A071141 -id:A071141 - cp's OEIS frontend

A071312 Squarefree numbers k such that the largest prime factor of k is equal to the sum of the other prime factors of k.

30, 70, 286, 646, 1798, 3135, 3526, 3570, 6279, 7198, 8855, 8970, 10366, 10626, 10695, 11571, 16095, 16530, 17255, 17391, 20615, 20706, 20735, 20806, 23326, 24738, 24882, 26691, 28083, 31031, 36519, 36890, 38086, 38130, 41151, 41615

Offset: 1

Benoit Cloitre, Jun 11 2002

If k = p(1)*p(2)*...p(r) is in the sequence, where p(r) is the largest prime factor, then p(r) = p(1)+p(2)+...+p(r-1).

			20706 = 2*3*7*17*29 and 2+3+7+17 = 29 hence 20706 is in the sequence.

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

Cf. A005117, A071141.

Select[Range[40000], SquareFreeQ[#] && Plus @@ (f = FactorInteger[#][[;;,1]]) == 2 * f[[-1]] &] (* Amiram Eldar, Apr 23 2022 *)

for(n=2,100000,if(issquarefree(n)*sum(i=1,omega(n)-1, component(component(factor(n),1),i))==vecmax(factor(n,1)),print1(n,",")))

A083263 Numbers k such that the difference of the largest and smallest prime factors of k divides k.

Original entry on oeis.org

6, 12, 18, 24, 30, 36, 48, 54, 60, 70, 72, 90, 96, 108, 120, 140, 144, 150, 162, 180, 192, 198, 210, 216, 240, 270, 280, 286, 288, 300, 324, 350, 360, 384, 396, 420, 432, 450, 480, 486, 490, 510, 540, 560, 572, 576, 594, 600, 630, 646, 648, 700, 720, 750, 768

Offset: 1

Labos Elemer, May 12 2003

nonn

			Every number k of the form 2^i * 3^j * m is a term because 3 - 2 = 1 is always a divisor of k.
Every number k of the form 2 * p * (p+2) * m is a term if p and p+2 form a twin prime pair.
Other terms include some in which the difference d = gpf(k) - lpf(k) > 2 is prime (e.g., 30 = 2*3*5 = 3*10; d = 5 - 2 = 3) and some in which it is composite (e.g., 8710 = 2*5*13*67 = 65*134; d = 67 - 2 = 65).
All terms are even. - _Jon E. Schoenfield_, Jul 10 2018

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

Cf. A033845, A071141, A006530, A020639.

ffi[x_] := Flatten[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; lf[x_] := Length[FactorInteger[x]]; ma[x_] := Max[ba[x]]; mi[x_] := Min[ba[x]] Do[s=ma[ba[n]]-mi[ba[n]]; If[Mod[n, s]==0, Print[{n, ba[n], s}]], {n, 1, 10000}]

Solutions to x mod (A006530(x) - A020639(x)) = 0.

Edited by Jon E. Schoenfield, Jul 10 2018

A384498 Squarefree numbers whose distinct prime factors can be partitioned into two sets with equal sums.

Original entry on oeis.org

1, 30, 70, 286, 646, 1798, 2145, 2310, 2730, 3135, 3526, 3570, 4641, 4845, 5005, 5610, 6006, 6279, 6630, 7198, 7410, 7854, 8778, 8855, 8970, 9177, 10366, 10374, 10626, 10695, 11305, 11571, 11730, 13110, 13485, 13566, 13585, 15470, 16095, 16302, 16422, 16530

Offset: 1

Alois P. Heinz, May 31 2025

nonn

			2145 = 3*5*11*13 is a term because it is squarefree and 3+13 = 5+11.
16422 = 2*3*7*17*23 is squarefree and 2+7+17 = 3+23.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Intersection of A005117 and A221054.

Cf. A071141, A071142, A071312.

q:= n-> (l-> {l[.., 2][]} minus {1}={} and (s->
        (m-> m::even and coeff(mul(1+x^j, j=s), x, m/2)>0)
        (add(i, i=s)))({l[.., 1][]}))(ifactors(n)[2]):
select(q, [$1..20000])[];

Join[{1},Select[Range[16600],SquareFreeQ[#]&&MemberQ[Total/@Subsets[First/@FactorInteger[#]],Total[First/@FactorInteger[#]]/2]&]] (* James C. McMahon, Jun 02 2025 *)

A083264 Numbers k such that the difference d of the largest and smallest prime factors of k is a composite divisor of k.

Original entry on oeis.org

198, 396, 510, 594, 792, 966, 990, 1020, 1188, 1386, 1530, 1566, 1584, 1782, 1932, 1980, 2040, 2178, 2376, 2550, 2590, 2772, 2898, 2970, 3060, 3132, 3168, 3198, 3564, 3570, 3864, 3960, 4080, 4158, 4230, 4356, 4590, 4698, 4752, 4830, 4950, 5100, 5180

Offset: 1

Labos Elemer, May 12 2003

From David A. Corneth, Jul 14 2018: (Start)
No term k is a perfect power (or 1). If k is a perfect power then it's divisible by 0, a contradiction. Hence a term k has at least two prime factors.
All terms are even. Suppose a term k is odd. Then the smallest prime factor is > 2. Since k has at least two prime factors which are odd, the difference between the largest and smallest prime factor is even hence k is even. A contradiction, hence all terms are even.
All terms are of the form 2 * (p - 2) * p * m where p - 2 is composite, p is prime and m has all, if any, of its prime factors between 2 and p (inclusive). (End)

			198 = 2*3*3*11 = 2*9*11 is in the sequence where d = 11 - 2 = 9 is composite.

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

Cf. A025584, A033845, A071141, A006530, A020639, A083263.

ffi[x_] := Flatten[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; lf[x_] := Length[FactorInteger[x]]; ma[x_] := Max[ba[x]]; mi[x_] := Min[ba[x]] Do[s=ma[ba[n]]-mi[ba[n]]; If[Mod[n, s]==0&&Greater[s, 2]&&!PrimeQ[s], Print[n]], {n, 1, 20000}]
dllpfQ[n_]:=Module[{c=Transpose[FactorInteger[n]][[1]],d},d=Last[c]-First[ c];If[d==0,d=1];Divisible[n,d]&&d>2&&CompositeQ[d]]; Select[ Range[ 6000],dllpfQ] (* Harvey P. Dale, Sep 26 2014 *)

isok(n) = if (n>1, my(f=factor(n)[,1], d = vecmax(f) - vecmin(f)); (d > 1) && !isprime(d) && !(n % d)); \\ Michel Marcus, Jul 09 2018

Solutions to x mod d = 0 where d = A006530(x) - A020639(x) is composite.

Name, Formula, and Example simplified by Jon E. Schoenfield, Jul 14 2018

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A071312 Squarefree numbers k such that the largest prime factor of k is equal to the sum of the other prime factors of k.

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A083263 Numbers k such that the difference of the largest and smallest prime factors of k divides k.

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A384498 Squarefree numbers whose distinct prime factors can be partitioned into two sets with equal sums.

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A083264 Numbers k such that the difference d of the largest and smallest prime factors of k is a composite divisor of k.

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