A225643 -id:A225643 - cp's OEIS frontend

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

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

Antti Karttunen, May 15 2013

nonn

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

Cf. A225645 (partial sums).

Cf. A225634, A225653, A225654, A225639, A226056.

(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))))))))

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

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

A225650 The greatest common divisor of Landau g(n) and n.

Original entry on oeis.org

1, 1, 2, 3, 4, 1, 6, 1, 1, 1, 10, 1, 12, 1, 14, 15, 4, 1, 6, 1, 20, 21, 2, 1, 24, 5, 2, 1, 14, 1, 30, 1, 4, 3, 2, 35, 36, 1, 2, 39, 40, 1, 42, 1, 44, 15, 2, 1, 24, 7, 10, 3, 52, 1, 18, 55, 56, 3, 2, 1, 60, 1, 2, 21, 8, 65, 66, 1, 4, 3, 70, 1, 72, 1, 2, 15, 76, 77, 78, 1

Offset: 0

Antti Karttunen, May 11 2013

nonn

R. J. Mathar, Table of n, a(n) for n = 0..10000

A225648 gives the position of ones, and likewise A225651 gives the positions of fixed points, that is, a(A225651(n)) = A225651(n) for all n.

Cf. A225640-A225643, A225654.

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]]]]]; a[n_] := GCD[n, g[n]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 02 2016, after Alois P. Heinz *)

(define (A225650 n) (gcd (A000793 n) n))
;; Scheme-code for A000793 can be found in the Program section of that entry.

a(n) = gcd(n, A000793(n)).

A225633 Number of steps to reach a fixed point (A003418(n)), when starting from partition {1+1+1+...+1} of n and continuing with the process described in A225632.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, May 13 2013

nonn

a(0)=0, as its only partition is an empty partition {}, and by convention lcm()=1, thus it takes no steps to reach from 1 to A003418(0)=1.

The records occur at positions 0, 2, 3, 5, 9, 11, 13, 19, 27, 31, 38, 43, 47, 61, 73, 81, ... and they seem to occur in order, i.e., as A001477. Thus the record-positions probably also give the left inverse function for this sequence. It also seems that each integer occurs only finite times in this sequence, so there should be a right inverse function as well.

Cf. A225634, A225643, A225629, A225654, A226056.

a(n) = A225634(n) - 1.

cp's OEIS Frontend

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

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A225650 The greatest common divisor of Landau g(n) and n.

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A225633 Number of steps to reach a fixed point (A003418(n)), when starting from partition {1+1+1+...+1} of n and continuing with the process described in A225632.

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