

Cf. A000010, A000203, A006881.

Cf. A324331, A324332, A324333 (complement).

A154932 Decimal expansion of Sum_{q in A006881} 1/(q(q-1)) over the squarefree semiprimes q.

Original entry on oeis.org

0, 7, 1, 6, 0, 6, 0, 1, 5, 3, 6, 4, 0, 6, 2, 9, 5, 0, 6, 8, 9, 0, 1, 4, 9, 0, 5, 2, 3, 3, 2, 7, 8, 5, 7, 0, 0, 3, 2, 9, 7, 7, 5, 7, 7, 4, 9, 6, 7, 6, 4, 7, 6, 6, 9, 9, 6, 8, 8, 1, 5

Offset: 0

R. J. Mathar, Jan 17 2009

A variant of A152447.

			Equals 0.071606015364062950... = 1/(6*5) + 1/(10*9) + 1/(14*13) + 1/(15*14) + ...

R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], 2008-2009; see Table 9.

Equals Sum_{j>=1} 1/(A006881(j)*(A006881(j)-1)) = A152447 - Sum_{p in A000040} 1/(p^2*(p^2-1)) = A152447 + A085548 - Sum_{p in A000040} 1/(p^2-1).

cp's OEIS Frontend

A338899 Concatenated sequence of prime indices of squarefree semiprimes (A006881).

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A270650 Min(i, j), where p(i)*p(j) is the n-th term of A006881.

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A270652 Max(i,j), where p(i)*p(j) is the n-th term of A006881.

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A168472 Partial sums of products of two distinct primes (A006881).

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Maple

Mathematica

A166237 Differences between consecutive products of two distinct primes: a(n) = A006881(n+1) - A006881(n).

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Magma

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A048639 Binary encoding of A006881, numbers with two distinct prime divisors.

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A228578 Sum of the distinct prime factors of the squarefree semiprimes (A006881).

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A324333 Numbers d such that A324331(x) = d^2 only when x is a squarefree semiprime (A006881).

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