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 1-6 of 6 results.

A212813 Number of steps for n to reach 8 under iteration of the map i -> A036288(i), or -1 if 8 is never reached.

Original entry on oeis.org

-1, -1, -1, -1, -1, -1, 1, 0, 2, 1, 2, 1, 3, 2, 3, 3, 4, 3, 3, 2, 3, 3, 3, 2, 3, 4, 2, 2, 4, 3, 4, 3, 4, 3, 4, 3, 5, 4, 5, 2, 5, 4, 5, 4, 2, 5, 3, 2, 4, 4, 4, 4, 3, 2, 5, 3, 4, 4, 5, 4, 5, 4, 3, 4, 4, 5, 5, 4, 3, 4, 5, 4, 4, 3, 3, 3, 4, 4, 4, 3, 4, 5, 5, 4, 4, 6, 5, 4, 4, 3, 4, 3, 5, 5, 4, 3, 6, 5, 4, 4, 5, 4, 4, 3, 4, 4, 4, 3, 5, 4, 6, 4, 5, 4, 5, 4, 3, 5, 4
Offset: 1

Views

Author

N. J. A. Sloane, May 30 2012

Keywords

Comments

It is known that a(n) >= 0 for n >= 7. Bellamy and Cadogan call a(n) the "class number" of n, but this is not a good idea as this term is already overworked.
a(A212911(n)) = n and a(m) < n for m < A212911(n). - Reinhard Zumkeller, May 30 2012

References

  • Bellamy, O. S.; Cadogan, C. C. Subsets of positive integers: their cardinality and maximality properties. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 167--178, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561043 (82b:10006)
  • R. Honsberger, Mathematical Morsels, MAA, 1978, p. 223.

Links

Crossrefs

Cf. A036288, A212814, A212815, A212816, A212908, A212909, A212911.

Programs

  • Haskell
    a212813 n | n < 7     = -1
          | otherwise = fst $ (until ((== 8) . snd))
                              (\(s, x) -> (s + 1, a036288 x)) (0, n)
-- Reinhard Zumkeller, May 30 2012
  • Maple
    Simple-minded Maple program from N. J. A. Sloane, May 30 2012:
f:=proc(n) local i,t1; t1:=ifactors(n)[2]; 1+add( t1[i][1]*t1[i][2], i=1..nops(t1)); end; # this is A036288
g:=proc(n) local i,t1; global f; t1:=n; for i from 1 to 1000 do if t1=8 then RETURN(i-1); fi; t1:=f(t1); od; -1; end; # this is A212813
  • Mathematica
    imax = 11 (* = max term plus 1 *);
a36288[n_] := If[n == 1, 1, Total[Times @@@ FactorInteger[n]] + 1];
a[n_] := Module[{i, k}, For[k = n; i = 1, i <= imax, i++, If[k == 8, Return[i - 1]]; k = a36288[k]]; If[n > 6, Print["imax ", imax, " probably too small"]]; -1];
Array[a, 120] (* Jean-François Alcover, Aug 01 2018 *)
  • PARI
    A212813(n)={ n>8 & for(c=1,9e9,(n=A036288(n))==8 & return(c));(n==7)-(n<7) }  \\ M. F. Hasler, May 30 2012

A036288 a(n) = 1 + integer log of n: if the prime factorization of n is n = Product (p_j^k_j) then a(n) = 1 + Sum (p_j * k_j) (cf. A001414).

Original entry on oeis.org

1, 3, 4, 5, 6, 6, 8, 7, 7, 8, 12, 8, 14, 10, 9, 9, 18, 9, 20, 10, 11, 14, 24, 10, 11, 16, 10, 12, 30, 11, 32, 11, 15, 20, 13, 11, 38, 22, 17, 12, 42, 13, 44, 16, 12, 26, 48, 12, 15, 13, 21, 18, 54, 12, 17, 14, 23, 32, 60, 13, 62, 34, 14, 13, 19, 17, 68, 22
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

If this function is iterated then, starting at any number n >= 7, we will always reach an 8 - see A212813, A212814, A212815. - N. J. A. Sloane, May 30 2012
a(n) = 1 + Sum_{k=1..A001221(n)} A027748(k) * A124010(k). - Reinhard Zumkeller, May 30 2012

Examples

			12 = 2^2 * 3 so a(12) = 1 + 2^2 + 3 = 8.

References

  • Bellamy, O. S.; Cadogan, C. C. Subsets of positive integers: their cardinality and maximality properties. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 167--178, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561043 (82b:10006) - From N. J. A. Sloane, May 30 2012
  • R. Honsberger, Problem 89, Another Curious Sequence, Mathematical Morsels, MAA, 1978, pp. 223-227.

Links

Crossrefs

Equals A001414 + 1.
Cf. A212813, A212814, A212815, A212816, A212908, A212909.

Programs

  • Haskell
    a036288 n = 1 + sum (zipWith (*)
            (a027748_row n) (map fromIntegral $ a124010_row n))
-- Reinhard Zumkeller, May 30 2012
  • Maple
    f:=proc(n) local i,t1; t1:=ifactors(n)[2]; 1+add( t1[i][1]*t1[i][2], i=1..nops(t1)); end; # N. J. A. Sloane, May 30 2012
  • Mathematica
    f[1]=1;f[n_]:=Total[Apply[Times,FactorInteger[n],1]]+1;f/@Range@68 (* Ivan N. Ianakiev, Apr 18 2016 *)
  • PARI
    A036288(n)=1+(n=factor(n))[,1]~*n[,2]  \\ M. F. Hasler, May 30 2012

Extensions

Edited by N. J. A. Sloane, Jun 01 2012

A212814 a(n) = number of integers k >= 7 such that A212813(k) = n.

Original entry on oeis.org

1, 3, 11, 2632
Offset: 0

Views

Author

N. J. A. Sloane, May 30 2012. I added Hans Havermann's comment May 31 2012

Keywords

Comments

The next term may be very large, see A212815.
Comment from Hans Havermann, Sequence Fans Mailing List, May 31 2012: The 11 numbers k for which A212813(k)=2 are 9, 11, 14, 20, 24, 27, 28, 40, 45, 48, 54. Empirically, it appears that 2632 is the sum of the number of prime partitions (A000607) of the eleven numbers 8, 10, 13, 19, 23, 26, 27, 39, 44, 47, 53. I hesitate turning this into a conjecture only because the 3 numbers k for which A212813(k)=1 are 7, 10, 12 and the sum of the number of prime partitions of the three numbers 6, 9, 11 is twelve, not eleven (the extra partition being, I think, 2+2+2).

Examples

			The 11 numbers k for which A212813(k)=2 are 9, 11, 14, 20, 24, 27, 28, 40, 45, 48, 54 (see A212816).

References

  • Bellamy, O. S.; Cadogan, C. C. Subsets of positive integers: their cardinality and maximality properties. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 167--178, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561043 (82b:10006)

Links

Crossrefs

Cf. A036288, A212813, A212815, A212816, A212908, A212909.

A212815 a(n) = largest number k >= 7 such that A212813(k) = n.

Original entry on oeis.org

8, 12, 54, 258280326
Offset: 0

Views

Author

N. J. A. Sloane, May 30 2012

Keywords

Comments

Bellamy and Cadogan say that a(4) = 2*3^86093441, which is too large to include here.

References

  • Bellamy, O. S.; Cadogan, C. C. Subsets of positive integers: their cardinality and maximality properties. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 167--178, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561043 (82b:10006)

Crossrefs

Cf. A036288, A212813, A212814, A212816, A212908, A212909.

A212908 Numbers n such that A212813(n) = 3.

Original entry on oeis.org

13, 15, 16, 18, 19, 21, 22, 23, 25, 30, 32, 34, 36, 47, 53, 56, 63, 69, 74, 75, 76, 80, 90, 92, 96, 104, 108, 117, 123, 133, 136, 153, 165, 169, 172, 176, 190, 198, 228, 238, 245, 259, 273, 285, 286, 294, 304, 325, 328, 340, 342, 350, 357, 369, 370, 376, 385, 390, 403, 408, 416, 420, 423, 425, 429, 444, 448, 459, 462, 465, 468, 484, 496, 500
Offset: 1

Views

Author

N. J. A. Sloane, May 30 2012

Keywords

References

  • Bellamy, O. S.; Cadogan, C. C. Subsets of positive integers: their cardinality and maximality properties. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 167--178, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561043 (82b:10006)

Links

Crossrefs

Cf. A212813, A212814, A212815, A212816, A212909.

Programs

  • Haskell
    import Data.List (elemIndices)
a212908 n = a212908_list !! (n-1)
a212908_list = map (+ 1) $ elemIndices 3 a212813_list
-- Reinhard Zumkeller, May 30 2012
  • Mathematica
    nmax = 258280326 (* = last term = a(2632) *);
kmax = 100 (* = number of terms to compute *);
a36288[n_] := a36288[n] = If[n==1, 1, Total[Times @@@ FactorInteger[n]]+1];
a212813[n_] := Module[{i, k = n}, For[i = 1, i <= 4, i++, If[k == 8, Return[i-1]]; k = a36288[k]]; -1];
k = 0;
Do[If[a212813[n] == 3, k++; If[k > kmax, Break[]]; a[k] = n; Print["a(", k, ") = ", n]], {n, 1, nmax}];
Array[a, kmax] (* Jean-François Alcover, Aug 02 2018 *)

Extensions

Keyword "full" added by Donovan Johnson, Jun 02 2012

A212909 Numbers n such that A212813(n) = 4.

Original entry on oeis.org

17, 26, 29, 31, 33, 35, 38, 42, 44, 49, 50, 51, 52, 57, 58, 60, 62, 64, 65, 68, 70, 72, 73, 77, 78, 79, 81, 84, 85, 88, 89, 91, 95, 99, 100, 102, 103, 105, 106, 107, 110, 112, 114, 116, 119, 120, 121, 124, 125, 126, 128, 129, 130, 132, 135, 142, 143, 144, 146, 147, 150, 154, 156, 160, 162, 170, 175, 177, 178, 180, 182, 184, 192, 195, 196, 197, 202, 204, 205
Offset: 1

Views

Author

N. J. A. Sloane, May 30 2012

Keywords

References

  • Bellamy, O. S.; Cadogan, C. C. Subsets of positive integers: their cardinality and maximality properties. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 167--178, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561043 (82b:10006)

Links

Crossrefs

Cf. A212813, A212814, A212815, A212816, A212908.

Programs

  • Haskell
    import Data.List (elemIndices)
a212909 n = a212909_list !! (n-1)
a212909_list = map (+ 1) $ elemIndices 4 a212813_list
-- Reinhard Zumkeller, May 30 2012
Showing 1-6 of 6 results.

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

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