cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A071140 Numbers n such that sum of distinct primes dividing n is divisible by largest prime dividing n; n is neither a prime, nor a true power of prime.

Original entry on oeis.org

30, 60, 70, 90, 120, 140, 150, 180, 240, 270, 280, 286, 300, 350, 360, 450, 480, 490, 540, 560, 572, 600, 646, 700, 720, 750, 810, 900, 960, 980, 1080, 1120, 1144, 1200, 1292, 1350, 1400, 1440, 1500, 1620, 1750, 1798, 1800, 1920, 1960, 2160, 2240, 2250
Offset: 1

Views

Author

Labos Elemer, May 13 2002

Keywords

Comments

a(n) are the numbers such that the difference between the largest and the smallest prime divisor equals the sum of the other distinct prime divisors. - Michel Lagneau, Nov 13 2011
The statement above is only true for 966 of the first 1000 terms. The first counterexample is a(140) = 15015. - Donovan Johnson, Apr 10 2013
Lagneau's definition can be simplified to the largest prime divisor equals the sum of the other distinct prime divisors. - Christian N. K. Anderson, Apr 15 2013

Examples

			n = 70 = 2*5*7 has a form of 2pq, where p and q are twin primes; n = 3135 = 3*5*11*19, sum = 3+5+11+19 = 38 = 2*19, divisible by 19.

Links

Crossrefs

Cf. A008472, A006530, A000961, A025475, A037074, A071139-A071147.
Subsequence of A024619.

Programs

  • Haskell
    a071140 n = a071140_list !! (n-1)
a071140_list = filter (\x -> a008472 x `mod` a006530 x == 0) a024619_list
-- Reinhard Zumkeller, Apr 18 2013
  • Mathematica
    ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sb[x_] := Apply[Plus, ba[x]] ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] Do[s=sb[n]/ma[n]; If[IntegerQ[s]&&Greater[s, 1], Print[{n, ba[n]}]], {n, 2, 1000000}]
(* Second program: *)
Select[Range@ 2250, And[Length@ # > 1, Divisible[Total@ #, Last@ #]] &[FactorInteger[#][[All, 1]] ] &] (* Michael De Vlieger, Jul 18 2017 *)

Formula

A008472(n)/A006530(n) is an integer and n has at least 3 distinct prime factors.
A008472(a(n)) mod A006530(a(n)) = 0 and A010055(a(n)) = 0. - Reinhard Zumkeller, Apr 18 2013

A071142 Numbers of the form 2*p*q where (p,q) is a twin prime pair.

Original entry on oeis.org

30, 70, 286, 646, 1798, 3526, 7198, 10366, 20806, 23326, 38086, 44998, 64798, 73726, 78406, 103966, 115198, 145798, 159046, 194686, 242206, 352798, 373246, 426886, 544966, 649798, 719998, 763846, 824326, 871198, 1312198, 1351366, 1371166, 1472326, 1555846
Offset: 1

Views

Author

Labos Elemer, May 13 2002

Keywords

Comments

For each term k, A008472(k)/A006530(k) = (2+p+q)/q = (q+q)/q = 2.

Examples

			a(1) = 2 * (product of 1st twin prime pair) = 2*3*5 = 30.

Links

Crossrefs

Cf. A008472, A006530, A000961, A025475, A037074, A071139-A071147.

Programs

  • Mathematica
    ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sb[x_] := Apply[Plus, ba[x]] ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] amo[x_] := Abs[MoebiusMu[x]] Do[s=sb[n]/ma[n]; If[IntegerQ[s]&&Equal[lf[n], 3]&& !Equal[amo[n], 0], Print[{n, ba[n]}]], {n, 2, 1000000}]

Formula

a(n) = 2*A037074(n).

Extensions

Edited by Jon E. Schoenfield, Sep 30 2023

A221054 Numbers whose distinct prime factors can be partitioned into two equal sums.

Original entry on oeis.org

1, 30, 60, 70, 90, 120, 140, 150, 180, 240, 270, 280, 286, 300, 350, 360, 450, 480, 490, 540, 560, 572, 600, 646, 700, 720, 750, 810, 900, 960, 980, 1080, 1120, 1144, 1200, 1292, 1350, 1400, 1440, 1500, 1620, 1750, 1798, 1800, 1920, 1960, 2145, 2160, 2240, 2250, 2288, 2310, 2400, 2430, 2450, 2584, 2700, 2730, 2800, 2880, 3000, 3135
Offset: 1

Views

Author

Kevin L. Schwartz and Christian N. K. Anderson, Apr 15 2013

Keywords

Comments

This is a superset of 2*product of twin primes, A071142.

Links

Crossrefs

Cf. A175592 (multiplicity of prime factors allowed).
Cf. A071139-A071147, especially A071140.
Cf. A008472, A037074, A071142.
Cf. A027748, A005843, A083207.

Programs

  • Haskell
    a221054 n = a221054_list !! (n-1)
a221054_list = filter (z 0 0 . a027748_row) $ tail a005843_list where
   z u v []     = u == v
   z u v (p:ps) = z (u + p) v ps || z u (v + p) ps
-- Reinhard Zumkeller, Apr 18 2013
  • Mathematica
    q[n_] := Module[{p = FactorInteger[n][[;; , 1]], sum, x}, sum = Total[p]; EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, p}], x][[1 + sum/2]] > 0]; Select[Range[3200], q] (* Amiram Eldar, May 31 2025 *)
  • PARI
    isok(k) = my(f=factor(k), nb=#f~); for (i=0,2^nb-1, my(v=Vec(Vecrev(binary(i)), nb)); if (sum(k=1, nb, if (v[k], f[k,1])) == sum(k=1, nb, if (!v[k], f[k,1])), return(1));); \\ Michel Marcus, May 31 2025

Extensions

Missing terms inserted by Michel Marcus, May 31 2025

A071141 Numbers n such that sum of distinct primes dividing n is divisible by the largest prime dividing n. Also n is neither a prime, nor a true power of prime and n is squarefree. Squarefree solutions of A071140.

Original entry on oeis.org

30, 70, 286, 646, 1798, 3135, 3526, 3570, 6279, 7198, 8855, 8970, 10366, 10626, 10695, 11571, 15015, 16095, 16530, 17255, 17391, 20615, 20706, 20735, 20806, 23326, 24738, 24882, 26691, 28083, 31031, 36519, 36890, 38086, 38130, 41151, 41615, 44330, 44998
Offset: 1

Views

Author

Labos Elemer, May 13 2002

Keywords

Examples

			n = 286 = 2*11*13 has a form of 2pq, where p and q are twin primes;
n = 5414430 = 2*3*5*7*19*23*59, sum = 2+3+5+7+19+23+59 = 118 = 2*59.

Links

Crossrefs

Cf. A008472, A006530, A000961, A025475, A037074, A071139-A071147.

Programs

  • Mathematica
    ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sb[x_] := Apply[Plus, ba[x]] ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] amo[x_] := Abs[MoebiusMu[x]] Do[s=sb[n]/ma[n]; If[IntegerQ[s]&&Greater[lf[n], 1]&& !Equal[amo[n], 1], Print[{n, ba[n]}]], {n, 2, 1000000}]
(* Second program: *)
Select[Range@ 45000, Function[n, And[Length@ # > 1, SquareFreeQ@ n, Divisible[Total@ #, Last@ #]] &[FactorInteger[n][[All, 1]] ]]] (* Michael De Vlieger, Jul 18 2017 *)

Formula

A008472(n)/A006530(n) is an integer, n has at least 3 distinct prime factors and n is squarefree.

A071146 Numbers n such that sum of distinct primes dividing n is divisible by the largest prime dividing n. Also n has exactly 7 distinct prime factors and n is squarefree.

Original entry on oeis.org

1231230, 2062830, 2181270, 3327870, 3594990, 4224990, 4320030, 4671030, 5162430, 5411406, 5414430, 6767670, 7052430, 7432230, 7870830, 7947030, 8150142, 8273265, 8287230, 8569470, 8804334, 9378390, 10630830, 10705695, 10757838, 10776990, 10900230
Offset: 1

Views

Author

Labos Elemer, May 13 2002

Keywords

Examples

			n = pqrstu, p<q<r<s<t<u, primes, p+q+r+s+t+u = ku; n = 9378390 = 2*3*5*7*17*37*71; sum = 2+3+5+7+17+37+71 = 142 = 2*71

Links

Crossrefs

Cf. A008472, A006530, A000961, A025475, A037074, A071139-A071147.

Programs

  • Mathematica
    ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sb[x_] := Apply[Plus, ba[x]] ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] amo[x_] := Abs[MoebiusMu[x]] Do[s=sb[n]/ma[n]; If[IntegerQ[s]&&Equal[lf[n], 7]&& !Equal[amo[n], 0], Print[{n, ba[n]}]], {n, 2, 1000000}]

Formula

A008472(n)/A006530(n) is an integer; A001221(n) = 7, n is squarefree.

A071143 Numbers n such that (i) the sum of the distinct primes dividing n is divisible by the largest prime dividing n and (ii) n has exactly 4 distinct prime factors and (iii) n is squarefree.

Original entry on oeis.org

3135, 6279, 8855, 10695, 11571, 16095, 17255, 17391, 20615, 20735, 26691, 28083, 31031, 36519, 41151, 41615, 45695, 46655, 47859, 48495, 50439, 54131, 56823, 57239, 59295, 61295, 66215, 72611, 76055, 76479, 80135, 84135, 88595, 89999, 90951, 93651, 94611
Offset: 1

Views

Author

Labos Elemer, May 13 2002

Keywords

Examples

			n = pqrs, p<q<r<s, p+q+r+s = ks; n = 6279 = 3*7*13*23, sum = 3+7+13+23 = 2*23

Links

Crossrefs

Cf. A008472, A006530, A000961, A025475, A037074, A071139-A071147.

Programs

  • Mathematica
    ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sb[x_] := Apply[Plus, ba[x]] ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] amo[x_] := Abs[MoebiusMu[x]] Do[s=sb[n]/ma[n]; If[IntegerQ[s]&&Equal[lf[n], 4]&& !Equal[amo[n], 0], Print[{n, ba[n]}]], {n, 2, 1000000}]
s = {}; Do[Length[f=FactorInteger@n] == 4 && Max[(t = Transpose@f)[[2]]] == 1 && Mod[Plus @@ t[[1]], t[[1,-1]]] == 0 && AppendTo[s,n], {n, 3, 10^6, 2}]; s (* 12 times faster, Giovanni Resta, Apr 10 2013 *)
sdpQ[n_]:=Module[{fi=FactorInteger[n][[All,1]]},Divisible[Total[fi], Last[ fi]] &&Length[fi]==4&&SquareFreeQ[n]]; Select[Range[100000],sdpQ] (* Harvey P. Dale, May 01 2018 *)

Formula

A008472(n)/A006530(n) is an integer; A001221(n) = 4, n is squarefree.

Extensions

Definition clarified by Harvey P. Dale, May 01 2018

A071144 Numbers n such that sum of distinct primes dividing n is divisible by the largest prime dividing n. Also n has exactly 5 distinct prime factors and n is squarefree.

Original entry on oeis.org

3570, 8970, 10626, 15015, 16530, 20706, 24738, 24882, 36890, 38130, 44330, 49938, 51051, 52170, 54834, 55986, 59570, 62985, 68370, 73554, 74613, 77330, 79458, 81770, 87290, 91266, 96162, 96866, 103730, 106314, 116466, 123234, 128570, 129426, 129930, 138890
Offset: 1

Views

Author

Labos Elemer, May 13 2002

Keywords

Examples

			n = pqrst, p<q<r<s<t, primes, p+q+r+s+t = kt; n = 8970 = 2*3*5*13*23, sum = 46 = 2*23.

Links

Crossrefs

Cf. A008472, A006530, A000961, A025475, A037074, A071139-A071147.

Programs

  • Mathematica
    ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sb[x_] := Apply[Plus, ba[x]] ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] amo[x_] := Abs[MoebiusMu[x]] Do[s=sb[n]/ma[n]; If[IntegerQ[s]&&Equal[lf[n], 5]&& !Equal[amo[n], 0], Print[{n, ba[n]}]], {n, 2, 1000000}]
sdpQ[n_]:=Module[{fi=FactorInteger[n][[;;,1]]},Length[fi]==5&&SquareFreeQ[n]&&Mod[Total[ fi],Max[fi]]==0]; Select[Range[150000],sdpQ] (* Harvey P. Dale, May 04 2023 *)

Formula

A008472(n)/A006530(n) is an integer; A001221(n) = 5, n is squarefree.

A071145 Numbers n such that sum of distinct primes dividing n is divisible by the largest prime dividing n. Also n has exactly 6 distinct prime factors and n is squarefree.

Original entry on oeis.org

72930, 106590, 190190, 222870, 335478, 397670, 620310, 836418, 844305, 884442, 1008678, 1195670, 1218945, 1247290, 1704794, 1761110, 1799798, 2086238, 2206022, 2328410, 2485830, 2496585, 2517258, 2863718, 2903538, 3024021, 3157665, 3172785, 3291890
Offset: 1

Views

Author

Labos Elemer, May 13 2002

Keywords

Examples

			n = pqrstw, p<q<r<s<t<w, primes, p+q+r+s+t+w = kt; n = 106590 = 2*3*5*11*17*19; sum = 2+3+5+11+17+19 = 57 = 3*19 (quotient=3) (Corrected Mar 06 2006.)

Links

Crossrefs

Cf. A008472, A006530, A000961, A025475, A037074, A071139-A071147.

Programs

  • Mathematica
    ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sb[x_] := Apply[Plus, ba[x]] ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] amo[x_] := Abs[MoebiusMu[x]] Do[s=sb[n]/ma[n]; If[IntegerQ[s]&&Equal[lf[n], 6]&& !Equal[amo[n], 0], Print[{n, ba[n]}]], {n, 2, 1000000}]

Formula

A008472(n)/A006530(n) is an integer; A001221(n) = 6, n is squarefree.
Showing 1-8 of 8 results.

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