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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.

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A225639 a(n) is the index of the first row in column n of A225640 where A226055(n) occurs.

Original entry on oeis.org

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

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Author

Antti Karttunen, May 21 2013

Keywords

Comments

Consider an algorithm which finds a maximal value lcm(p1,p2,...,pk,prevmax) among all partitions {p1+p2+...+pk} of n, where the "seed number" prevmax is a maximal value from the previous iteration.
a(n) gives the number of such iterations needed when starting from the initial seed value n, for the process to reach the first identical value (A226055(n)) that is eventually produced when the same algorithm is started with the initial seed value of 1.
The records occur at positions 0, 5, 11, 14, 21, 26, 30, 38, 50, 62, 74, 80, ...

Examples

			Looking at A225632 and A225642, which are just arrays A225630 and A225640 transposed and eventually repeating values removed, we see that:
row 11 of A225632 is 1, 30, 420, 4620, 13860, 27720;
row 11 of A225642 is 11, 330, 4620, 13860, 27720;
their first common term, 4620 (= A226055(11)), occurs as two positions after the initial 11 of that row in A225642, thus a(11)=2.
Equivalently, 4620 occurs as the element A(2,11) of array A225640.

Programs

  • Scheme
    (define (A225639 n) (if (zero? n) n (let ((fun1 (lambda (seed) (let ((max1 (list 0))) (fold_over_partitions_of n 1 lcm (lambda (p) (set-car! max1 (max (car max1) (lcm seed p))))) (car max1)))) (fun2 (lambda (seed) (let ((max2 (list 0))) (fold_over_partitions_of n (max 1 n) lcm (lambda (p) (set-car! max2 (max (car max2) (lcm seed p))))) (car max2))))) (steps-to-convergence-nondecreasing fun2 fun1 n 1))))
(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))))))))
(define (steps-to-convergence-nondecreasing fun1 fun2 initval1 initval2) (let loop ((steps 0) (a1 initval1) (a2 initval2)) (cond ((equal? a1 a2) steps) ((< a1 a2) (loop (+ steps 1) (fun1 a1) a2)) (else (loop steps a1 (fun2 a2))))))

Formula

a(n) = A225638(n) - A225654(n) = A225644(n) - A226056(n). (But please see the given Scheme-program for how this sequence can actually be computed.)
A226055(n) = A225640(a(n),k) = A225630(A225638(n),k).

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).

A225627 a(n) = lcm(A000793(n),p1,p2,...,pk) for such a partition {p1+p2+...+pk} of n that maximizes this value among all partitions of n.

Original entry on oeis.org

1, 1, 2, 6, 12, 30, 30, 84, 120, 180, 210, 420, 660, 780, 1260, 4620, 5460, 5460, 5460, 9240, 13860, 13860, 16380, 32760, 120120, 180180, 180180, 235620, 180180, 471240, 1021020, 1021020, 1141140, 1141140, 2282280, 2282280, 4476780, 4476780, 6846840, 6846840
Offset: 0

Views

Author

Antti Karttunen, May 13 2013

Keywords

Comments

Row 2 of A225630.
This could be called a "twice-iterated Landau's function."

Links

Crossrefs

Cf. A000793, A225630, A225628, A225629, A225636.

Programs

  • Scheme
    (define (A225627 n) (let ((maxlcm (list 0))) (fold_over_partitions_of n (A000793 n) lcm (lambda (p) (set-car! maxlcm (max (car maxlcm) p)))) (car maxlcm)))
(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) = A225636(n)*A000793(n).

A225637 a(n) = A003418(n)/A225629(n).

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 2, 5, 7, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 3, 2, 2, 2, 3, 3, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
Offset: 0

Views

Author

Antti Karttunen, May 13 2013

Keywords

Comments

For n >= 2, a(n) is the final factor by which the A225629(n) needs to be multiplied that it finally reaches the fixed point A003418(n) of the column n of A225630.
The first composite, 4, occurs at n=20. The first composite which is not power of prime, namely 6, occurs at n=61.
For all n >= 3, a(n) divides A225558(n).

Crossrefs

Cf. A003418, A225629, A225636, A225558, A225652, A225656.

Programs

A225628 a(n) = lcm(A225627(n),p1,p2,...,pk) for such a partition {p1+p2+...+pk} of n which maximizes this value among all partitions of n.

Original entry on oeis.org

1, 1, 2, 6, 12, 60, 60, 420, 840, 1260, 840, 4620, 4620, 8580, 16380, 60060, 60060, 92820, 92820, 175560, 263340, 360360, 360360, 753480, 2762760, 6126120, 6126120, 8953560, 6846840, 13665960, 58198140, 58198140, 78738660, 78738660, 157477320, 157477320
Offset: 0

Views

Author

Antti Karttunen, May 13 2013

Keywords

Comments

Row 3 of A225630.
This could be called a "thrice-iterated Landau's function."

Links

Crossrefs

Cf. A225630, A225627, A225629.

Programs

  • Scheme
    (define (A225628 n) (let ((maxlcm (list 0))) (fold_over_partitions_of n (A225627 n) lcm (lambda (p) (set-car! maxlcm (max (car maxlcm) p)))) (car maxlcm)))
;; Adapted by AK from Kreher & Stinson, CAGES-book, p. 68, Algorithm 3.1:
(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))))))))
Previous Showing 11-15 of 15 results.

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