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 1-5 of 5 results.

A078174 Numbers with an integer arithmetic mean of distinct prime factors.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 33, 35, 37, 39, 41, 42, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 64, 65, 67, 69, 71, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 110, 111, 113, 114, 115
Offset: 1

Views

Author

Reinhard Zumkeller, Nov 20 2002

Keywords

Comments

A008472(a(n)) == 0 modulo A001221(a(n)).

Examples

			42=2*3*7: (2+3+7)/3=4, therefore 42 is a term.

Links

Crossrefs

Union of A246655 and A070005.
Positions of 1's in A323172.
The version counting multiplicity is A078175.
The version for prime indices is A326621.
The average of the set of distinct prime factors is A323171/A323172.
The average of the multiset of prime factors is A123528/A123529.
Cf. A001221, A001414, A067629, A112798, A316413, A316428, A326567/A326568, A328966.

Programs

  • Haskell
    a078174 n = a078174_list !! (n-1)
a078174_list = filter (\x -> a008472 x `mod` a001221 x == 0) [2..]
-- Reinhard Zumkeller, Jun 01 2013
  • Mathematica
    Select[Range[2,200],IntegerQ[Mean[Transpose[FactorInteger[#]][[1]]]]&] (* Harvey P. Dale, Apr 18 2016 *)
  • PARI
    is(n)=my(f=factor(n)[,1]);sum(i=1,#f,f[i])%#f==0 \\ Charles R Greathouse IV, May 30 2013

Formula

a(n) << n log n/(log log n)^k for any k. - Charles R Greathouse IV, May 30 2013

A070006 Composite numbers that are not a prime power and whose distinct prime divisors' arithmetic mean is a prime.

Original entry on oeis.org

21, 33, 57, 63, 69, 85, 93, 99, 105, 129, 133, 145, 147, 171, 177, 189, 195, 205, 207, 213, 217, 231, 237, 249, 253, 265, 279, 297, 309, 315, 363, 387, 393, 417, 425, 441, 445, 465, 469, 483, 489, 493, 505, 513, 517, 525, 531, 553, 565, 567, 573, 585, 597
Offset: 1

Views

Author

Labos Elemer, Apr 11 2002

Keywords

Comments

Subsequence of A070005.

Examples

			n = 993 = 3*331, mean = 334/2 = 167, a prime.

Links

Crossrefs

Cf. A001414, A008472, A070005.

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]}] Do[s=Apply[Plus, ba[n]]/lf[n]; If[PrimeQ[s]&&Greater[lf[n], 1], Print[n]], {n, 2, 1000}]
(* Second program: *)
Select[Range@ 600, And[! PrimePowerQ@ #, PrimeQ@ Mean[FactorInteger[#][[All, 1]]]] &] (* Michael De Vlieger, Jul 18 2017 *)
  • PARI
    lista(nn) = {for (n=2, nn, f = factor(n); if ((#f~ != 1) && (type(q=sum(k=1, #f~, f[k,1])/#f~) == "t_INT") && isprime(q), print1(n, ", ")););} \\ Michel Marcus, Mar 28 2015

A070007 Arithmetic mean of distinct primes dividing n is a square number.

Original entry on oeis.org

15, 42, 45, 65, 75, 77, 84, 87, 126, 135, 141, 168, 225, 247, 252, 258, 261, 285, 294, 301, 325, 335, 336, 357, 375, 378, 405, 410, 423, 429, 481, 504, 516, 539, 588, 589, 591, 618, 671, 672, 675, 717, 756, 767, 774, 783, 785, 820, 845, 847, 855, 882, 986
Offset: 1

Views

Author

Labos Elemer, Apr 11 2002

Keywords

Comments

Subset of A078174. [R. J. Mathar, Sep 20 2008]

Examples

			n=1972=2*17*29: mean=(2+17+29)/3=48/3=16, a square.

Links

Crossrefs

Cf. A070005, A070006, A001414, A008472.

Programs

  • Mathematica
    Select[Range[2, 1000], IntegerQ@ Sqrt[Total[First /@ FactorInteger@ #]/PrimeNu@ #] &] (* Michael De Vlieger, Mar 28 2015 *)
  • PARI
    lista(nn) = {for (n=2, nn, f = factor(n); if ((#f~ != 1) && (type(q=sum(k=1, #f~, f[k,1])/#f~) == "t_INT") && issquare(q), print1(n, ", ")););} \\ Michel Marcus, Mar 28 2015

A070008 Arithmetic mean of distinct primes dividing n is a power of 2 (powers of 2 are not left out).

Original entry on oeis.org

2, 4, 8, 15, 16, 32, 39, 42, 45, 55, 64, 75, 84, 87, 114, 117, 126, 128, 135, 168, 170, 183, 225, 228, 247, 252, 256, 258, 261, 275, 294, 295, 336, 340, 342, 351, 375, 378, 405, 410, 456, 504, 507, 512, 516, 549, 583, 588, 605, 672, 675, 680, 684, 756, 774
Offset: 1

Views

Author

Labos Elemer, Apr 11 2002

Keywords

Examples

			n=9189=3*1021, mean=(3+1021)/2=512.

Links

Crossrefs

Cf. A070005, A070006, A070007, A008472, A001414.

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]}] Do[s=Apply[Plus, ba[n]]/lf[n]; If[IntegerQ[Log[2, s]], Print[{n, s}]], {n, 2, 10000}]
fQ[n_] := Block[{pf = FactorInteger@ n}, Length@ pf == 1 && pf[[1, 1]] == 2]; Select[Range[2, 780], fQ[Total[First /@ FactorInteger@#]/PrimeNu@#] &] (* Michael De Vlieger, Mar 28 2015 *)

A070009 Least number m such that the arithmetic mean of the distinct prime divisors of m is equal to 2^n.

Original entry on oeis.org

2, 15, 39, 87, 183, 2071, 1255, 1527, 3063, 18402, 12279, 106327, 49143, 622231, 589794, 1703767, 1310695, 9961111, 3145719, 31457210, 12582903, 310377127, 50331639, 2046816631, 335544295, 10603194271, 8858369762, 1610612727, 44023413103, 40802188951
Offset: 1

Views

Author

Labos Elemer, Apr 11 2002

Keywords

Comments

Are there any terms with more than 3 prime factors? - David Wasserman, May 05 2003

Examples

			a(15) = 589794 because m = 2*3*98299; mean = (2+3+98299)/3 = 32768 = 2^15.

Crossrefs

Cf. A070005, A070006, A070007, A070008, A070009, A008472, A001414, A001221.

Programs

  • Mathematica
    a = Table[0, {21}]; Do[b = Transpose[ FactorInteger[n]][[1]]; c = Log[2, Apply[ Plus, b] / Length[b]]; If[ IntegerQ[c] && a[[c]] == 0, a[[c]] = n], {n, 2, 10^8/3}]; a

Formula

a(n) = Min{x; A008472(x)/A001221(x)=2^n}.

Extensions

Edited and extended by Robert G. Wilson v, Apr 30 2002
More terms from David Wasserman, May 05 2003
a(29)-a(30) from Donovan Johnson, Aug 06 2012
Showing 1-5 of 5 results.

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