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 1-4 of 4 results.

A070005 Arithmetic mean of prime factors of n is an integer and n is neither a prime nor power of a prime.

Original entry on oeis.org

15, 21, 33, 35, 39, 42, 45, 51, 55, 57, 63, 65, 69, 75, 77, 78, 84, 85, 87, 91, 93, 95, 99, 105, 110, 111, 114, 115, 117, 119, 123, 126, 129, 133, 135, 141, 143, 145, 147, 153, 155, 156, 159, 161, 168, 170, 171, 175, 177, 183, 185, 186, 187, 189, 195, 201, 203
Offset: 1

Views

Author

Labos Elemer, Apr 11 2002

Keywords

Examples

			n=33=3*11, mean=(3+11)/2=6.

Links

Crossrefs

Cf. A070006, A001414, A008472.
Cf. A000961, A010055; subsequence of A078174.

Programs

  • Haskell
    a070005 n = a070005_list !! (n-1)
a070005_list = filter ((== 0) . a010055) a078174_list
-- Reinhard Zumkeller, Jun 01 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]}] ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] Do[s=Apply[Plus, ba[n]]/lf[n]; If[IntegerQ[s]&&Greater[lf[n], 1], Print[n]], {n, 2, 1000}]
  • PARI
    lista(nn) = {for (n=2, nn, f = factor(n); if ((#f~ != 1) && (sum(k=1, #f~, f[k,1]) % #f~ == 0), 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 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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