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.

A080688 Resort the index of A064553 using A080444 and maintaining ascending order within each grouping: seen as a triangle read by rows, the n-th row contains the A001055(n) numbers m with A064553(m)=n.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 11, 13, 8, 10, 17, 9, 19, 14, 23, 29, 12, 15, 22, 31, 37, 26, 41, 21, 43, 16, 20, 25, 34, 47, 53, 18, 33, 38, 59, 61, 28, 35, 46, 67, 39, 71, 58, 73, 79, 24, 30, 44, 51, 55, 62, 83, 49, 89, 74, 97, 27, 57, 101, 52, 65, 82
Offset: 1

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Author

Alford Arnold, Mar 23 2003

Keywords

Comments

The number 12 can be written as 3*2*2, 4*3, 6*2 and 12 corresponding to each of the four values (12,15,22,31) in the example. Note that A001055(12) = 4. Since A001055(n) depends only on the least prime signature, the values 1,2,4,6,8,12,16,24,30,32,36,... A025487 are of special interest when counting multisets. (see for example, A035310 and A035341).
A064553(T(n,k)) = A080444(n,k) = n for k=1..A001055(n); T(n,1) = A064554(n); T(n,A001055(n)) = A064554(n). - Reinhard Zumkeller, Oct 01 2012
Row n is the sorted list of shifted Heinz numbers of factorizations of n into factors > 1, where the shifted Heinz number of a factorization (y_1, ..., y_k) is prime(y_1 - 1) * ... * prime(y_k - 1). - Gus Wiseman, Sep 05 2018

Examples

			a(18),a(19),a(20) and a(21) are 12,15,22 and 31 because A064553(12,15,22,31) = (12,12,12,12) similarly, A064553(36,45,66,76,93,95,118,121,149) = (36,36,36,36,36,36,36,36,36)
From _Gus Wiseman_, Sep 05 2018: (Start)
Triangle begins:
   1
   2
   3
   4  5
   7
   6 11
  13
   8 10 17
   9 19
  14 23
  29
  12 15 22 31
  37
  26 41
  21 43
  16 20 25 34 47
Corresponding triangle of factorizations begins:
  (),
  (2),
  (3),
  (2*2), (4),
  (5),
  (2*3), (6),
  (7),
  (2*2*2), (2*4), (8),
  (3*3), (9),
  (2*5), (10),
  (11),
  (2*2*3), (3*4), (2*6), (12).
(End)

Links

Crossrefs

Cf. A001055, A025487, A035310, A035341, A064553, A080444.
Cf. A007716, A056239, A162247, A215366, A275024, A317144, A317145, A318871.

Programs

  • Haskell
    a080688 n k = a080688_row n !! (k-1)
a080688_row n = map (+ 1) $ take (a001055 n) $
                elemIndices n $ map fromInteger a064553_list
a080688_tabl = map a080688_row [1..]
a080688_list = concat a080688_tabl
-- Reinhard Zumkeller, Oct 01 2012
  • Mathematica
    facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
Table[Sort[Table[Times@@Prime/@(f-1),{f,facs[n]}]],{n,20}] (* Gus Wiseman, Sep 05 2018 *)

Extensions

More terms from Sean A. Irvine, Oct 05 2011
Keyword tabf added and definition complemented accordingly by Reinhard Zumkeller, Oct 01 2012

A064553 a(1) = 1, a(prime(i)) = i + 1 for i > 0 and a(u * v) = a(u) * a(v) for u, v > 0.

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 5, 8, 9, 8, 6, 12, 7, 10, 12, 16, 8, 18, 9, 16, 15, 12, 10, 24, 16, 14, 27, 20, 11, 24, 12, 32, 18, 16, 20, 36, 13, 18, 21, 32, 14, 30, 15, 24, 36, 20, 16, 48, 25, 32, 24, 28, 17, 54, 24, 40, 27, 22, 18, 48, 19, 24, 45, 64, 28, 36, 20, 32, 30, 40, 21, 72, 22, 26
Offset: 1

Views

Author

Reinhard Zumkeller, Sep 21 2001

Keywords

Comments

a(n) <= n for all n and a(x) = x iff x = 2^i * 3^j for i, j >= 0: a(A003586(n)) = A003586(n) for n > 0. By definition a is completely multiplicative and also surjective. a(p) < a(q) for primes p < q.
Completely multiplicative with a(prime(i)) = i + 1. - Charles R Greathouse IV, Sep 07 2012
a(A080688(n,k)) = A080444(n,k) = n for k=1..A001055(n). - Reinhard Zumkeller, Oct 01 2012

Examples

			a(5) = a(prime(3)) = 3 + 1 = 4; a(14) = a(2*7) = a(prime(1)* prime(4)) = (1+1)*(4+1) = 10.

Links

Crossrefs

Cf. A000005, A000040, A003961, A003963, A049084, A020639, A064554, A064555, A001055, A003586, A064557, A064558, A027746, A027748, A124010, A181819.
A left inverse of A290641.

Programs

  • Haskell
    a064553 1 = 1
a064553 n = product $ map ((+ 1) . a049084) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012, Feb 17 2012, Jan 28 2011
  • Maple
    A064553 := proc(n)
    local a,f,p,e ;
    a := 1 ;
    for f in ifactors(n)[2] do
        p :=op(1,f) ;
        e :=op(2,f) ;
        a := a*(numtheory[pi](p)+1)^e ;
    end do:
    a ;
end proc: # R. J. Mathar, Sep 07 2012
  • Mathematica
    nn=100; a=Table[0, {nn}]; a[[1]]=1; Do[If[PrimeQ[i], a[[i]]=PrimePi[i]+1, p=FactorInteger[i][[1,1]]; a[[i]] = a[[p]]*a[[i/p]]], {i, 2, nn}]; a (* T. D. Noe, Dec 12 2004, revised Sep 27 2011 *)
Array[Apply[Times, Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger[ #]] /. p_ /; PrimeQ@ p :> PrimePi@ p + 1] &, 74] (* Michael De Vlieger, Aug 22 2017 *)
  • PARI
    A064553(n)={n=factor(n);n[,1]=apply(f->1+primepi(f),n[,1]);factorback(n)} \\ M. F. Hasler, Aug 28 2012
  • Scheme
    (define (A064553 n) (if (= 1 n) n (* (+ 1 (A055396 n)) (A064553 (A032742 n))))) ;; Antti Karttunen, Aug 22 2017

Formula

a(A000040(n)) = n+1.
Let the prime factorization of n be p1^e1...pk^ek, then a(n) = (pi(p1)+1)^e1...(pi(pk)+1)^ek, where pi(p) is the index of prime p. - T. D. Noe, Dec 12 2004
From Antti Karttunen, Aug 22 2017: (Start)
a(n) = A003963(A003961(n)).
a(A181819(n)) = A000005(n).
a(A290641(n)) = n. (End)

Extensions

Displayed values double-checked with new PARI code by M. F. Hasler, Aug 28 2012

A146288 Number of divisors of the n-th prime signature number (A025487(n)).

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 5, 8, 8, 6, 9, 10, 12, 7, 12, 12, 16, 8, 15, 18, 14, 16, 16, 20, 9, 18, 24, 16, 24, 20, 24, 10, 21, 30, 18, 32, 24, 27, 28, 11, 32, 24, 36, 25, 36, 20, 40, 28, 36, 32, 12, 40, 27, 32, 48, 30, 42, 22, 48, 32, 45, 36, 13, 48, 30, 48, 60, 35, 48, 48, 24, 54, 50, 56
Offset: 1

Views

Author

Matthew Vandermast, Nov 11 2008

Keywords

Examples

			a(4) = 4 because 4 positive integers divide evenly into A025487(4) = 6: 1, 2, 3 and 6.

Links

Crossrefs

a(n) = sum of the n-th row of A146290, A146292.
A rearrangement of A080444.
Cf. A000005 (number of divisors), A025487.

Programs

  • Haskell
    a146288 = a000005 . a025487  -- Reinhard Zumkeller, Sep 17 2014
  • Mathematica
    s = {1}; Do[If[GreaterEqual @@ (f = FactorInteger[n])[[;; , 2]] && PrimePi[f[[-1, 1]]] == Length[f], AppendTo[s, DivisorSigma[0, n]]], {n, 2, 10000}]; s (* Amiram Eldar, Aug 05 2024 *)

Formula

a(n) = A000005(A025487(n)).
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