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.

A225634 a(n) = Number of distinct values in column n of A225630.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 7, 6, 6, 6, 7, 7, 8, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 9, 10, 11, 11, 10, 11, 10, 12, 12, 12, 12, 13, 12, 13, 13, 13, 12, 13, 13, 13, 12, 12, 12, 13, 13, 14, 13, 13, 14, 14, 13, 14, 13, 13, 13, 14, 14
Offset: 0

Views

Author

Antti Karttunen, May 13 2013

Keywords

Comments

Also, for n>=1, a(n) = the length of n-th row of A225632.
For the positions of records, and other remarks, see comments at A225633.

Crossrefs

Cf. A225635 (partial sums).
Cf. A225644, A225653, A225654.

Programs

  • Scheme
    (define (A225634 n) (count_number_of_distinct_lcms_of_partitions_until_fixed_point_met n 1))
(define (count_number_of_distinct_lcms_of_partitions_until_fixed_point_met n initial_value) (let loop ((lcms (list initial_value initial_value))) (fold_over_partitions_of n 1 lcm (lambda (p) (set-car! lcms (max (car lcms) (lcm (second lcms) p))))) (if (= (car lcms) (second lcms)) (length (cdr lcms)) (loop (cons (car lcms) lcms)))))
(define (fold_over_partitions_of m initval addpartfun colfun) (let recurse ((m m) (b m) (n 0) (partition initval)) (cond ((zero? m) (colfun partition)) (else (let loop ((i 1)) (recurse (- m i) i (+ 1 n) (addpartfun i partition)) (if (< i (min b m)) (loop (+ 1 i))))))))

Formula

a(n) = A225638(n)+A226056(n).
a(n) = A225633(n) + 1.

A225644 a(n) = number of distinct values in column n of A225640.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 6, 5, 5, 5, 6, 5, 7, 6, 7, 6, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 9, 10, 9, 9, 10, 10, 10, 10, 10, 11, 12, 11, 11, 12, 12, 11, 11, 13, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 12, 14, 13, 13, 13, 13, 13, 13, 14, 14, 15
Offset: 0

Views

Author

Antti Karttunen, May 15 2013

Keywords

Comments

For the positions of records, and other remarks, see comments at A225643.

Crossrefs

Cf. A225645 (partial sums).
Cf. A225634, A225653, A225654, A225639, A226056.

Programs

  • Scheme
    (define (A225644 n) (count_number_of_distinct_lcms_of_partitions_until_fixed_point_met n n))
(define (count_number_of_distinct_lcms_of_partitions_until_fixed_point_met n initial_value) (let loop ((lcms (list initial_value initial_value))) (fold_over_partitions_of n 1 lcm (lambda (p) (set-car! lcms (max (car lcms) (lcm (second lcms) p))))) (if (= (car lcms) (second lcms)) (length (cdr lcms)) (loop (cons (car lcms) lcms)))))
(define (fold_over_partitions_of m initval addpartfun colfun) (let recurse ((m m) (b m) (n 0) (partition initval)) (cond ((zero? m) (colfun partition)) (else (let loop ((i 1)) (recurse (- m i) i (+ 1 n) (addpartfun i partition)) (if (< i (min b m)) (loop (+ 1 i))))))))

Formula

a(n) = A225643(n) + 1.
a(n) = A225639(n) + A226056(n).

A225654 a(n) = the number of surplus elements on the n-th row of A225632 compared to the n-th row of A225642.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 2, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1
Offset: 0

Views

Author

Antti Karttunen, May 17 2013

Keywords

Comments

a(n) = how many more iterations is required to reach fixed point A003418(n) with the process described in A225632 and A225642 when starting from partition {1+1+...+1} of n, than when starting from partition {n} of n.
a(0)=0 by convention.

Crossrefs

Cf. A225653 (positions of zeros).
Cf. A225632, A225642, A225630, A225640, A225634, A225644, A225630, A225640.

Programs

Formula

a(n) = A225634(n) - A225644(n).
a(n) = A225638(n) - A225639(n).

A225651 Numbers k that divide A000793(k).

Original entry on oeis.org

1, 2, 3, 4, 6, 10, 12, 14, 15, 20, 21, 24, 30, 35, 36, 39, 40, 42, 44, 52, 55, 56, 60, 65, 66, 70, 72, 76, 77, 78, 84, 85, 90, 91, 95, 99, 102, 105, 110, 114, 115, 117, 119, 120, 126, 130, 132, 133, 136, 138, 140, 143, 152, 153, 154, 155, 156, 161, 165, 170
Offset: 1

Views

Author

Antti Karttunen, May 16 2013

Keywords

Comments

After 1, a subset of A225649.
Also, for all n, A225650(a(n)) = a(n) and A225655(a(n)) = A000793(a(n)).

Links

Crossrefs

Cf. A225648, A225649, A225650, A225653, A225655-A225657.

Programs

  • Maple
    b:= proc(n, i) option remember; local p;
      p:= `if`(i<1, 1, ithprime(i));
      `if`(n=0 or i<1, 1, max(b(n, i-1),
           seq(p^j*b(n-p^j, i-1), j=1..ilog[p](n))))
    end:
g:=n->b(n, `if`(n<8, 3, numtheory[pi](ceil(1.328*isqrt(n*ilog(n)))))):
a:= proc(n) option remember; local k;
      for k from 1+`if`(n=1, 0, a(n-1))
      while not irem(g(k), k)=0 do od; k
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, May 22 2013
  • Mathematica
    Reap[For[n=1, n <= 40, n++, If[Divisible[Max[LCM @@@ IntegerPartitions[n] ], n], Sow[n]]]][[2, 1]]
(* or, for a large number of terms: *)
b[n_, i_] := b[n, i] = Module[{p}, p = If[i<1, 1, Prime[i]]; If[n==0 || i<1, 1, Max[b[n, i-1], Table[p^j*b[n - p^j, i-1], {j, 1, Log[p, n] // Floor}]]]]; g[n_] := b[n, If[n<8, 3, PrimePi[Ceiling[1.328*Sqrt[n*Log[n] // Floor]]]]]; Reap[For[k=1, k <= 1000, k++, If[Divisible[g[k], k], Sow[ k]]]][[2, 1]] (* Jean-François Alcover, Feb 28 2016, after Alois P. Heinz *)
Showing 1-4 of 4 results.

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