A252461 -id:A252461

Search: a252461 -id:a252461

Displaying 1-5 of 5 results found.

page 1

A064989

Multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

+10
480

1, 1, 2, 1, 3, 2, 5, 1, 4, 3, 7, 2, 11, 5, 6, 1, 13, 4, 17, 3, 10, 7, 19, 2, 9, 11, 8, 5, 23, 6, 29, 1, 14, 13, 15, 4, 31, 17, 22, 3, 37, 10, 41, 7, 12, 19, 43, 2, 25, 9, 26, 11, 47, 8, 21, 5, 34, 23, 53, 6, 59, 29, 20, 1, 33, 14, 61, 13, 38, 15, 67, 4, 71, 31, 18, 17, 35, 22, 73, 3, 16

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

From Antti Karttunen, May 12 2014: (Start)

a(A003961(n)) = n for all n. [This is a left inverse function for the injection A003961.]

Bisections are A064216 (the terms at odd indices) and A064989 itself (the terms at even indices), i.e., a(2n) = a(n) for all n.

(End)

From Antti Karttunen, Dec 18-21 2014: (Start)

When n represents an unordered integer partition via the indices of primes present in its prime factorization (for n >= 2, n corresponds to the partition given as the n-th row of A112798) this operation subtracts one from each part. If n is of the form 2^k (a partition having just k 1's as its parts) the result is an empty partition (which is encoded by 1, having an "empty" factorization).

For all odd numbers n >= 3, a(n) tells which number is located immediately above n in square array A246278. Cf. also A246277.

(End)

Alternatively, if numbers are represented as the multiset of indices of prime factors with multiplicity, this operation subtracts 1 from each element and discards the 0's. - M. F. Hasler, Dec 29 2014

LINKS

Harry J. Smith (first 1000 terms) & Antti Karttunen, Table of n, a(n) for n = 1..10000

FORMULA

From Antti Karttunen, Dec 18 2014: (Start)

If n = product A000040(k)^e(k) then a(n) = product A008578(k)^e(k) [where A000040(n) gives the n-th prime, and A008578(n) gives 1 for 1 and otherwise the (n-1)-th prime].

a(1) = 1; for n > 1, a(n) = A008578(A055396(n)) * a(A032742(n)). [Above formula represented as a recurrence. Cf. A252461.]

a(1) = 1; for n > 1, a(n) = A008578(A061395(n)) * a(A052126(n)). [Compare to the formula of A252462.]

This prime-shift operation is used in the definitions of many other sequences, thus it can be expressed in many alternative ways:

a(n) = A200746(n) / n.

a(n) = A242424(n) / A105560(n).

a(n) = A122111(A122111(n)/A105560(n)) = A122111(A052126(A122111(n))). [In A112798-partition context: conjugate, remove the largest part (the largest prime factor), and conjugate again.]

a(1) = 1; for n > 1, a(2n) = a(n), a(2n+1) = A163511((A243071(2n+1)-1) / 2).

a(n) = A249818(A250470(A249817(n))). [A250470 is an analogous operation for "going one step up" in the square array A083221 (A083140).]

(End)

Product_{k=1..n} a(k) = n! / A307035(n). - Vaclav Kotesovec, Mar 21 2019

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-q(p))) = 0.220703928... , where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Nov 18 2022

EXAMPLE

a(20) = a(2^2*5) = a(2^2)*a(5) = prevprime(5) = 3.

MAPLE

q:= proc(p) prevprime(p) end proc: q(2):= 1:

[seq(mul(q(f[1])^f[2], f = ifactors(n)[2]), n = 1 .. 1000)]; # Robert Israel, Dec 21 2014

MATHEMATICA

Table[Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n, {n, 81}] (* Michael De Vlieger, Jan 04 2016 *)

PROG

(Haskell)

a064989 1 = 1

a064989 n = product $ map (a008578 . a049084) $ a027746_row n

-- Reinhard Zumkeller, Apr 09 2012

(MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library)

(require 'factor)

(define (A064989 n) (if (= 1 n) n (apply * (map (lambda (k) (if (zero? k) 1 (A000040 k))) (map -1+ (map A049084 (factor n)))))))

;; Antti Karttunen, May 12 2014

(definec (A064989 n) (if (= 1 n) n (* (A008578 (A055396 n)) (A064989 (A032742 n))))) ;; One based on given recurrence and utilizing memoizing definec-macro.

(definec (A064989 n) (cond ((= 1 n) n) ((even? n) (A064989 (/ n 2))) (else (A163511 (/ (- (A243071 n) 1) 2))))) ;; Corresponds to one of the alternative formulas, but is very unpractical way to compute this sequence. - Antti Karttunen, Dec 18 2014

(PARI) { for (n=1, 1000, f=factor(n)~; a=1; j=1; if (n>1 && f[1, 1]==2, j=2); for (i=j, length(f), a*=precprime(f[1, i] - 1)^f[2, i]); write("b064989.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 02 2009

(PARI) a(n) = {my(f = factor(n)); for (i=1, #f~, if ((p=f[i, 1]) % 2, f[i, 1] = precprime(p-1), f[i, 1] = 1); ); factorback(f); } \\ Michel Marcus, Dec 18 2014

(PARI) A064989(n)=factorback(Mat(apply(t->[max(precprime(t[1]-1), 1), t[2]], Vec(factor(n)~))~)) \\ M. F. Hasler, Dec 29 2014

(Python)

from sympy import factorint, prevprime

from operator import mul

from functools import reduce

def a(n):

f=factorint(n)

return 1 if n==1 else reduce(mul, [1 if i==2 else prevprime(i)**f[i] for i in f])

print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 15 2017

(Python)

from math import prod

from sympy import prevprime, factorint

def A064989(n): return prod(prevprime(p)**e for p, e in factorint(n>>(~n&n-1).bit_length()).items()) # Chai Wah Wu, Jan 05 2023

CROSSREFS

Cf. A064216 (odd bisection), A003961 (inverse), A151799.

Cf. A000040, A008578, A027746, A032742, A049084, A052126, A055396, A061395, A112798, A249817, A249818.

Other sequences whose definition involve or are some other way related with this sequence: A105560, A108951, A118306, A122111, A156552, A163511, A200746, A241909, A243070, A243071, A243072, A243073, A244319, A245605, A245607, A246165, A246266, A246268, A246277, A246278, A246361, A246362, A246371, A246372, A246373, A246374, A246376, A246380, A246675, A246682, A249745, A250470.

Similar prime-shifts towards smaller numbers: A252461, A252462, A252463.

KEYWORD

nonn,look,mult

AUTHOR

Vladeta Jovovic, Oct 30 2001

STATUS

approved

A252463

Hybrid shift: a(1) = 1, a(2n) = n, a(2n+1) = A064989(2n+1); shift the even numbers one bit right, shift the prime factorization of odd numbers one step towards smaller primes.

+10
82

1, 1, 2, 2, 3, 3, 5, 4, 4, 5, 7, 6, 11, 7, 6, 8, 13, 9, 17, 10, 10, 11, 19, 12, 9, 13, 8, 14, 23, 15, 29, 16, 14, 17, 15, 18, 31, 19, 22, 20, 37, 21, 41, 22, 12, 23, 43, 24, 25, 25, 26, 26, 47, 27, 21, 28, 34, 29, 53, 30, 59, 31, 20, 32, 33, 33, 61, 34, 38, 35, 67, 36, 71, 37, 18, 38, 35, 39, 73, 40, 16

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

For any node n >= 2 in binary trees A005940 and A163511, a(n) gives the parent node of n. (Here we assume that their initial root 1 is its own parent).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..8192

FORMULA

a(1) = 1, a(2n) = n, a(2n+1) = A064989(2n+1).

Other identities. For all n >= 1:

a(2n-1) = A064216(n).

A001222(a(n)) = A001222(n) - (1 - A000035(n)).

Above means: if n is odd, A001222(a(n)) = A001222(n) and if n is even, A001222(a(n)) = A001222(n) - 1.

Sum_{k=1..n} a(k) ~ c * n^2, where c = 1/8 + (1/2) * Product_{p prime > 2} ((p^2-p)/(p^2-q(p))) = 0.2905279467..., where q(p) = prevprime(p) (A151799). - Amiram Eldar, Jan 21 2023

MATHEMATICA

Table[Which[n == 1, 1, EvenQ@ n, n/2, True, Times @@ Power[

Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n], {n, 81}] (* Michael De Vlieger, Sep 16 2017 *)

PROG

(Scheme) (define (A252463 n) (cond ((<= n 1) n) ((even? n) (/ n 2)) (else (A064989 n))))

(Python)

from sympy import factorint, prevprime

from operator import mul

def a064989(n):

f = factorint(n)

return 1 if n==1 else reduce(mul, [1 if i==2 else prevprime(i)**f[i] for i in f])

def a(n): return 1 if n==1 else n//2 if n%2==0 else a064989(n)

print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Sep 15 2017

(PARI) a064989(n) = factorback(Mat(apply(t->[max(precprime(t[1]-1), 1), t[2]], Vec(factor(n)~))~)); \\ A064989

a(n) = if (n==1, 1, if (n%2, a064989(n), n/2)); \\ Michel Marcus, Oct 13 2021

CROSSREFS

A252464 gives the number of iterations needed to reach 1 from n.

Bisections: A000027 and A064216.

Cf. A000035, A001222, A004526, A005940, A064989, A151799, A163511, A246277, A252461, A252462.

KEYWORD

nonn,look

AUTHOR

Antti Karttunen, Dec 20 2014

STATUS

approved

A252464

a(1) = 0, a(2n) = 1 + a(n), a(2n+1) = 1 + a(A064989(2n+1)); also binary width of terms of A156552 and A243071.

+10
66

0, 1, 2, 2, 3, 3, 4, 3, 3, 4, 5, 4, 6, 5, 4, 4, 7, 4, 8, 5, 5, 6, 9, 5, 4, 7, 4, 6, 10, 5, 11, 5, 6, 8, 5, 5, 12, 9, 7, 6, 13, 6, 14, 7, 5, 10, 15, 6, 5, 5, 8, 8, 16, 5, 6, 7, 9, 11, 17, 6, 18, 12, 6, 6, 7, 7, 19, 9, 10, 6, 20, 6, 21, 13, 5, 10, 6, 8, 22, 7, 5, 14, 23, 7, 8, 15, 11, 8, 24, 6, 7, 11, 12, 16, 9, 7, 25, 6, 7, 6, 26, 9, 27

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

a(n) tells how many iterations of A252463 are needed before 1 is reached, i.e., the distance of n from 1 in binary trees like A005940 and A163511.

Similarly for A253553 in trees A253563 and A253565. - Antti Karttunen, Apr 14 2019

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..8192

FORMULA

a(1) = 0; for n > 1: a(n) = 1 + a(A252463(n)).

a(n) = A029837(1+A243071(n)). [a(n) = binary width of terms of A243071.]

a(n) = A029837(A005941(n)) = A029837(1+A156552(n)). [Also binary width of terms of A156552.]

Other identities. For all n >= 1:

a(A000040(n)) = n.

a(A001248(n)) = n+1.

a(A030078(n)) = n+2.

And in general, a(prime(n)^k) = n+k-1.

a(A000079(n)) = n. [I.e., a(2^n) = n.]

For all n >= 2:

a(n) = A001222(n) + A061395(n) - 1 = A001222(n) + A252735(n) = A061395(n) + A252736(n) = 1 + A252735(n) + A252736(n).

a(n) = A325134(n) - 1. - Gus Wiseman, Apr 02 2019

From Antti Karttunen, Apr 14 2019: (Start)

a(1) = 0; for n > 1: a(n) = 1 + a(A253553(n)).

a(n) = A001221(n) + A297167(n) = A297113(n) + A297155(n).

(End).

EXAMPLE

From Gus Wiseman, Apr 02 2019: (Start)

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so a(n) is the size of the inner lining of the integer partition with Heinz number n, which is also the size of the largest hook of the same partition. For example, the partition with Heinz number 715 is (6,5,3), with diagram

o o o o o o

o o o o o

o o o

which has inner lining

o o

o o o

and largest hook

o o o o o o

both of which have size 8, so a(715) = 8.

(End)

MATHEMATICA

Table[If[n==1, 1, PrimeOmega[n]+PrimePi[FactorInteger[n][[-1, 1]]]]-1, {n, 100}] (* Gus Wiseman, Apr 02 2019 *)

PROG

(Scheme, two different versions)

;; Memoization-macro definec can be found from Antti Karttunen's IntSeq-library

(definec (A252464 n) (if (<= n 1) 0 (+ 1 (A252464 (A252463 n)))))

(define (A252464 n) (A029837 (+ 1 (A243071 n))))

(define (A252464 n) (A029837 (A005941 n)))

(PARI)

A061395(n) = if(n>1, primepi(vecmax(factor(n)[, 1])), 0);

A252464(n) = (bigomega(n) + A061395(n) - 1); \\ Antti Karttunen, Apr 14 2019

(Python)

from sympy import primepi, primeomega, primefactors

def A252464(n): return primeomega(n)+primepi(max(primefactors(n)))-1 if n>1 else 0 # Chai Wah Wu, Jul 17 2023

CROSSREFS

Cf. A000040, A000079, A001221, A001222, A005940, A029837, A061395, A156552 (A005941), A163511, A243071, A252461, A252463, A252735, A252736, A252759, A253553, A253563, A253565, A297113, A297155, A297167, A324870, A324872.

Right edge of irregular triangle A265146.

Cf. also A246348.

Cf. A052126, A056239, A093641, A112798, A257990, A325133, A325134, A325135.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Dec 20 2014

STATUS

approved

A129129

An irregular triangular array of natural numbers read by rows, with shape sequence A000041(n) related to sequence A060850.

+10
34

1, 2, 3, 4, 5, 6, 8, 7, 10, 9, 12, 16, 11, 14, 15, 20, 18, 24, 32, 13, 22, 21, 28, 25, 30, 40, 27, 36, 48, 64, 17, 26, 33, 44, 35, 42, 56, 50, 45, 60, 80, 54, 72, 96, 128, 19, 34, 39, 52, 55, 66, 88, 49, 70, 63, 84, 112, 75, 100, 90, 120, 160, 81, 108, 144, 192, 256

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

The tree begins (at height n, n >= 0, nodes represent partitions of n)

0: 1

1: 2

2: 3 4

3: 5 6 8

4: 7 10 9 12 16

5: 11 14 15 20 18 24 32

...

and hence differs from A114622.

Ordering [graded reverse lexicographic order] of partitions (positive integer representation) of nonnegative integers, where part of size i [as summand] is mapped to i-th prime [as multiplicand], where the empty partition for 0 yields the empty product, i.e., 1. Permutation of positive integers, since bijection [1-1 and onto map] between the set of all partitions of nonnegative integers and positive integers. - Daniel Forgues, Aug 07 2018

These are all Heinz numbers of integer partitions in graded reverse-lexicographic order, where The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This is the so-called "Mathematica" order (sum/revlex) of partitions (A080577). Partitions in lexicographic order (sum/lex) are A193073, with Heinz numbers A334434. - Gus Wiseman, May 19 2020

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..11731 (rows 0 <= n <= 26).

OEIS Wiki, Partitions#Orderings of partitions.

Wikiversity, Lexicographic and colexicographic order.

FORMULA

From Gus Wiseman, May 19 2020: (Start)

A001222(a(n)) = A238966(n).

A001221(a(n)) = A115623(n).

A056239(a(n)) = A036042(n).

A061395(a(n)) = A331581(n).

(End)

EXAMPLE

The array is a tree structure as described by A128628. If a node value has only one branch the value is twice that of its parent node. If it has two branches one is twice that of its parent node but the other is defined as indicated below:

(1) pick an odd number (e.g., 135)

(2) calculate its prime factorization (135 = 5*3*3*3)

(3) note the least prime factor (LPF(135) = 3)

(4) note the index of the LPF (index(3) = 2)

(5) subtract one from the index (2-1 = 1)

(6) calculate the prime associated with the value in step five (prime(1) = 2)

(7) The parent node of the odd number 135 is (2/3)*135 = 90 = A252461(135).

From Daniel Forgues, Aug 07 2018: (Start)

Partitions of 4 in graded reverse lexicographic order:

{4}: p_4 = 7;

{3,1}: p_3 * p_1 = 5 * 2 = 10;

{2,2}: p_2 * p_2 = 3^2 = 9;

{2,1,1}: p_2 * p_1 * p_1 = 3 * 2^2 = 12;

{1,1,1,1}: p_1 * p_1 * p_1 * p_1 = 2^4 = 16. (End)

From Gus Wiseman, May 19 2020: (Start)

The sequence together with the corresponding partitions begins:

1: () 24: (2,1,1,1) 35: (4,3)

2: (1) 32: (1,1,1,1,1) 42: (4,2,1)

3: (2) 13: (6) 56: (4,1,1,1)

4: (1,1) 22: (5,1) 50: (3,3,1)

5: (3) 21: (4,2) 45: (3,2,2)

6: (2,1) 28: (4,1,1) 60: (3,2,1,1)

8: (1,1,1) 25: (3,3) 80: (3,1,1,1,1)

7: (4) 30: (3,2,1) 54: (2,2,2,1)

10: (3,1) 40: (3,1,1,1) 72: (2,2,1,1,1)

9: (2,2) 27: (2,2,2) 96: (2,1,1,1,1,1)

12: (2,1,1) 36: (2,2,1,1) 128: (1,1,1,1,1,1,1)

16: (1,1,1,1) 48: (2,1,1,1,1) 19: (8)

11: (5) 64: (1,1,1,1,1,1) 34: (7,1)

14: (4,1) 17: (7) 39: (6,2)

15: (3,2) 26: (6,1) 52: (6,1,1)

20: (3,1,1) 33: (5,2) 55: (5,3)

18: (2,2,1) 44: (5,1,1) 66: (5,2,1)

(End)

MAPLE

b:= (n, i)-> `if`(n=0 or i=1, [2^n], [map(x-> x*ithprime(i),

b(n-i, min(n-i, i)))[], b(n, i-1)[]]):

T:= n-> b(n$2)[]:

seq(T(n), n=0..10); # Alois P. Heinz, Feb 14 2020

MATHEMATICA

Array[Times @@ # & /@ Prime@ IntegerPartitions@ # &, 9, 0] // Flatten (* Michael De Vlieger, Aug 07 2018 *)

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {2^n}, Join[(# Prime[i]&) /@ b[n - i, Min[n - i, i]], b[n, i - 1]]];

T[n_] := b[n, n];

T /@ Range[0, 10] // Flatten (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

CROSSREFS

Cf. A080577 (the partitions), A252461, A114622, A128628, A215366 (sorted rows).

Row lengths are A000041.

Compositions under the same order are A066099.

The opposite version (sum/lex) is A334434.

The length-sensitive version (sum/length/revlex) is A334438.

The version for reversed (weakly increasing) partitions is A334436.

Lexicographically ordered reversed partitions are A026791.

Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.

Sum of prime indices is A056239.

Sorting reversed partitions by Heinz number gives A112798.

Partitions in lexicographic order are A193073.

Sorting partitions by Heinz number gives A296150.

Cf. A036037, A185974, A211992, A334301, A334433, A334435, A334437, A334439.

KEYWORD

nonn,tabf

AUTHOR

Alford Arnold, Mar 31 2007

STATUS

approved

A252462

Shift one instance of the largest prime one step towards smaller primes: a(1) = 1, a(2^n) = 2^(n-1), and for other numbers: a(n) = (n / prime(g)) * prime(g-1), where g = A061395(n), index of the greatest prime dividing n.

+10
7

1, 1, 2, 2, 3, 4, 5, 4, 6, 6, 7, 8, 11, 10, 9, 8, 13, 12, 17, 12, 15, 14, 19, 16, 15, 22, 18, 20, 23, 18, 29, 16, 21, 26, 25, 24, 31, 34, 33, 24, 37, 30, 41, 28, 27, 38, 43, 32, 35, 30, 39, 44, 47, 36, 35, 40, 51, 46, 53, 36, 59, 58, 45, 32, 55, 42, 61, 52, 57, 50, 67, 48, 71, 62, 45, 68, 49, 66, 73, 48, 54, 74, 79, 60

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Iterating from any n as a(n), a(a(n)), a(a(a(n))), etc. reaches 1 after A056239(n) iterations.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

FORMULA

a(1) = 1; for n>1: a(n) = A008578(A061395(n)) * A052126(n). [Compare to the similar formula given for A064989.]

a(n) = A008578(1 + A252735(n)) * A052126(n).

Other identities. For all n >= 1:

a(2^n) = 2^(n-1).

For n >= 2, A001222(a(n)) = A001222(n) - A209229(n). [Number of prime divisors decreases only when n is a power of 2 larger than 1.]

MATHEMATICA

a252462[n_Integer] := Block[{a008578, a052126, a061395, a},

a008578[x_] := If[x == 1, 1, Prime[x - 1]];

a052126[x_] := If[x == 1, 1, x/FactorInteger[x][[-1]][[1]]];

a061395[x_] := PrimePi[FactorInteger[x][[-1]][[1]]];

a[1] = 1;

a[x_] := a008578[a061395[x]]*a052126[x];

Array[a, n]]; a252462[84] (* Michael De Vlieger, Dec 21 2014 *)

PROG

(Scheme) (define (A252462 n) (if (= 1 n) n (* (A008578 (A061395 n)) (A052126 n))))

CROSSREFS

Variant: A252461.

Cf. A001222, A008578, A052126, A061395, A056239, A064989, A209229, A252735.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Dec 20 2014

STATUS

approved

page 1

Search completed in 0.011 seconds