A242378 -id:A242378 - cp's OEIS frontend

A182944 Square array A(i,j), i >= 1, j >= 1, of prime powers prime(i)^j, by descending antidiagonals.

2, 4, 3, 8, 9, 5, 16, 27, 25, 7, 32, 81, 125, 49, 11, 64, 243, 625, 343, 121, 13, 128, 729, 3125, 2401, 1331, 169, 17, 256, 2187, 15625, 16807, 14641, 2197, 289, 19, 512, 6561, 78125, 117649, 161051, 28561, 4913, 361, 23

Offset: 1

Clark Kimberling, Dec 14 2010

We alternatively refer to this sequence as a triangle T(.,.), with T(n,k) = A(k,n-k+1) = prime(k)^(n-k+1).
The monotonic ordering of this sequence, prefixed by 1, is A000961.
The joint-rank array of this sequence is A182869.
Main diagonal gives A062457. - Omar E. Pol, Sep 11 2018

			Square array A(i,j) begins:
  i \ j: 1      2      3      4      5  ...
  ---\-------------------------------------
  1:     2,     4,     8,    16,    32, ...
  2:     3,     9,    27,    81,   243, ...
  3:     5,    25,   125,   625,  3125, ...
  4:     7,    49,   343,  2401, 16807, ...
  ...
The triangle T(n,k) begins:
  n\k:  1     2     3     4     5     6  ...
  1:    2
  2:    4     3
  3:    8     9     5
  4:   16    27    25     7
  5:   32    81   125    49    11
  6:   64   243   625   343   121    13
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1..150, flattened)
Michael De Vlieger, Diagram showing row n of triangle in a semicircle as noted, with a color function associated with the magnitude of T(n,k) compared to 2^n in light blue, where prime(n) is the smallest and the prime power indicated in red the largest in the row.

Cf. A000961, A006939 (row products of triangle), A062457, A182945, A332979 (row maxima of triangle).
Columns: A000040 (1), A001248 (2), A030078 (3), A030514 (4), A050997 (5), A030516 (6), A092759 (7), A179645 (8), A179665 (9), A030629 (10).
A319075 extends the array with 0th powers.
Subtable of A242378, A284457, A329332.

TableForm[Table[Prime[n]^j,{n,1,14},{j,1,8}]]

From Peter Munn, Dec 29 2019: (Start)
A(i,j) = A182945(j,i) = A319075(j,i).
A(i,j) = A242378(i-1,2^j) = A329332(2^(i-1),j).
A(i,i) = A062457(i).
(End)

Clarified in respect of alternate reading as a triangle by Peter Munn, Aug 28 2022

A242419 Reverse both the exponents and the deltas of the indices of distinct primes present in the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 11, 18, 13, 35, 10, 16, 17, 12, 19, 75, 21, 77, 23, 54, 25, 143, 27, 245, 29, 30, 31, 32, 55, 221, 14, 36, 37, 323, 91, 375, 41, 105, 43, 847, 50, 437, 47, 162, 49, 45, 187, 1859, 53, 24, 33, 1715, 247, 667, 59, 150, 61, 899, 147, 64, 65

Offset: 1

Antti Karttunen, May 17 2014

nonn

This self-inverse permutation (involution) of natural numbers preserves both the total number of prime divisors and the (index of) largest prime factor of n, i.e. for all n it holds that A001222(a(n)) = A001222(n) and A006530(a(n)) = A006530(n) [equally: A061395(a(n)) = A061395(n)].
Considered as an operation on partitions encoded by the indices of primes in the prime factorization of n (as in table A112798), this implements a bijection which reverses the order of "steps" in Young (or Ferrers) diagram of a partition (but keeps the horizontal line segment of each step horizontal and the vertical line segment vertical). Please see the last example in the example section.
To understand the given recursive formula, it helps to see that in the above context (Young diagrams drawn with French notation), the sequences employed effect the following operations:
  A001222: gives the height of whole diagram,
  A051119: removes the bottommost step from the diagram,
  A241919: gives the length of the horizontal line segment of the bottom step, i.e. its width,
  A071178: gives the length of the vertical line segment of the bottom step, i.e. its height,
  A242378(k,n): increases the width of whole Young diagram encoded by n by adding a rectangular area A001222(n) squares high and k squares wide to its left,
  and finally, multiplying by A000040(a)^b adds a new topmost step whose width is a and height is b. Particularly, multiplying by (A000040(A241919(n))^A071178(n)) transfers the bottommost step to the top.

			For n = 10 = 2*5 = p_1^1 * p_3^1, we get p_(3-1)^1 * p_3^1 = 3 * 5 = 15, thus a(10) = 15.
For n = 20 = 2*2*5 = p_1^2 * p_3^1, we get p_(3-1)^1 * p_3^2 = 3^1 * 5^2 = 3*25 = 75, thus a(20) = 75.
For n = 84 = 2*2*3*7 = p_1^2 * p_2 * p_4, when we reverse the deltas of indices, and reverse also the order of exponents, we get p_(4-2) * p_(4-1) * p_4^2 = 3 * 5 * 7^2 = 735, thus a(84) = 735.
For n = 2200, we see that it encodes the partition (1,1,1,3,3,5) in A112798 as 2200 = p_1 * p_1 * p_1 * p_3 * p_3 * p_5 = 2^3 * 5^2 * 11. This in turn corresponds to the following Young diagram in French notation:
   _
  | |
  | |
  | |_ _
  |     |
  |     |_ _
  |_ _ _ _ _|
Reversing the order of "steps", so that each horizontal and vertical line segment centered around a "convex corner" moves as a whole, means that the first stair from the top (one unit wide and three units high) is moved to the last position, the second one (two units wide and two units high) stays in the middle, and the original bottom step (two units wide and one unit high) will be the new topmost step, thus we get the following Young diagram:
   _ _
  |   |_ _
  |       |
  |       |_
  |         |
  |         |
  |_ _ _ _ _|
which represents the partition (2,4,4,5,5,5), encoded in A112798 by p_2 * p_4^2 * p_5^3 = 3 * 7^2 * 11^3 = 195657, thus a(2200) = 195657.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Wikipedia, Young diagram
Index entries for sequences that are permutations of the natural numbers

Fixed points: A242417.
Cf. A112798, A122111, A153212, A000040, A241919, A071178, A242378, A051119, A243057, A105119, A242420.
{A000027, A122111, A153212, A242419} form a 4-group.
{A000027, A069799, A242415, A242419} form also a 4-group.

If n = p_a^e_a * p_b^e_b * ... * p_h^e_h * p_i^e_i * p_j^e_j * p_k^e_k, where p_a < ... < p_k are distinct primes (sorted into ascending order) in the prime factorization of n, and e_a .. e_k are their nonzero exponents, then a(n) = p_{k-j}^e_k * p_{k-i}^e_j  * p_{k-h}^e_i * ... * p_{k-a}^e_b * p_k^e_a.
As a recurrence:
a(1) = 1, and for n>1, a(n) = (A000040(A241919(n))^A071178(n)) * A242378(A241919(n), a(A051119(n))).
By composing related permutations:
a(n) = A122111(A153212(n)) = A153212(A122111(n)).
a(n) = A069799(A242415(n)) = A242415(A069799(n)).
a(n) = A105119(A242420(n)).

A242415 Reverse the deltas of indices of distinct primes in the prime factorization of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 11, 12, 13, 35, 10, 16, 17, 18, 19, 45, 21, 77, 23, 24, 25, 143, 27, 175, 29, 30, 31, 32, 55, 221, 14, 36, 37, 323, 91, 135, 41, 105, 43, 539, 20, 437, 47, 48, 49, 75, 187, 1573, 53, 54, 33, 875, 247, 667, 59, 60, 61, 899, 63, 64, 65

Offset: 1

Antti Karttunen, May 24 2014

nonn

This self-inverse permutation (involution) of natural numbers preserves both the total number of prime divisors and the (index of) largest prime factor of n, i.e., for all n it holds that A001222(a(n)) = A001222(n) and A006530(a(n)) = A006530(n) [equally: A061395(a(n)) = A061395(n)]. It also preserves the exponent of the largest prime: A053585(a(n)) = A053585(n).
From the above it follows, that this fixes prime powers (A000961), among other numbers.
Considered as a function on partitions encoded by the indices of primes in the prime factorization of n (as in table A112798), this implements an operation which reverses the order of horizontal line segments of the "steps" in Young (or Ferrers) diagram of a partition, but keeps the order of vertical line segments intact. Please see the last example in the example section and compare also to the comments given in A242419.

			For n = 10 = 2*5 = p_1 * p_3, we get p_(3-1) * p_3 = 3 * 5 = 15, thus a(10) = 15.
For n = 20 = 2*2*5 = p_1^2 * p_3^1, we get p_(3-1)^2 * p_3^1 = 3^2 * 5 = 45, thus a(20) = 45.
For n = 84 = 2*2*3*7 = p_1^2 * p_2 * p_4, when we reverse the deltas of indices, but keep the exponents same, we get p_(4-2)^2 * p_(4-1) * p_4 = p_2^2 * p_3 * p_4 = 3^2 * 5 * 7 = 315, thus a(84) = 315.
For n = 2200, we see that it encodes the partition (1,1,1,3,3,5) in A112798 as 2200 = p_1 * p_1 * p_1 * p_3 * p_3 * p_5 = 2^3 * 5^2 * 11. This in turn corresponds to the following Young diagram in French notation:
   _
  | |
  | |
  | |_ _
  |     |
  |     |_ _
  |_ _ _ _ _|
Reversing the order of horizontal line segment lengths (1,2,2) to (2,2,1), but keeping the order of vertical line segment lengths as (3,2,1), we get a new Young diagram
   _ _
  |   |
  |   |
  |   |_ _
  |       |
  |       |_
  |_ _ _ _ _|
which represents the partition (2,2,2,4,4,5), encoded in A112798 by p_2^3 * p_4^2 * p_5^1 = 3^3 * 7^2 * 11 = 14553, thus a(2200) = 14553.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Wikipedia, Young diagram
Index entries for sequences that are permutations of the natural numbers

Fixed points: A242413.
Cf. A112798, A122111, A153212, A000040, A241919, A067029, A242378, A051119, A225891.
{A000027, A069799, A242415, A242419} form a 4-group.

If n = p_a^e_a * p_b^e_b * ... * p_h^e_h * p_i^e_i * p_j^e_j * p_k^e_k, where p_a < ... < p_k are distinct primes (sorted into ascending order) in the prime factorization of n, and e_a .. e_k are their nonzero exponents, then a(n) = p_{k-j}^e_a * p_{k-i}^e_b  * p_{k-h}^e_c * ... * p_{k-a}^e_j * p_k^e_k.
As a recurrence: a(1) = 1, and for n>1, a(n) = (A000040(A241919(n))^A067029(n)) * A242378(A241919(n), a(A051119(A225891(n)))).
By composing/conjugating related permutations:
a(n) = A069799(A242419(n)) = A242419(A069799(n)).
a(n) = A122111(A069799(A122111(n))) = A153212(A069799(A153212(n))).

A285102 a(n) = A007913(A285101(n)).

Original entry on oeis.org

2, 6, 210, 72930, 620310, 278995269860970, 12849025509071310, 492608110538467706074890, 1342951001046021018427857601026746070, 37793589449865555275592120894959094883390892772270, 728982633030274864467458719371654181886452163442582606072870, 28339554655955912942523491885490197708224606885407444005070

Offset: 0

Antti Karttunen, Apr 15 2017

nonn

Cf. A003961, A007913, A048675, A068052, A242378, A285101, A285103.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A242378(k,n) = { while(k>0,n = A003961(n); k = k-1); n; };
A285102(n) = { if(0==n,2,lcm(A285102(n-1),A242378(n,A285102(n-1)))/gcd(A285102(n-1),A242378(n,A285102(n-1)))); };

# uses [A003961, A242378]
from sympy import factorint, prime, primepi
from sympy.ntheory.factor_ import core
from operator import mul
def a003961(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [prime(primepi(i) + 1)**f[i] for i in f])
def a242378(k, n):
    while k>0:
        n=a003961(n)
        k-=1
    return n
l=[2]
for n in range(1, 12):
    x=l[n - 1]
    l.append(x*a242378(n, x))
print([core(j) for j in l]) # Indranil Ghosh, Jun 27 2017

(definec (A285102 n) (if (zero? n) 2 (/ (lcm (A285102 (- n 1)) (A242378bi n (A285102 (- n 1)))) (gcd (A285102 (- n 1)) (A242378bi n (A285102 (- n 1)))))))

a(0) = 2, for n > 0, a(n) = lcm(a(n-1),A242378(n,a(n-1))) / gcd(a(n-1),A242378(n,a(n-1))).
a(n) = A007913(A285101(n)).
Other identities. For all n >= 0:
A001221(a(n)) = A001222(a(n)) = A285103(n).
A048675(a(n)) = A068052(n).

A337470 Square array read by falling antidiagonals, where A(n,k) = primorial inflation of k prime shifted n times with A003961.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 4, 15, 5, 1, 30, 9, 35, 7, 1, 12, 105, 25, 77, 11, 1, 210, 45, 385, 49, 143, 13, 1, 8, 1155, 175, 1001, 121, 221, 17, 1, 36, 27, 5005, 539, 2431, 169, 323, 19, 1, 60, 225, 125, 17017, 1573, 4199, 289, 437, 23, 1, 2310, 315, 1225, 343, 46189, 2873, 7429, 361, 667, 29, 1, 24, 15015, 1925, 5929, 1331, 96577, 5491, 12673, 529, 899, 31

Offset: 0

Antti Karttunen, Aug 28 2020

Array is read by descending antidiagonals with n >= 0 and k >= 1 ranging as: (0, 1), (0, 2), (1, 1), (0, 3), (1, 2), (2, 1), (0, 4), (1, 3), (2, 2), (3, 1), ...

			The top left corner of the array begins as:
n/k | 1   2    3    4     5     6      7     8      9     10
----|------------------------------------------------------------
  0 | 1,  2,   6,   4,   30,   12,   210,    8,    36,    60, ...
  1 | 1,  3,  15,   9,  105,   45,  1155,   27,   225,   315, ...
  2 | 1,  5,  35,  25,  385,  175,  5005,  125,  1225,  1925, ...
  3 | 1,  7,  77,  49, 1001,  539, 17017,  343,  5929,  7007, ...
  4 | 1, 11, 143, 121, 2431, 1573, 46189, 1331, 20449, 26741, ...
  5 | 1, 13, 221, 169, 4199, 2873, 96577, 2197, 48841, 54587, ...
etc.

Cf. A108951 (row 0), A337471 (row 1).
Cf. A003961, A242378.
Cf. also A337205, A337472.

up_to = 105-1; \\ Or 78-1.
Ashifted_primorial(n,d) = prod(i=1, primepi(n), prime(i+d));
A337470sq(n, k) = { my(f=factor(k)); prod(i=1, #f~, Ashifted_primorial(f[i, 1], n)^f[i, 2]); };
A337470list(up_to) = { my(v = vector(1+up_to), i=0); for(a=0, oo, for(b=1, a, i++; if(i > #v, return(v)); v[i] = A337470sq(b-1, (a-(b-1))))); (v); };
v337470 = A337470list(up_to);
A337470(n) = v337470[1+n];

A(n,k) = A242378(n,A108951(k)).

A284262 a(n) = where A284259 for the first time obtains value n (positions of its records).

Original entry on oeis.org

1, 2, 6, 105, 5005, 85085, 1616615, 37182145, 6685349671, 247357937827, 10141675450907, 436092044389001, 20496326086283047, 9156001667401012567, 558516101711461766587, 37420578814667938361329, 2656861095841423623654359, 193950859996423924526768207, 15322117939717490037614688353, 1271735788996551673122019133299

Offset: 0

Antti Karttunen, Mar 24 2017

nonn

G. C. Greubel, Table of n, a(n) for n = 0..335

Cf. A001221, A001222, A002110, A003961, A242378, A284259 (a left inverse), A284263.

Cf. also A109819.

A[n_]:= If[n<1, 0, Block[{k=1}, While[Prime[n + k  - 1] > Prime[k]^2, k++]; k - 1]]; a[n_]:=If[n<2, n + 1, Product[Prime[i], {i, A[n] + 1, A[n] + n}]]; Table[a[n], {n, 0, 51}] (* Indranil Ghosh, Mar 24 2017 *)

A(n) = { my(k=1); if(0==n, 0, while(prime(n+k-1) > (prime(k)^2), k = k+1); (k-1)); };
a(n) = prod(i=A(n) + 1, A(n) + n, prime(i));
for(n=0, 51, print1(a(n),", ")) \\ Indranil Ghosh, after Antti Karttunen, Mar 24 2017

from sympy import prime
from operator import mul
from functools import reduce
def A(n):
    if n<1: return 0
    k=1
    while prime(n + k - 1)>prime(k)**2:k+=1
    return k - 1
def a(n): return n + 1 if n<2 else reduce(mul, [prime(i) for i in range(A(n) + 1, A(n) + n + 1)])
print([a(n) for n in range(21)]) # Indranil Ghosh, Mar 24 2017

(define (A284262 n) (A242378bi (A284263 n) (A002110 n))) ;; Where A242378bi(k,n) applies prime shift A003961(n) k times. See A242378.

For n > 1, a(n) = Product_{i = A284263(n)+1 .. A284263(n)+n} prime(i); a(0) = 1, a(1) = 2.
a(n) = A242378(A284263(n), A002110(n)) [shift the prime factorization of the n-th primorial A284263(n) steps towards larger primes].
Other identities. For all n >= 0:
A001221(a(n)) = A001222(a(n))  = n.
A284259(a(n)) = n.

cp's OEIS Frontend

A182944 Square array A(i,j), i >= 1, j >= 1, of prime powers prime(i)^j, by descending antidiagonals.

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A242419 Reverse both the exponents and the deltas of the indices of distinct primes present in the prime factorization of n.

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A242415 Reverse the deltas of indices of distinct primes in the prime factorization of n.

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A285102 a(n) = A007913(A285101(n)).

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A337470 Square array read by falling antidiagonals, where A(n,k) = primorial inflation of k prime shifted n times with A003961.

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A284262 a(n) = where A284259 for the first time obtains value n (positions of its records).

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