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.

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A164865 Sum of the distinct semiprime divisors of the n-th number with two or more distinct semiprime divisors.

Original entry on oeis.org

10, 15, 14, 10, 18, 31, 19, 14, 41, 26, 24, 10, 35, 30, 15, 18, 35, 30, 61, 38, 59, 19, 40, 42, 71, 14, 45, 26, 40, 50, 10, 63, 42, 39, 91, 30, 71, 19, 87, 18, 101, 62, 48, 35, 66, 50, 101, 65, 24, 38, 121, 63, 19, 70, 78, 56, 42, 60, 113, 75, 14, 15, 86, 103, 45, 129, 66, 90
Offset: 1

Views

Author

Jonathan Vos Post, Aug 28 2009

Keywords

Comments

The sum of semiprime divisors of all k such that A086971(k) > 1.
This sum is prime for k = 30, 36, 60, 72, and infinitely more values (every prime power of every primitive element).

Examples

			a(1) = 10 because the 1st number with 2 or more distinct semiprime divisors is k=12=A102467(2), as A001358(1) = 4, 4|12, A001358(2) = 6, 6|12, and 4+6 = 10.
a(6) = 31 because the 6th number with multiple distinct semiprime factors is k=30=A102467(7), the semiprimes 6, 10, and 15 divide 30, and 6 + 10 + 15 = 31.

Crossrefs

Cf. A001358, A068318, A086971, A102467.

Programs

  • Maple
    isA001358 := proc(n) RETURN( numtheory[bigomega](n) =2 ) ; end:
A086971 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do if isA001358(d) then a := a+1; fi; od; a ; end:
A102467 := proc(n) local a; if n = 1 then 1; else for a from procname(n-1)+1 do if A086971(a) >= 2 then RETURN(a) ; fi; od: fi; end:
A076290 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do if isA001358(d) then a := a+d; fi; od; a ; end:
A164865 := proc(n) A076290( A102467(n+1)) ; end: seq(A164865(n),n=1..120) ; # R. J. Mathar, Aug 31 2009
  • Mathematica
    sdsd[n_]:=Module[{spd=Select[Divisors[n],PrimeOmega[#]==2&]}, If[ Length[ spd]> 1,Total[spd],0]]; DeleteCases[Array[sdsd,200],0] (* Harvey P. Dale, Oct 29 2015 *)

Formula

a(n) = Sum_(k|A102467(n) and k in A001358) k.
a(n) = A076290( A102467(n+1)). - R. J. Mathar, Aug 31 2009

Extensions

Corrected and extended by R. J. Mathar, Aug 31 2009

A133477 Sum of cubefree divisors of n excluding 1.

Original entry on oeis.org

0, 2, 3, 6, 5, 11, 7, 6, 12, 17, 11, 27, 13, 23, 23, 6, 17, 38, 19, 41, 31, 35, 23, 27, 30, 41, 12, 55, 29, 71, 31, 6, 47, 53, 47, 90, 37, 59, 55, 41, 41, 95, 43, 83, 77, 71, 47, 27, 56, 92, 71, 97, 53, 38, 71, 55, 79, 89, 59, 167, 61, 95, 103, 6, 83, 143, 67, 125, 95, 143, 71, 90
Offset: 1

Views

Author

Jonathan Vos Post, Nov 29 2007

Keywords

Examples

			a(8) = 6 because the divisors of 8 are {1,2,4,8}, the cubefree divisors are 1, 2, 4 so we get a(8) = 2 + 4 = 6.

Links

Crossrefs

Cf. A004709, A005117, A039653, A048250, A068318, A083681.
Cf. A073184, A073185.

Programs

  • Mathematica
    scfd[n_]:=Total[Select[Divisors[n],Max[Transpose[FactorInteger[#]][[2]]]<3&]]; Array[scfd,80]-1 (* Harvey P. Dale, Nov 30 2014 *)
f[p_, e_] := 1 + p + If[e > 1, p^2, 0]; a[1] = 0; a[n_] := -1 + Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 09 2022 *)

Formula

a(n) = A073185(n) - 1. - N. J. A. Sloane, Nov 30 2007

Extensions

Edited by N. J. A. Sloane, Nov 30 2007

A176347 n-th semiprime minus sum of its prime factors.

Original entry on oeis.org

0, 1, 3, 3, 5, 7, 11, 9, 15, 11, 19, 15, 23, 17, 23, 21, 35, 31, 39, 35, 27, 29, 47, 43, 35, 59, 39, 63, 41, 55, 71, 59, 45, 71, 51, 71, 87, 57, 95, 99, 59, 79, 83, 107, 65, 91, 69, 119, 111, 71, 119, 77, 103, 131, 81, 143, 115, 87, 119, 143, 159, 95, 131, 99, 167, 159, 101
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Apr 15 2010

Keywords

Programs

  • Maple
    From R. J. Mathar, Apr 20 2010: (Start)
isA001358 := proc(n) numtheory[bigomega](n) = 2 ; end proc:
A001358 := proc(n) option remember ; if n = 1 then return 4 ; else for a from procname(n-1)+1 do if isA001358(a) then return a; end if; end do; end if; end proc:
A084126 := proc(n) min(op(numtheory[factorset](A001358(n)))) ; end proc:
A084127 := proc(n) max(op(numtheory[factorset](A001358(n)))) ; end proc:
A068318 := proc(n) A084126(n)+A084127(n) ; end proc:
A176347 := proc(n) A001358(n)-A068318(n) ; end proc: seq(A176347(n),n=1..80) ; (End)

Formula

a(n) = A001358(n) - A068318(n).

Extensions

Entries checked by R. J. Mathar, Apr 20 2010
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