A110187 -id:A110187 - cp's OEIS frontend

A110188 3-almost primes p * q * r not relatively prime to p+q+r.

8, 18, 27, 30, 42, 50, 66, 70, 78, 98, 102, 105, 110, 114, 125, 130, 138, 154, 170, 174, 182, 186, 190, 195, 222, 230, 231, 238, 242, 246, 255, 258, 266, 282, 285, 286, 290, 310, 318, 322, 338, 343, 354, 357, 366, 370, 374, 402, 406, 410, 418, 426, 429, 430

Offset: 1

Jonathan Vos Post, Jul 15 2005

A110187 is the converse, 3-almost primes p * q * r which are relatively prime to p+q+r.

			a(1) = 8 because 8 = 2^3, which has a prime factor 2 in common with prime 2 + 2 + 2 = 6.
30 is in the sequence, since 30 = 2 * 3 * 5, which is in fact divisible by 2 + 3 + 5 = 10.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A014612, A110187, A110227, A110228, A110229, A110230, A110231, A110232, A110289, A110290, A110296, A110297.

list(lim)=my(v=List()); forprime(p=2, lim\4, forprime(q=2, min(p, lim\2\p), my(pq=p*q, t); forprime(r=2, min(lim\pq, q), t=r*pq; if(gcd(t, p+q+r)>1, listput(v, t))))); Set(v) \\ Charles R Greathouse IV, Jan 31 2017

Extended by Ray Chandler, Jul 20 2005

A110227 4-almost primes p * q * r * s relatively prime to p + q + r + s.

Original entry on oeis.org

40, 54, 56, 88, 90, 104, 135, 136, 152, 184, 189, 198, 210, 225, 232, 248, 250, 294, 296, 297, 306, 328, 344, 350, 351, 376, 390, 414, 424, 441, 459, 462, 472, 488, 513, 522, 536, 546, 550, 568, 570, 584, 621, 632, 664, 686, 712, 714, 735, 738, 765, 776

Offset: 1

Jonathan Vos Post, Jul 16 2005

p, q, r, s are not necessarily distinct. The converse to this is A110228: 4-almost primes p * q * r * s not relatively prime to p+q+r+s.

			104 is in this sequence because 104 = 2^3 * 13, which is relatively prime to 2 + 2 + 2 + 13 = 19, which is prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A014613, A110187, A110188, A110228, A110229, A110230, A110231, A110232, A110289, A110290, A110296, A110297.

list(lim)=my(v=List()); forprime(p=2,lim\8, forprime(q=2,min(p,lim\4\p), my(pq=p*q); forprime(r=2,min(lim\pq\2,q), my(pqr=pq*r,t); forprime(s=2,min(lim\pqr,r), t=pqr*s; if(gcd(t,p+q+r+s)==1, listput(v,t)))))); Set(v) \\ Charles R Greathouse IV, Jan 31 2017

Corrected and extended by Ray Chandler, Jul 20 2005

A110228 4-almost primes p * q * r * s not relatively prime to p + q + r + s.

Original entry on oeis.org

16, 24, 36, 60, 81, 84, 100, 126, 132, 140, 150, 156, 196, 204, 220, 228, 234, 260, 276, 308, 315, 330, 340, 342, 348, 364, 372, 375, 380, 444, 460, 476, 484, 490, 492, 495, 510, 516, 525, 532, 558, 564, 572, 580, 585, 620, 625, 636, 644, 650, 666, 676, 690

Offset: 1

Jonathan Vos Post, Jul 16 2005

p, q, r, s are not necessarily distinct. The converse to this is A110227: 4-almost primes p * q * r * s which are relatively prime to p+q+r+s.

			84 is in this sequence because 84 = 2^2 * 3 * 7 and the sum of these prime factors is 2 + 2 + 3 + 7 = 14 = 2 * 7, which is a divisor of 84.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A014613, A110187, A110188, A110227, A110229, A110230, A110231, A110232, A110289, A110290, A110296, A110297.

list(lim)=my(v=List()); forprime(p=2,lim\8, forprime(q=2,min(p,lim\4\p), my(pq=p*q); forprime(r=2,min(lim\pq\2,q), my(pqr=pq*r,t); forprime(s=2,min(lim\pqr,r), t=pqr*s; if(gcd(t,p+q+r+s)>1, listput(v,t)))))); Set(v) \\ Charles R Greathouse IV, Jan 31 2017

Corrected and extended by Ray Chandler, Jul 20 2005

A110229 5-almost primes p * q * r * s * t relatively prime to p + q + r + s + t.

Original entry on oeis.org

48, 80, 108, 112, 176, 208, 252, 272, 300, 304, 368, 405, 420, 464, 468, 496, 500, 567, 592, 656, 660, 675, 684, 688, 752, 848, 891, 924, 944, 976, 980, 1020, 1053, 1072, 1116, 1136, 1140, 1168, 1264, 1300, 1323, 1328, 1332, 1372, 1377, 1424, 1428, 1452

Offset: 1

Jonathan Vos Post, Jul 17 2005

p, q, r, s, t are not necessarily distinct. The converse to this is A110230: 5-almost primes p * q * r * s * t which are not relatively prime to p+q+r+s+t. A014614 is the 5-almost primes.

			48 is in this sequence because 48 = 2^4 * 3, which has no factors in common with 2 + 2 + 2 + 2 + 3 = 11.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A014614, A002033, A110187, A110188, A110227, A110228, A110230, A110231, A110232, A110289, A110290, A110296, A110297.

list(lim)=my(v=List()); forprime(p=2, lim\16, forprime(q=2, min(p, lim\8\p), my(pq=p*q); forprime(r=2, min(lim\pq\4, q), my(pqr=pq*r); forprime(s=2, min(lim\pqr\2, r), my(pqrs=pqr*s,n); forprime(t=2,min(lim\pqrs,s), n=pqrs*t; if(gcd(n, p+q+r+s+t)==1, listput(v, n))))))); Set(v) \\ Charles R Greathouse IV, Jan 31 2017

Extended by Ray Chandler, Jul 20 2005

Incorrect formula and comment of Sep 2009 related to A002033 deleted - R. J. Mathar, Oct 14 2009

A110230 5-almost primes p * q * r * s * t not relatively prime to p + q + r + s + t.

Original entry on oeis.org

32, 72, 120, 162, 168, 180, 200, 243, 264, 270, 280, 312, 378, 392, 396, 408, 440, 450, 456, 520, 552, 588, 594, 612, 616, 630, 680, 696, 700, 702, 728, 744, 750, 760, 780, 828, 882, 888, 918, 920, 945, 952, 968, 984, 990, 1026, 1032, 1044, 1050, 1064, 1092

Offset: 1

Jonathan Vos Post, Jul 17 2005

p, q, r, s, t are not necessarily distinct. The converse to this is A110229: 5-almost primes p * q * r * s * t which are relatively prime to p+q+r+s+t. A014614 is the 5-almost primes.

			180 is in this sequence because 180 = 2^2 * 3^2 * 5, the sum of the prime factors being 2 + 2 + 3 + 3 + 5 = 15 = 3 * 5 which has two prime factors in common with 180.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A014614, A110187, A110188, A110227, A110228, A110229, A110231, A110232, A110289, A110290, A110296, A110297.

list(lim)=my(v=List()); forprime(p=2, lim\16, forprime(q=2, min(p, lim\8\p), my(pq=p*q); forprime(r=2, min(lim\pq\4, q), my(pqr=pq*r); forprime(s=2, min(lim\pqr\2, r), my(pqrs=pqr*s,n); forprime(t=2,min(lim\pqrs,s), n=pqrs*t; if(gcd(n, p+q+r+s+t)>1, listput(v, n))))))); Set(v) \\ Charles R Greathouse IV, Jan 31 2017

Extended by Ray Chandler, Jul 20 2005

A110231 6-almost primes p * q * r * s * t * u relatively prime to p+q+r+s+t+u.

Original entry on oeis.org

96, 224, 352, 360, 416, 486, 504, 544, 600, 608, 736, 792, 810, 928, 936, 992, 1000, 1176, 1184, 1224, 1312, 1368, 1376, 1400, 1504, 1656, 1696, 1701, 1782, 1888, 1890, 1952, 2025, 2040, 2088, 2144, 2184, 2200, 2232, 2250, 2272, 2336, 2528, 2600, 2646

Offset: 1

Jonathan Vos Post, Jul 17 2005

p, q, r, s, t, u are not necessarily distinct. The converse to this is A110232: 6-almost primes p * q * r * s * t * u which are not relatively prime to p+q+r+s+t+u. A046306 is the 6-almost primes.

			96 is an element of this sequence because 96 = 2^5 * 3, the sum of whose prime factors is 2 + 2 + 2 + 2 + 2 + 3 = 13, which has no prime factors in common with 96.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A046306, A110187, A110188, A110227, A110228, A110229, A110230, A110232, A110289, A110290, A110296, A110297.

list(lim)=my(v=List()); forprime(p=2, lim\32, forprime(q=2, min(p, lim\16\p), my(pq=p*q); forprime(r=2, min(lim\pq\8, q), my(pqr=pq*r); forprime(s=2, min(lim\pqr\4, r), my(pqrs=pqr*s); forprime(t=2,min(lim\pqrs\2,s), my(pqrst=pqrs*t,n); forprime(u=2,min(lim\pqrst,t), n=pqrst*u; if(gcd(n, p+q+r+s+t+u)==1, listput(v, n)))))))); Set(v) \\ Charles R Greathouse IV, Jan 31 2017

Extended by Ray Chandler, Jul 20 2005

A110232 6-almost primes p * q * r * s * t * u not relatively prime to p+q+r+s+t+u.

Original entry on oeis.org

64, 144, 160, 216, 240, 324, 336, 400, 528, 540, 560, 624, 729, 756, 784, 816, 840, 880, 900, 912, 1040, 1104, 1134, 1188, 1215, 1232, 1260, 1320, 1350, 1360, 1392, 1404, 1456, 1488, 1500, 1520, 1560, 1764, 1776, 1836, 1840, 1848, 1904, 1936, 1960, 1968

Offset: 1

Jonathan Vos Post, Jul 17 2005

p, q, r, s, t, u are not necessarily distinct. The converse to this is A110231: 6-almost primes p * q * r * s * t * u which are relatively prime to p+q+r+s+t+u. A046306 is the 6-almost primes.

			160 is in this sequence because 160 = 2^5 * 5, the sum of whose prime factors is 2 + 2 + 2 + 2 + 2 + 5 = 15 = 3 * 5, which has a prime factor in common with 160.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A046306, A110187, A110188, A110227, A110228, A110229, A110230, A110231, A110289, A110290, A110296, A110297.

list(lim)=my(v=List()); forprime(p=2, lim\16, forprime(q=2, min(p, lim\8\p), my(pq=p*q); forprime(r=2, min(lim\pq\4, q), my(pqr=pq*r); forprime(s=2, min(lim\pqr\2, r), my(pqrs=pqr*s); forprime(t=2,min(lim\pqrs,s), my(pqrst=pqrs*t,n); forprime(u=2,min(lim\pqrst,t), n=pqrst*u; if(gcd(n, p+q+r+s+t+u)>1, listput(v, n)))))))); Set(v) \\ Charles R Greathouse IV, Jan 31 2017

Extended by Ray Chandler, Jul 20 2005

A110289 7-almost primes pqrstuv relatively prime to p+q+r+s+t+u+v.

Original entry on oeis.org

320, 432, 448, 704, 720, 832, 972, 1088, 1216, 1472, 1584, 1680, 1856, 1984, 2000, 2268, 2352, 2368, 2448, 2624, 2700, 2752, 3008, 3120, 3312, 3392, 3645, 3696, 3776, 3904, 3920, 4176, 4212, 4288, 4368, 4400, 4544, 4672, 5056, 5103, 5200, 5312, 5488

Offset: 1

Jonathan Vos Post, Jul 18 2005

The primes p, q, r, s, t, u, v are not necessarily distinct. The 7-almost primes are A046308. The converse, A110290, is 7-almost primes p*q*r*s*t*u*v which are not relatively prime to p+q+r+s+t+u+v.

Contains p*q^6 if p and q are distinct primes, p >= 5. - Robert Israel, Jan 13 2017

			832 = 2^6 * 13 is in this sequence because its sum of prime factors is 2 + 2 + 2 + 2 + 2 + 2 + 13 = 25 = 5^2, which has no factor in common with 832.

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

Cf. A046308, A110187, A110188, A110227, A110228, A110229, A110230, A110231, A110232, A110290, A110296, A110297.

Cf. A001414 (sopfr(n)).

N:= 10^4: # to get all terms <= N
P:= select(isprime, [$1..N/2^6]):
nP:= nops(P):
Res:= {}:
for p in P do
  for q in P while q <= p and p*q*2^5 <= N do
    for r in P while r <= q and p*q*r*2^4 <= N do
      for s in P while s <= r and p*q*r*s*2^3 <= N do
        for t in P while t <= s and p*q*r*s*t*2^2 <= N do
          for u in P while u <= t and p*q*r*s*t*u*2 <= N do
            for v in P while v <= u and p*q*r*s*t*u*v <= N do
              if igcd(p+q+r+s+t+u+v,p*q*r*s*t*u*v) = 1 then
                  Res:= Res union {p*q*r*s*t*u*v} fi
od od od od od od od:
sort(convert(Res,list)); # Robert Israel, Jan 13 2017

Select[Range[6000],PrimeOmega[#]==7&&CoprimeQ[Total[ Times@@@ FactorInteger[ #]],#]&] (* Harvey P. Dale, Nov 19 2019 *)

sopfr(n)=local(f);if(n<1,0,f=factor(n);sum(k=1,matsize(f)[1],f[k,1]*f[k,2]))
isok(n)=bigomega(n)==7&&gcd(n, sopfr(n))==1 \\ Rick L. Shepherd, Jul 20 2005

Extended by Ray Chandler and Rick L. Shepherd, Jul 20 2005

A110290 7-almost primes pqrstuv not relatively prime to p+q+r+s+t+u+v.

Original entry on oeis.org

128, 192, 288, 480, 648, 672, 800, 1008, 1056, 1080, 1120, 1200, 1248, 1458, 1512, 1568, 1620, 1632, 1760, 1800, 1824, 1872, 2080, 2187, 2208, 2376, 2430, 2464, 2520, 2640, 2720, 2736, 2784, 2800, 2808, 2912, 2976, 3000, 3040, 3402, 3528, 3552, 3564

Offset: 1

Jonathan Vos Post, Jul 18 2005

The primes p, q, r, s, t, u, v are not necessarily distinct. The 7-almost primes are A046308. The converse, A110289, is 7-almost primes p*q*r*s*t*u*v which are relatively prime to p+q+r+s+t+u+v.

			800 = 2^5 * 5^2 is in this sequence because the sum of prime factors 2 + 2 + 2 + 2 + 2 + 5 + 5 = 20 is not relatively prime to 800 (in fact, it is a divisor of 800).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A046308, A110187, A110188, A110227, A110228, A110229, A110230, A110231, A110232, A110289, A110296, A110297.

Cf. A001414 (sopfr(n)).

Select[Range[4000],PrimeOmega[#]==7&&!CoprimeQ[Total[Flatten[Table[ #[[1]], #[[2]]]&/@ FactorInteger[#]]],#]&] (* Harvey P. Dale, Apr 30 2018 *)

sopfr(n)=local(f);if(n<1,0,f=factor(n);sum(k=1,matsize(f)[1],f[k,1]*f[k,2]))
isok(n)=bigomega(n)==7&&gcd(n, sopfr(n))>1 \\ Rick L. Shepherd, Jul 20 2005

Extended by Ray Chandler and Rick L. Shepherd, Jul 20 2005

A110296 8-almost primes pqrstuv*w relatively prime to p+q+r+s+t+u+v+w.

Original entry on oeis.org

384, 640, 864, 1408, 1664, 2016, 2176, 2400, 2432, 2944, 3240, 3712, 3744, 3968, 4374, 4536, 4736, 5248, 5280, 5472, 5504, 5600, 6016, 6240, 6784, 7128, 7392, 7552, 7808, 7840, 8424, 8576, 8800, 8928, 9088, 9120, 9344, 10112, 10400, 10584, 10624

Offset: 1

Jonathan Vos Post, Jul 18 2005

The primes p, q, r, s, t, u, v, w are not necessarily distinct. The 8-almost primes are A046310. The converse, A110297, is 8-almost primes p*q*r*s*t*u*v*w which are not relatively prime to p+q+r+s+t+u+v+w.

			864 is an element of this sequence because 864 = 2^5 * 3^3, so the sum of prime factors is 2 + 2 + 2 + 2 + 2 + 3 + 3 + 3 = 19 which is prime, hence relatively prime to 864. That is the same sum of prime factors as 640 = 2^7 * 5, hence 640 is also a member of this sequence. The sum of prime factors need not be prime for this membership, for example, 2432 = 2^7 * 19 has sum of prime factors 2 + 2 + 2 + 2 + 2 + 2 + 2 + 19 = 33 = 3 * 11, which is composite, yet relatively prime to 2432.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A046310, A110187, A110188, A110227, A110228, A110229, A110230, A110231, A110232, A110289, A110290, A110297.

list(lim)=my(v=List()); forprime(p=2, lim\128, forprime(q=2, min(p, lim\64\p), my(pq=p*q); forprime(r=2, min(lim\pq\32, q), my(pqr=pq*r); forprime(s=2, min(lim\pqr\16, r), my(pqrs=pqr*s); forprime(t=2, min(lim\pqrs\8, s), my(pqrst=pqrs*t); forprime(u=2, min(lim\pqrst\4, t), my(pqrstu=pqrst*u); forprime(w=2,min(lim\pqrstu\2,u), my(pqrstuw=pqrstu*w,n); forprime(x=2,min(lim\pqrstuw,w), n=pqrstuw*x; if(gcd(n, p+q+r+s+t+u+w+x)==1, listput(v, n)))))))))); Set(v) \\ Charles R Greathouse IV, Feb 01 2017

Corrected and extended by Ray Chandler, Jul 20 2005

cp's OEIS Frontend

A110188 3-almost primes p * q * r not relatively prime to p+q+r.

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A110227 4-almost primes p * q * r * s relatively prime to p + q + r + s.

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A110228 4-almost primes p * q * r * s not relatively prime to p + q + r + s.

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A110229 5-almost primes p * q * r * s * t relatively prime to p + q + r + s + t.

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A110230 5-almost primes p * q * r * s * t not relatively prime to p + q + r + s + t.

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A110231 6-almost primes p * q * r * s * t * u relatively prime to p+q+r+s+t+u.

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A110232 6-almost primes p * q * r * s * t * u not relatively prime to p+q+r+s+t+u.

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A110289 7-almost primes p*q*r*s*t*u*v relatively prime to p+q+r+s+t+u+v.

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A110289 7-almost primes pqrstuv relatively prime to p+q+r+s+t+u+v.

A110290 7-almost primes pqrstuv not relatively prime to p+q+r+s+t+u+v.

A110296 8-almost primes pqrstuv*w relatively prime to p+q+r+s+t+u+v+w.