A252462 -id:A252462 - cp's OEIS frontend

A253553 a(1) = 1; for n>1, if A241917(n) = 0 [i.e., n is a term of A070003], a(n) = A052126(n), otherwise a(n) = A252462(n).

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

Offset: 1

Antti Karttunen, Jan 12 2015

nonn

If the exponent of the largest prime dividing n is larger than one, subtract one from that exponent. Otherwise, shift that "lonely largest prime" one step towards smaller primes.

For any number n >= 2 in binary trees A253563 and A253565, a(n) gives the number which is the parent of n.

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

Cf. A252464 (the number of iterations of n -> a(n) needed to reach 1 from n.)

Cf. A052126, A122111, A241917, A243287, A252462, A252463, A253550, A253563, A253565, A336120, A336125.

A253553(n) = if(n<=2,1,my(f=factor(n), k=#f~); if(f[k,2]>1,f[k,2]--,f[k,1] = precprime(f[k,1]-1)); factorback(f)); \\ Antti Karttunen, Jul 17 2020

(define (A253553 n) (cond ((<= n 1) n) ((zero? (A241917 n)) (A052126 n)) (else (A252462 n))))

a(1) = 1; for n>1, if A241917(n) = 0 [i.e., n is a term of A070003], a(n) = A052126(n), otherwise a(n) = A252462(n).

a(n) = A122111(A252463(A122111(n))). - Antti Karttunen, Jul 14 2020

A064989 Multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

Original entry on oeis.org

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

Offset: 1

Vladeta Jovovic, Oct 30 2001

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

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

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

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.

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

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

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

{ 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

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

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

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

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

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

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.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 20 2014

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).

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

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.

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 *)

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

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

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

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

A253550 Shift one instance of the largest prime one step towards larger primes: a(1) = 1, for n>1: a(n) = (n / prime(g)) * prime(g+1), where g = A061395(n), index of the greatest prime dividing n.

Original entry on oeis.org

1, 3, 5, 6, 7, 10, 11, 12, 15, 14, 13, 20, 17, 22, 21, 24, 19, 30, 23, 28, 33, 26, 29, 40, 35, 34, 45, 44, 31, 42, 37, 48, 39, 38, 55, 60, 41, 46, 51, 56, 43, 66, 47, 52, 63, 58, 53, 80, 77, 70, 57, 68, 59, 90, 65, 88, 69, 62, 61, 84, 67, 74, 99, 96, 85, 78, 71, 76, 87, 110, 73, 120, 79, 82, 105, 92, 91, 102, 83, 112, 135, 86, 89

Offset: 1

Antti Karttunen, Jan 03 2015

nonn

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

Inverse: A252462.
Cf. A102750 (same terms, but with 2 instead of 1, sorted into ascending order).
Cf. A003961, A052126, A065091, A061395, A122111, A253553, A253554, A253560, A253563, A253565.

(define (A253550 n) (if (= 1 n) n (* (A065091 (A061395 n)) (A052126 n))))

a(1) = 1; for n>1: a(n) = A065091(A061395(n)) * A052126(n).
Other identities. For all n >= 1:
A252462(a(n)) = n. [A252462 works as an inverse function for this injection.]
a(n) <= A253560(n).

A252735 a(1) = 0; for n > 1: a(2n) = a(n), a(2n+1) = 1 + a(A064989(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 21 2014

nonn

Consider the binary tree illustrated in A005940: If we start from any n, computing successive iterations of A252463 until 1 is reached (i.e., we are traversing level by level towards the root of the tree, starting from that vertex of the tree where n is located at), a(n) gives the number of odd numbers > 1 encountered on the path (i.e., excluding the final 1 from the count but including the starting n if it was odd).

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Essentially one less than A061395.
Cf. A252464, A252736.
Cf. A000120, A005940, A064989, A080791, A156552, A243071, A252462, A252464, A252736.
Cf. also A246369.

a252735[n_] := Prepend[Rest@Array[PrimePi[FactorInteger[#][[-1]][[1]]] - 1 &, n], 0]; a252735[108] (* Michael De Vlieger, Dec 21 2014, after Stefan Steinerberger at A061395 *)

a(1) = 0; for n > 1: a(2n) = a(n), a(2n+1) =  1 + a(A064989(n)).
a(n) = A080791(A156552(n)). [Number of nonleading 0-bits in A156552(n).]
Other identities:
For all n >= 2:
a(n) = A061395(n) - 1.
a(n) = A000120(A243071(n)) - 1. [One less than the binary weight of A243071(n).]
a(n) = A252464(n) - A252736(n) - 1.

A252461 Shift one instance of the smallest prime one step towards smaller primes: a(1) = 1, a(2n) = n, and for odd numbers > 1: a(n) = (n / prime(s)) * prime(s-1), where s = A055396(n), index of the smallest prime dividing n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 20 2014

nonn

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

Even bisection gives the natural numbers A000027, the odd bisection from the third term onward is A129128: 2, 3, 5, 6, 7, 11, 10, ...

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

Variants: A252462, A252463.
Bisections: A000027, A129128.
Cf. A001222, A008578, A032742, A055396, A056239, A064989.

a252461[n_Integer] := Block[{a008578, a032742, a055396, a},
  a008578[x_] := If[x == 1, 1, Prime[x - 1]];
  a032742[x_] := If[x == 1, 1, Divisors[x][[-2]]];
  a055396[x_] := PrimePi[FactorInteger[x][[1]][[1]]];
  a[1] = 1;
  a[x_] := a008578[a055396[x]]*a032742[x];
Array[a, n]]; a252461[84] (* Michael De Vlieger, Dec 21 2014 *)

(define (A252461 n) (if (= 1 n) n (* (A008578 (A055396 n)) (A032742 n))))

a(1) = 1; for n>1: a(n) = A008578(A055396(n)) * A032742(n). [Compare to the similar formula of A064989.]
Other identities. For all n >= 1:
a(2n) = n.
If n is odd, A001222(a(n)) = A001222(n).
If n is even, A001222(a(n)) = A001222(n) - 1.

cp's OEIS Frontend

A253553 a(1) = 1; for n>1, if A241917(n) = 0 [i.e., n is a term of A070003], a(n) = A052126(n), otherwise a(n) = A252462(n).

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A064989 Multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

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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.

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A253550 Shift one instance of the largest prime one step towards larger primes: a(1) = 1, for n>1: a(n) = (n / prime(g)) * prime(g+1), where g = A061395(n), index of the greatest prime dividing n.

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A252735 a(1) = 0; for n > 1: a(2n) = a(n), a(2n+1) = 1 + a(A064989(n)).

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A252461 Shift one instance of the smallest prime one step towards smaller primes: a(1) = 1, a(2n) = n, and for odd numbers > 1: a(n) = (n / prime(s)) * prime(s-1), where s = A055396(n), index of the smallest prime dividing n.

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