A006881 -id:A006881 - cp's OEIS frontend

A280683 Number of ways to write n as an ordered sum of two positive squarefree semiprimes (A006881).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 2, 0, 2, 3, 2, 1, 2, 4, 0, 0, 2, 6, 2, 0, 2, 4, 4, 1, 4, 5, 4, 0, 4, 8, 6, 2, 0, 5, 4, 4, 4, 6, 4, 0, 4, 8, 10, 0, 2, 4, 6, 3, 6, 9, 4, 3, 6, 14, 8, 2, 4, 5, 8, 3, 10, 8, 4, 0, 8, 12, 4, 4, 4, 8, 6, 8, 12, 11, 6, 2, 10, 12, 12, 4, 8, 12, 12, 5, 12, 10, 4, 6

Offset: 1

Ilya Gutkovskiy, Jan 07 2017

Conjecture: a(n) > 0 for n > 82 (see comment in A006881 from Richard R. Forberg).

			a(20) = 3 because we have [14, 6], [10, 10] and [6, 14].

Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Semiprime
Eric Weisstein's World of Mathematics, Squarefree

Cf. A001222, A001358, A005117, A006881, A008683, A073610, A098235, A199331, A280710.

nmax = 106; Rest[CoefficientList[Series[(Sum[MoebiusMu[k]^2 Floor[PrimeOmega[k]/2] Floor[2/PrimeOmega[k]] x^k, {k, 2, nmax}])^2, {x, 0, nmax}], x]]

G.f.: (Sum_{k>=2} mu(k)^2*floor(bigomega(k)/2)*floor(2/bigomega(k))*x^k)^2, where mu(k) is the Moebius function (A008683) and bigomega(k) is the number of prime divisors of k counted with multiplicity (A001222).

a(n) = Sum_{k=1..n-1} c(k) * c(n-k), where c = A280710. - Wesley Ivan Hurt, Jan 07 2024

A324332 Numbers m such that A324331(m) = (m-1)^2 - phi(m)*sigma(m) is a square, even though they are not squarefree semiprimes (A006881).

Original entry on oeis.org

12, 20, 24, 40, 42, 44, 45, 48, 63, 72, 80, 96, 104, 105, 108, 132, 135, 160, 189, 190, 192, 200, 216, 275, 320, 342, 384, 385, 399, 405, 429, 452, 456, 465, 567, 575, 610, 637, 639, 640, 648, 693, 768, 783, 848, 969, 988, 1000, 1015, 1044, 1098, 1105, 1127, 1210, 1215

Offset: 1

Michel Marcus, Feb 23 2019

nonn

If m is a squarefree semiprime, then A324331(m) is a square. But the converse is not always true.

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

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

Cf. A000010, A000203, A006881.

Cf. A324331, A324333, A324334.

f(n) = (n-1)^2 - eulerphi(n)*sigma(n); \\ A324331
isok(n) = !((bigomega(n) == 2) && issquarefree(n)) && issquare(f(n));

A339191 Partial products of squarefree semiprimes (A006881).

Original entry on oeis.org

6, 60, 840, 12600, 264600, 5821200, 151351200, 4994589600, 169816046400, 5943561624000, 225855341712000, 8808358326768000, 405184483031328000, 20664408634597728000, 1136542474902875040000, 64782921069463877280000, 3757409422028904882240000

Offset: 1

Gus Wiseman, Nov 30 2020

nonn

A squarefree semiprime is a product of any two distinct prime numbers.

Do all terms belong to A242031 (weakly decreasing prime signature)?

			The sequence of terms together with their prime indices begins:
          6: {1,2}
         60: {1,1,2,3}
        840: {1,1,1,2,3,4}
      12600: {1,1,1,2,2,3,3,4}
     264600: {1,1,1,2,2,2,3,3,4,4}
    5821200: {1,1,1,1,2,2,2,3,3,4,4,5}
  151351200: {1,1,1,1,1,2,2,2,3,3,4,4,5,6}
The sequence of terms together with their prime signatures begins:
                   6: (1,1)
                  60: (2,1,1)
                 840: (3,1,1,1)
               12600: (3,2,2,1)
              264600: (3,3,2,2)
             5821200: (4,3,2,2,1)
           151351200: (5,3,2,2,1,1)
          4994589600: (5,4,2,2,2,1)
        169816046400: (6,4,2,2,2,1,1)
       5943561624000: (6,4,3,3,2,1,1)
     225855341712000: (7,4,3,3,2,1,1,1)
    8808358326768000: (7,5,3,3,2,2,1,1)
  405184483031328000: (8,5,3,3,2,2,1,1,1)

A000040 lists the primes, with partial products A002110 (primorials).
A001358 lists semiprimes, with partial products A112141.
A002100 counts partitions into squarefree semiprimes (restricted: A338903)
A000142 lists factorial numbers, with partial products A000178.
A005117 lists squarefree numbers, with partial products A111059.
A006881 lists squarefree semiprimes, with partial sums A168472.
A166237 gives first differences of squarefree semiprimes.
A320655 counts factorizations into semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A338898/A338912/A338913 give prime indices of semiprimes.
A338899/A270650/A270652 give prime indices of squarefree semiprimes.
A338901 gives first appearances in the list of squarefree semiprimes.
A339113 gives products of primes of squarefree semiprime index.
Cf. A001221, A112798, A167171, A320732, A320891, A320892, A320894, A320911, A338900, A338902, A339003, A339004.

FoldList[Times,Select[Range[20],SquareFreeQ[#]&&PrimeOmega[#]==2&]]

A340380 Numbers whose odd part is a squarefree semiprime (A006881); numbers of the form 2^k * p * q, with k >= 0, and distinct odd primes p and q.

Original entry on oeis.org

15, 21, 30, 33, 35, 39, 42, 51, 55, 57, 60, 65, 66, 69, 70, 77, 78, 84, 85, 87, 91, 93, 95, 102, 110, 111, 114, 115, 119, 120, 123, 129, 130, 132, 133, 138, 140, 141, 143, 145, 154, 155, 156, 159, 161, 168, 170, 174, 177, 182, 183, 185, 186, 187, 190, 201, 203, 204, 205, 209, 213, 215, 217, 219, 220, 221, 222, 228, 230

Offset: 1

Antti Karttunen, Jan 06 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..24567; terms < 2^16

Cf. A000265, A006881, A280710, A340370 (characteristic function).

Subsequence of A285800, from which this differs for the first time at n=25, where a(25) = 110, while A285800(25) = 105, which is missing from this sequence.

semiQ[n_] := FactorInteger[n][[;;,2]] == {1, 1}; Select[Range[230], semiQ[#/2^IntegerExponent[#, 2]] &] (* Amiram Eldar, Jan 03 2022 *)

isA340380(n) = A340370(n); \\ Uses the program given in A340370.

Sum_{n>=1} 1/a(n)^s = (2^s/(2^s-1)) * ((1/2)*(P(s)^2 - P(2*s)) + 1/4^s - P(s)/2^s), for s>1, where P is the prime zeta function. - Amiram Eldar, Jan 03 2022

A176885 Let p*q = A006881(n) be the n-th number that is the product of two distinct primes, with p = prime(i), q=prime(j); a(n) = p^j - q^i.

Original entry on oeis.org

1, 3, 9, 2, 32, 21, 51, 122, 111, 282, 237, 560, 489, 1898, 1794, 6200, 995, 2017, 13428, 19154, 4059, 2166, 8151, 73212, 16341, 58208, 89088, 176186, 32721, 383766, 65483, 530072, 1940958, 131013, 740022, 262083, 1592642, 4781120, 5634480, 524221

Offset: 1

Juri-Stepan Gerasimov, Apr 28 2010

nonn

			For n=3, A006881(3) = 14 = 2*7, p=2, i=1, q=7, j=4; a(n) = 2^4-7^1 = 9.

Cf. A006881.

A176885 := proc(n) c := A006881(n) ; pm := A020639(c) ; pk := A006530(c) ; pm^ numtheory[pi](pk) -pk^numtheory[pi](pm) ; end proc:
seq(A176885(n),n=1..80) ; # R. J. Mathar, May 01 2010

a(14) and a(15) corrected and sequence extended by R. J. Mathar, May 01 2010

Definition clarified by N. J. A. Sloane, Feb 16 2019

A192599 Let N = pq (pA006881(n). Then a(n) = | (p^2-q)(q-1)/2 |.

Original entry on oeis.org

1, 2, 9, 8, 6, 35, 54, 10, 104, 54, 135, 24, 209, 64, 70, 90, 350, 405, 72, 154, 594, 190, 740, 64, 819, 280, 216, 330, 989, 54, 1274, 504, 22, 1595, 256, 1710, 640, 714, 270, 2079, 874, 2345, 648, 56, 2484, 90, 2925, 1144, 286, 3239, 1450, 3740, 1560, 216, 832, 4464, 1914, 4850, 280, 320, 5049

Offset: 1

J. M. Bergot, Jul 04 2011

nonn

This is the area of the triangle with vertices (1,p), (p,q), (q,pq).

A200050 gives the record values. [Arkadiusz Wesolowski, Apr 18 2012]

			For p*q=3*5=15, |(3^2 + 5^2 - 5*(3^2 +1))/2| = 8

Definition simplified by N. J. A. Sloane, Jul 10 2011

A271101 Squarefree semiprimes (A006881) whose average prime factor is prime.

Original entry on oeis.org

21, 33, 57, 69, 85, 93, 129, 133, 145, 177, 205, 213, 217, 237, 249, 253, 265, 309, 393, 417, 445, 469, 489, 493, 505, 517, 553, 565, 573, 597, 633, 669, 685, 697, 753, 781, 793, 813, 817, 865, 889, 913, 933, 949, 973, 985, 993, 1057, 1077, 1137, 1149, 1177, 1257, 1273, 1285, 1329

Offset: 1

Antonio Roldán, Mar 30 2016

nonn

Sum of factors of a(n) if semiprime (product 2*p, with p prime).
This sequence is subsequence of A006881, A089765, A187073, A108633 and A213015.
This sequence is also subsequence of A045835, because sopfr(omega(a(n))) = omega(sopfr(a(n))): sopfr(omega(a(n)))=sopfr(2)=2, and omega(sopfr(a(n)))=omega(2*p)=2  (p prime,  p>2, average prime factor).

			133 is in the sequence because 133 is a squarefree semiprime: 133=7*19, and (7+19)/2=13, a prime number.

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

Cf. A006881, A046315, A046388, A115585, A187073, A089765, A108633, A213015.

N:= 10000: # for terms <= N
Primes:= select(isprime, [seq(i, i=3..N/3)]):
SP:= [seq(seq([p, q], q = select(`<=`, Primes, min(p-1, N/p))), p=Primes)]:
B:= select(t -> isprime((t[1]+t[2])/2), SP):
sort(map(t -> t[1]*t[2], B)); # Robert Israel, Dec 14 2019

Select[Select[Range@ 1330, SquareFreeQ@ # && PrimeOmega@ # == 2 &], PrimeQ@ Mean[First /@ FactorInteger@ #] &] (* Michael De Vlieger, Mar 30 2016 *)

sopf(n)= { local(f, s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); return(s) }
{for (n=6, 2*10^3,  if(bigomega(n)==2&&omega(n)==2, m=sopf(n)/2;if(m==truncate(m),if(isprime(m), print1(n, ", ")))))}

A283929 Number of ways of writing n as a sum of a twin prime (A001097) and a squarefree semiprime (A006881).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Mar 18 2017

nonn

Conjecture: a(n) > 0 for all n > 30.

			a(17) = 3 because we have [14, 3], [11, 6] and [10, 7].

Robert Israel, Table of n, a(n) for n = 0..10000
Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Semiprime
Eric Weisstein's World of Mathematics, Squarefree
Eric Weisstein's World of Mathematics, Twin Primes

Cf. A001097, A006881, A098983, A100949, A282192, A282318.

N:= 200: # to get a(0) to a(N)
V:= Vector(N):
Primes:= select(isprime,[2,seq(i,i=3..N+2)]):
PS:= convert(Primes,set);
Twins:= PS intersect map(`-`,PS,2):
Twins:= Twins union map(`+`,Twins,2):
Twins:= sort(convert(Twins,list)):
for i from 1 to nops(Twins) do
  for j from 1 to nops(Primes) while Twins[i]+2*Primes[j] <= N do
    for k from 1 to j-1 do
      v:= Twins[i]+Primes[k]*Primes[j];
      if v > N then break fi;
      V[v]:= V[v]+1;
od od od:
0, seq(V[i],i=1..N); # Robert Israel, Mar 29 2017

nmax = 110; CoefficientList[Series[Sum[Boole[PrimeQ[k] && (PrimeQ[k - 2] || PrimeQ[k + 2])] x^k, {k, 1, nmax}] Sum[MoebiusMu[k]^2 Floor[2/PrimeOmega[k]] Floor[PrimeOmega[k]/2] x^k, {k, 2, nmax}], {x, 0, nmax}], x]

concat([0, 0, 0, 0, 0, 0, 0, 0, 0], Vec(sum(k=1, 110, (isprime(k) && (isprime(k - 2) || isprime(k + 2)))* x^k) * sum(k=2, 110, moebius(k)^2 * floor(2/bigomega(k)) * floor(bigomega(k)/2) * x^k) + O(x^111))) \\ Indranil Ghosh, Mar 18 2017

G.f.: (Sum_{k>=1} x^A001097(k))*(Sum_{k>=1} x^A006881(k)).

A342092 Odd numbers k such that if k = A001065(m) for some m then m is a squarefree semiprime (A006881).

Original entry on oeis.org

5, 9, 11, 17, 19, 23, 25, 27, 29, 35, 37, 39, 45, 47, 51, 53, 59, 61, 67, 69, 71, 75, 77, 79, 83, 85, 91, 93, 95, 99, 101, 103, 107, 111, 113, 115, 119, 125, 135, 139, 143, 147, 149, 151, 155, 159, 163, 165, 167, 171, 173, 179, 181, 187, 189, 197, 199, 207, 213

Offset: 1

Amiram Eldar, Feb 27 2021

nonn

Assuming that every even number above 6 is the sum of 2 distinct prime numbers, p + q (a slightly stronger version of the Goldbach conjecture), then every odd number m above 7 is of the form 1 + p + q, so A001065(p*q) = m. If this is true, then 5 is the only odd untouchable number (A005114).

Alanen (1972) suggested the study of odd numbers that are being "touched" only by Goldbach solutions, i.e., odd numbers k such that there is no solution m to A001065(m) = k which is not a squarefree semiprime. He suggested that perhaps these numbers deserved to be called "almost untouchable" numbers.

			9 is a term since the only solution to A001065(m) = 9 is m = 3 * 5 = 15.
13 is not a term since there are 2 solutions to A001065(m) = 9, m = 27 = 3^3 and m = 35 = 5*7, and the first solution is not a semiprime.

Amiram Eldar, Table of n, a(n) for n = 1..8251 (terms below 10^5)
Jack David Alanen, Empirical study of aliquot series, Ph.D Thesis, Yale University, 1972.
Eric Weisstein's World of Mathematics, Untouchable Number.
Wikipedia, Untouchable number.

Cf. A001065, A005114, A006881.

seq[max_] := Module[{v = Table[0, {max}]}, Do[If[! (PrimeOmega[n] == PrimeNu[n] == 2), k = DivisorSigma[1, n] - n; If[OddQ[k] && 2 <= k <= max, v[[k]]++]], {n, 1, max^2}]; Select[Rest[Position[v, _?(# == 0 &)] // Flatten], OddQ]]; seq[300]

A058894 Squarefree semiprimes A006881(m) such that |A006881(m)-A007304(m)| = 1.

Original entry on oeis.org

1280073, 1280438, 1280441, 1281166, 1281191, 1281242

Offset: 1

Naohiro Nomoto, Jan 08 2001

From Amiram Eldar, Jun 20 2025: (Start)
The corresponding values of m are 265608, 265682, 265683, 265832, 265837 and 265849, and the corresponding values of A007304(m) are 1280074, 1280437, 1280442, 1281165, 1281190 and 1281241.
a(7) > 10^8, if it exists. (End)

Cf. A006881, A007304.

Offset corrected by Amiram Eldar, Jun 20 2025

cp's OEIS Frontend

A280683 Number of ways to write n as an ordered sum of two positive squarefree semiprimes (A006881).

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A324332 Numbers m such that A324331(m) = (m-1)^2 - phi(m)*sigma(m) is a square, even though they are not squarefree semiprimes (A006881).

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A339191 Partial products of squarefree semiprimes (A006881).

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A340380 Numbers whose odd part is a squarefree semiprime (A006881); numbers of the form 2^k * p * q, with k >= 0, and distinct odd primes p and q.

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A176885 Let p*q = A006881(n) be the n-th number that is the product of two distinct primes, with p = prime(i), q=prime(j); a(n) = p^j - q^i.

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A192599 Let N = pq (pA006881(n). Then a(n) = | (p^2-q)(q-1)/2 |.

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A271101 Squarefree semiprimes (A006881) whose average prime factor is prime.

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A283929 Number of ways of writing n as a sum of a twin prime (A001097) and a squarefree semiprime (A006881).

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