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 1-3 of 3 results.

A164927 Sum of the odd prime divisors of numbers with all odd prime divisors of the form 4k+1.

Original entry on oeis.org

5, 5, 13, 17, 5, 5, 13, 29, 17, 37, 5, 41, 5, 13, 53, 29, 61, 18, 17, 73, 37, 5, 41, 22, 89, 97, 5, 101, 13, 53, 109, 113, 29, 61, 5, 18, 17, 137, 34, 73, 37, 149, 157, 5, 41, 13, 22, 173, 89, 181, 42, 193, 97, 197, 5, 101, 46, 13, 53, 109, 30, 113, 229, 29, 233, 241, 61, 5, 257
Offset: 1

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Author

Jonathan Vos Post, Aug 31 2009

Keywords

Comments

We define a sequence b(n) = 5, 10, 13, 17, 20, 25, 26, 29, 34, 37, 40, 41, 50, 52, 53, 58, 61, 65, 68, 73, ... to consist of those numbers where all odd prime factors are primes contained in A002144, and which have at least one prime factor in this class.
b(n) differs from A009003 which also contains numbers like 30=2*3*5 or 39=3*13, 3 not being in A002144.
b(n) essentially contains elements of A004613 multiplied by powers of 2.
a(n) is the sum of the distinct odd prime factors of b(n), where "distinct" means that the multiplicity (exponent) in the prime factorization of b(n) is ignored.
Sum of distinct Pythagorean prime divisors of integers whose only odd prime divisors are Pythagorean primes A002144.
Analogous sequence for primes of form 4k+3 is A164928.
Analogous sequence for primes of form 6k+1 is A164929.
Analogous sequence for primes of form 6k+5 is A164930.

Examples

			a(18) = 18 because b(18) = 65 = 5*13, and 5+13 = 18.
The smallest number, all of whose prime factors are of form 4n+1, whose sum of distinct prime factors is prime: 1885 = 5 * 13 * 29; and 5 + 13 + 29 = 47.

Crossrefs

Cf. A000040, A002144, A009003, A164927-A164930.

Programs

  • Maple
    isb := proc(n) fs := numtheory[factorset](n) minus {2} ; if fs = {} then RETURN(false); else for f in fs do if op(1,f) mod 4 <> 1 then RETURN(false) ; fi; od: RETURN(true) ; fi; end:
b := proc(n) if n = 1 then 5; else for a from procname(n-1)+1 do if isb(a) then RETURN(a) ; fi; od: fi; end:
A164927 := proc(n) local f; numtheory[factorset]( b(n)) minus {2} ; add(f,f=%) ; end: seq(A164927(n),n=1..120) ; # R. J. Mathar, Sep 09 2009

Extensions

Edited, definition clarified by R. J. Mathar, Sep 08 2009

A164928 Sum of the odd prime divisors of numbers whose odd prime divisors are all of the form 4k+3.

Original entry on oeis.org

3, 3, 7, 3, 11, 3, 7, 3, 19, 10, 11, 23, 3, 3, 7, 31, 14, 3, 19, 10, 43, 11, 23, 47, 3, 7, 3, 7, 22, 59, 31, 10, 14, 67, 26, 71, 3, 19, 18, 79, 3, 83, 10, 43, 11, 23, 34, 47, 3, 7, 14, 103, 107, 3, 7, 22, 59, 11, 31, 10, 127, 46, 131, 14, 26, 67, 26, 139, 50, 71, 3, 10, 151, 19, 18
Offset: 1

Views

Author

Jonathan Vos Post, Aug 31 2009

Keywords

Comments

We define a sequence b(n) = 3, 6, 7, 9, 11, 12, 14, 18, 19, 21, 22, 23, ... to consist of those numbers where all odd prime factors are primes contained in A002145, and which have at least one prime factor in this class; b(n) is basically A004144 without the powers of 2.
a(n) is the sum of the distinct odd prime factors of b(n), where "distinct" means that the multiplicity (exponent) in the prime factorization of b(n) is ignored.
Analogous sequence for primes of form 4k+1 is A164927.
Analogous sequence for primes of form 6k+1 is A164929.
Analogous sequence for primes of form 6k+5 is A164930.

Examples

			a(11) = 10 because b(11) = 21 = 3*7, and 3+7 = 10.
The smallest nonprime number, all of whose prime factors are of form 4n+3, whose sum of distinct prime factors is prime: b(181) = 3*7*19 = 399; 3+7+19 = 29.

Links

Crossrefs

Cf. A000040, A002145, A009003, A164927-A164930.

Programs

  • Maple
    isb := proc(n) fs := numtheory[factorset](n) minus {2} ; if fs = {} then RETURN(false); else for f in fs do if op(1,f) mod 4 <> 3 then RETURN(false) ; fi; od: RETURN(true) ; fi; end:
b := proc(n) if n = 1 then 3; else for a from procname(n-1)+1 do if isb(a) then RETURN(a) ; fi; od: fi; end:
A164928 := proc(n) local f; numtheory[factorset]( b(n)) minus {2} ; add(f,f=%) ; end: seq(A164928(n),n=1..120) ; # R. J. Mathar, Sep 08 2009
  • Mathematica
    sopd[n_]:=Module[{ff=Select[Transpose[FactorInteger[n]][[1]],OddQ]},If[ And@@ (Mod[#,4]==3&/@ff),Total[ff],0]]; Select[Array[sopd,200],#>0&] (* Harvey P. Dale, Dec 16 2013 *)

Extensions

Edited and extended by R. J. Mathar, Sep 08 2009

A164930 Sum of odd prime divisors of numbers with all odd prime divisors of the form 6k+5.

Original entry on oeis.org

5, 5, 11, 17, 5, 11, 23, 5, 29, 17, 5, 41, 11, 23, 47, 5, 53, 16, 29, 59, 17, 71, 5, 41, 83, 22, 11, 89, 23, 47, 5, 101, 53, 107, 16, 113, 28, 29, 59, 11, 5, 131, 17, 137, 71, 34, 149, 5, 41, 83, 167, 22, 173, 11, 89, 179, 23, 28, 47, 191, 197, 5, 101, 46, 53, 107, 16, 113, 227, 28
Offset: 1

Views

Author

Jonathan Vos Post, Aug 31 2009

Keywords

Comments

We define a sequence b(n) = 5, 10, 11, 17, 20, 22, 23, 25, 29, 34, 40, 41, 44, 46, 47, 50, 53, 55, 58, ... to consist of those numbers where all odd prime factors are primes contained in A007528, and which have at least one prime factor in this class. a(n) is the sum of the distinct odd prime factors of b(n), where "distinct" means that the multiplicity (exponent) in the prime factorization of b(n) is ignored.
Analogous sequence for primes of form 4k+1 is A164927.
Analogous sequence for primes of form 4k+3 is A164928.
Analogous sequence for primes of form 6k+1 is A164929.
The sum of an even number of primes of form 6k+1 is even (hence composite).
The sum of 3 primes of form 6n+5 is composite because (6a+5)+(6b+5)+(6c+5) = 3*(a+b+c+3).
However, the sum of 5 primes of form 6n+5 may be prime:
The smallest number, all of whose prime factors are of form 6k+5, whose sum of distinct prime factors is prime: 881705 = 5 * 11 * 17 * 23 * 41, and 5 + 11 + 17 + 23 + 41 = 97 is prime.

Examples

			a(18) = 16 because b(18)= 55 = 5*11, and 5+11 = 16.

Crossrefs

Cf. A000040, A007528, A164927-A164930.

Programs

  • Maple
    isb := proc(n) fs := numtheory[factorset](n) minus {2} ; if fs = {} then RETURN(false); else for f in fs do if op(1,f) mod 6 <> 5 then RETURN(false) ; fi; od: RETURN(true) ; fi; end:
b := proc(n) if n = 1 then 5; else for a from procname(n-1)+1 do if isb(a) then RETURN(a) ; fi; od: fi; end:
A164930 := proc(n) local f; numtheory[factorset]( b(n)) minus {2} ; add(f,f=%) ; end: seq(A164930(n),n=1..120) ; # R. J. Mathar, Sep 09 2009

Extensions

Edited and extended by R. J. Mathar, Sep 09 2009
Showing 1-3 of 3 results.

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