A087054 -id:A087054 - cp's OEIS frontend

A238404 Number of ways a prime from A087054 can be decomposed as a sum of the form pq+qr+r*p where p, q and r are distinct primes (p < q < r).

1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 3, 4, 1, 2, 1, 1, 1, 1, 3, 2, 2, 1, 1, 8, 1, 1, 2, 3, 2, 1, 1, 3, 2, 1, 1, 3, 1, 5, 4, 3, 1, 3, 1, 1, 4, 1, 1, 3, 2, 4, 1, 1, 3, 1, 1, 2, 1, 3, 2, 2, 1, 1, 3, 2, 5, 1, 1, 7, 8, 1, 3, 4, 1, 6, 3, 2, 12, 1, 1, 1, 1, 5, 2, 1, 9, 1, 1, 1, 2, 1, 5, 1, 2, 1, 3, 3, 1, 2, 7, 1

Offset: 1

Jean-François Alcover, Feb 26 2014

nonn

			A087054(5) = 71 = 3*5 + 5*7 + 7*3 = 2*3 + 3*13 + 13*2, therefore a(5) = 2.

Cf. A087053, A087054.

nn = 100; A087854 = Take[Select[ Union[Total[Times @@@ Subsets[#, {2}]] & /@ Subsets[Prime[Range[nn]], {3}]], PrimeQ], nn]; r[n_, p_] := Reduce[p < q < r && p*q+q*r+r*p == n, {q, r}, Primes]; a[n_] := (For[cnt = 0; p = 2, p <= Ceiling[(n-6)/5], p = NextPrime[p], rnp = r[n, p]; If[rnp =!= False, Which[rnp[[0]] === And, Print["n = ", n, " ", {p, q, r} /. ToRules[rnp]]; cnt++, rnp[[0]] === Or, Print["n = ", n, " ", {p, q, r} /. {ToRules[rnp]}]; cnt += Length[rnp], True, Print["error: n = ", n, " ", rnp]]]]; cnt); Reap[Do[ap = a[p]; If[ap > 0, Sow[ap]], {p, A087854}]][[2, 1]] (* after Harvey P. Dale *)

A087053 Numbers of the form pq + qr + rp where p, q and r are distinct primes, with multiplicity.

Original entry on oeis.org

31, 41, 61, 59, 71, 91, 71, 87, 101, 101, 121, 113, 103, 129, 151, 131, 161, 143, 119, 191, 171, 131, 167, 211, 151, 221, 185, 151, 241, 167, 191, 213, 227, 271, 221, 199, 301, 191, 311, 269, 243, 167, 211, 341, 275, 297, 269, 361, 215, 311, 293, 247, 371

Offset: 1

Reinhard Zumkeller, Aug 07 2003

nonn

Arithmetic derivative of numbers having exactly three primes that are distinct: a(n) = A003415(A007304(n)).

Cf. A087054, A238397.

is(n)=forprime(r=(sqrtint(3*n-3)+5)\3, (n-6)\5, forprime(q= sqrtint(r^2+n)-r+1, min((n-2*r)\(r+2), r-2), if((n-q*r)%(q+r)==0 && isprime((n-q*r)/(q+r)), return(1)))); 0 \\ Charles R Greathouse IV, Feb 26 2014

list(n)=my(v=List()); forprime(r=5, (n-6)\5, forprime(q=3, min((n-2*r)\(r+2), r-2), my(S=q+r, P=q*r); forprime(p=2, min((n-P)\S, q-1), listput(v, p*S+P))));  Set(v) \\ Charles R Greathouse IV, Feb 26 2014

from math import isqrt
from sympy import primepi, primerange, integer_nthroot, primefactors
def A087053(n):
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1),1) for b,m in enumerate(primerange(k+1,isqrt(x//k)+1),a+1)))
    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 (p:=primefactors(bisection(f)))[0]*(p[1]+p[2])+p[1]*p[2] # Chai Wah Wu, Aug 30 2024

A238397 Numbers of the form pq + qr + rp where p, q and r are distinct primes (sorted sequence without duplicates).

Original entry on oeis.org

31, 41, 59, 61, 71, 87, 91, 101, 103, 113, 119, 121, 129, 131, 143, 151, 161, 167, 171, 185, 191, 199, 211, 213, 215, 221, 227, 239, 241, 243, 247, 251, 263, 269, 271, 275, 281, 293, 297, 299, 301, 311, 321, 327, 331, 339, 341, 343, 347, 355

Offset: 1

Jean-François Alcover, Feb 26 2014

nonn

Numbers of the form e2(p, q, r) for distinct primes p, q, r, where e2 is the elementary symmetric polynomial of degree 2. Other sequences are obtained with different numbers of distinct primes and degrees: A000040 for 1 prime, A038609 and A006881 for 2 primes, A124867, this sequence, and A007304 for 3 primes. The 4-prime sequences are not presently in the OEIS with the exception of A046386. - Charles R Greathouse IV, Feb 26 2014

			71 = 3*5 + 5*7 + 7*3 = 2*3 + 3*13 + 13*2 is in the sequence (only once, though 2 solutions exist).

Jean-François Alcover, Table of n, a(n) for n = 1..1000

Cf. A087053, A087054.

terms = 50; dm (* initial number of primes *) = 10; f[p_, q_, r_] := p*q + q*r + r*p; Clear[A238397]; A238397[m_] := A238397[m] = Take[u = Union[f @@@ Subsets[Prime /@ Range[m], {3}]], Min[Length[u], terms]]; A238397[dm]; A238397[m = 2*dm]; While[Print["m = ", m]; A238397[m] != A238397[m - dm], m = m + dm]; A238397[m]

is(n)=forprime(r=(sqrtint(3*n-3)+5)\3, (n-6)\5, forprime(q= sqrtint(r^2+n)-r+1, min((n-2*r)\(r+2), r-2), if((n-q*r)%(q+r)==0 && isprime((n-q*r)/(q+r)), return(1)))); 0 \\ Charles R Greathouse IV, Feb 26 2014

list(n)=my(v=List()); forprime(r=5, (n-6)\5, forprime(q=3, min((n-2*r)\(r+2), r-2), my(S=q+r, P=q*r); forprime(p=2, min((n-P)\S, q-1), listput(v, p*S+P)))); Set(v) \\ Charles R Greathouse IV, Feb 26 2014

A189759 Numbers pqr such that pq + pr + qr is prime, where p, q, and r are primes.

Original entry on oeis.org

30, 42, 66, 70, 78, 105, 114, 130, 154, 165, 174, 182, 222, 231, 238, 246, 255, 273, 282, 285, 286, 310, 318, 345, 357, 366, 370, 385, 399, 418, 430, 434, 442, 455, 465, 474, 483, 494, 498, 518, 555, 561, 574, 582, 595, 602, 609, 618, 642, 645, 651, 663, 665

Offset: 1

T. D. Noe, Apr 27 2011

nonn

The number pq+pr+qr is prime only if p, q, and r are distinct. The primes of form pq+pr+qr are in A087054. A prime may have multiple representations as pq+pr+qr; for example, 2*3*13 and 3*5*7 both produce the prime 71.

As mentioned by Ufnarovski and Ahlander, if pq+pr+qr is prime, then the arithmetic derivative (A003415) of pqr is that prime. They conjecture that this sequence and A087054 are infinite.

T. D. Noe, Table of n, a(n) for n = 1..1000
Victor Ufnarovski and Bo Ahlander, How to Differentiate a Number, J. Integer Seqs., Vol. 6, 2003.

Cf. A003415, A087054, A356475.

pqr[nn_] := Module[{p=Prime[Range[PrimePi[nn/6]+1]],i,j,k,n,prod}, Sort[Reap[i=0; While[i++; p[[i]]p[[i+1]]p[[i+2]] <= nn, j=i; While[j++; p[[i]]p[[j]]p[[j+1]] <= nn, k=j; While[k++; prod=p[[i]]p[[j]]p[[k]]; prod <= nn, n=p[[i]]p[[j]]+p[[i]]p[[k]]+p[[j]]p[[k]]; If[PrimeQ[n], Sow[prod]]]]]][[2,1]]]]; pqr[1000]
Take[Union[Times@@@Select[Subsets[Prime[Range[30]],{3}],PrimeQ[ Total[ Times@@@Subsets[#,{2}]]]&]],60](* Harvey P. Dale, Dec 29 2011 *)

A227680 Numbers whose sum of semiprime divisors is a prime number.

Original entry on oeis.org

30, 36, 42, 66, 70, 72, 78, 105, 108, 114, 130, 144, 154, 165, 174, 182, 196, 210, 216, 222, 231, 238, 246, 255, 273, 282, 285, 286, 288, 310, 318, 324, 345, 357, 366, 370, 385, 392, 399, 418, 430, 432, 434, 441, 442, 455, 462, 465, 474, 483, 494, 498, 518

Offset: 1

Michel Lagneau, Jul 19 2013

nonn

There exists a subsequence of infinite squares {36, 144, 196, 324, 441, 576, 676, 784, 1089, 1225, 1296, 1764,...} because the numbers of the form n = (p*q)^2 with p and q primes are in the sequence if p^2 + p*q + q^2 is prime (subsequence of A007645), and the numbers p^2, p*q and q^2 are the three possible semiprime divisors of n. This numbers of the sequence are 6^2, 14^2, 21^2, 26^2, 33^2, 35^2, 51^2, 69^2,...
The numbers of the form n = (p^a*q^v)^2 are also in the sequence => the sequence is infinite.
There exists a subsequence of numbers having three distinct prime divisors p, q and r such that  p*q+q*r+r*p is prime (see A087054). This numbers are 30, 42, 66, 70, 78, 105, 114, ...

			30 is in the sequence because the semiprime divisors of 30 are 2*3, 2*5 and 3*5 and the sum 6+10+15 = 31 is a prime number.

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

Cf. A007645 (primes of the form x^2 + xy + y^2).

Cf. A087054 (primes of the form p*q + q*r + r*p where p, q and r are distinct prime numbers).

with(numtheory):for n from 2 to 600 do:x:=divisors(n):n1:=nops(x): y:=factorset(n):n2:=nops(y):s1:=0:s2:=0:for i from 1 to n1 do: if bigomega(x[i])=2 then s1:=s1+x[i]:else fi:od: s2:=sum('y[i]', 'i'=1..n2):if type(s1,prime)=true then printf(`%d, `,n):else fi:od:

semipSigma[n_] := DivisorSum[n, # &, PrimeOmega[#] == 2 &]; Select[Range[500], PrimeQ @ semipSigma[#] &] (* Amiram Eldar, May 10 2020 *)

A238403 Number of ways a number can be decomposed as a sum of the form pq + qr + rp where p < q < r are distinct primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Jean-François Alcover, Feb 26 2014

nonn

The average value of a(n) is >> sqrt(n)/log^3 n. - Charles R Greathouse IV, Feb 26 2014

			71 = 3*5 + 5*7 + 7*3 = 2*3 + 3*13 + 13*2, therefore a(71) = 2.

Jean-François Alcover, Table of n, a(n) for n = 1..1000

Cf. A238403, A087053, A087054.

r[n_, p_] := Reduce[p < q < r && p*q+q*r+r*p == n, {q, r}, Primes]; a[n_] := (For[cnt = 0; p = 2, p <= Ceiling[(n-6)/5], p = NextPrime[p], rnp = r[n, p]; If[rnp =!= False, Which[rnp[[0]] === And, Print["n = ", n, " ", {p, q, r} /. ToRules[rnp]]; cnt++, rnp[[0]] === Or, Print["n = ", n, " ", {p, q, r} /. {ToRules[rnp]}]; cnt += Length[rnp], True, Print["error: n = ", n, " ", rnp]]]]; cnt); Table[a[n], {n, 1, 100}]

list(n)=my(v=vector(n)); forprime(r=5,(n-6)\5, forprime(q=3, min((n-2*r)\(r+2),r-2), my(S=q+r,P=q*r); forprime(p=2,min((n-P)\S,q-1), v[p*S+P]++))); v \\ Charles R Greathouse IV, Feb 26 2014

a(n)=my(s);forprime(r=(sqrtint(3*n-3)+5)\3,(n-6)\5, forprime(q= sqrtint(r^2+n)-r+1,min((n-2*r)\(r+2),r-2),if((n-q*r)%(q+r)==0 && isprime((n-q*r)/(q+r)),s++)));s \\ Charles R Greathouse IV, Feb 26 2014

cp's OEIS Frontend

A238404 Number of ways a prime from A087054 can be decomposed as a sum of the form p*q+q*r+r*p where p, q and r are distinct primes (p < q < r).

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A087053 Numbers of the form pq + qr + rp where p, q and r are distinct primes, with multiplicity.

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A238397 Numbers of the form pq + qr + rp where p, q and r are distinct primes (sorted sequence without duplicates).

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A189759 Numbers pqr such that pq + pr + qr is prime, where p, q, and r are primes.

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A227680 Numbers whose sum of semiprime divisors is a prime number.

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A238403 Number of ways a number can be decomposed as a sum of the form pq + qr + rp where p < q < r are distinct primes.

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Mathematica

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A238404 Number of ways a prime from A087054 can be decomposed as a sum of the form pq+qr+r*p where p, q and r are distinct primes (p < q < r).