A250470 -id:A250470 - cp's OEIS frontend

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

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

A078898 Number of times the smallest prime factor of n is the smallest prime factor for numbers <= n; a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 7, 3, 8, 1, 9, 1, 10, 4, 11, 1, 12, 2, 13, 5, 14, 1, 15, 1, 16, 6, 17, 3, 18, 1, 19, 7, 20, 1, 21, 1, 22, 8, 23, 1, 24, 2, 25, 9, 26, 1, 27, 4, 28, 10, 29, 1, 30, 1, 31, 11, 32, 5, 33, 1, 34, 12, 35, 1, 36, 1, 37, 13, 38, 3, 39, 1, 40, 14, 41, 1, 42, 6, 43

Offset: 0

Reinhard Zumkeller, Dec 12 2002

nonn

From Antti Karttunen, Dec 06 2014: (Start)
For n >= 2, a(n) tells in which column of the sieve of Eratosthenes (see A083140, A083221) n occurs in. A055396 gives the corresponding row index.
(End)

Harvey P. Dale (terms 1 - 1000) & Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A002110, A008683, A008836, A020639, A032742, A054272, A055396, A078899, A078896, A083140, A083221, A243055, A246277, A249738, A249744, A249808, A249809, A249810, A249820, A249818, A250470, A250474, A250477, A250478, A251719, A251724, A251728.

Cf. A001248, A006094, A038110, A038111, A090076.

import Data.IntMap (empty, findWithDefault, insert)
a078898 n = a078898_list !! n
a078898_list = 0 : 1 : f empty 2 where
   f m x = y : f (insert p y m) (x + 1) where
           y = findWithDefault 0 p m + 1
           p = a020639 x
-- Reinhard Zumkeller, Apr 06 2015

N:= 1000: # to get a(0) to a(N)
Primes:= select(isprime, [2,seq(2*i+1,i=1..floor((N-1)/2))]):
A:= Vector(N):
for p in Primes do
  t:= 1:
  A[p]:= 1:
  for n from p^2 to N by p do
    if A[n] = 0 then
       t:= t+1:
       A[n]:= t
    fi
  od
od:
0,1,seq(A[i],i=2..N); # Robert Israel, Jan 04 2015

Module[{nn=90,spfs},spfs=Table[FactorInteger[n][[1,1]],{n,nn}];Table[ Count[ Take[spfs,i],spfs[[i]]],{i,nn}]] (* Harvey P. Dale, Sep 01 2014 *)

\\ Not practical for computing, but demonstrates the sum moebius formula:
A020639(n) = { if(1==n,n,vecmin(factor(n)[, 1])); };
A055396(n) = { if(1==n,0,primepi(A020639(n))); };
A002110(n) = prod(i=1, n, prime(i));
A078898(n) = { my(k,p); if(1==n, n, k = A002110(A055396(n)-1); p = A020639(n); sumdiv(k, d, moebius(d)*(n\(p*d)))); };
\\ Antti Karttunen, Dec 05 2014

;; With memoizing definec-macro.
(definec (A078898 n) (if (< n 2) n (+ 1 (A078898 (A249744 n)))))
;; Much better for computing. Needs also code from A249738 and A249744. - Antti Karttunen, Dec 06 2014

Ordinal transform of A020639 (Lpf). - Franklin T. Adams-Watters, Aug 28 2006
From Antti Karttunen, Dec 05-08 2014: (Start)
a(0) = 0, a(1) = 1, a(n) = 1 + a(A249744(n)).
a(0) = 0, a(1) = 1, a(n) = sum_{d | A002110(A055396(n)-1)} moebius(d) * floor(n / (A020639(n)*d)).
a(0) = 0, a(1) = 1, a(n) = sum_{d | A002110(A055396(n)-1)} moebius(d) * floor(A032742(n) / d).
[Instead of Moebius mu (A008683) one could use Liouville's lambda (A008836) in the above formulas, because all primorials (A002110) are squarefree. A020639(n) gives the smallest prime dividing n, and A055396 gives its index].
a(0) = 0, a(1) = 1, a(2n) = n, a(2n+1) = a(A250470(2n+1)). [After a similar recursive formula for A246277. However, this cannot be used for computing the sequence, unless a definition for A250470(n) is found which doesn't require computing the value of A078898(n).]
For n > 1: a(n) = A249810(n) - A249820(n).
(End)
Other identities:
a(2*n) = n.
For n > 1: a(n)=1 if and only if n is prime.
For n > 1: a(n) = A249808(n, A055396(n)) = A249809(n, A055396(n)).
For n > 1: a(n) = A246277(A249818(n)).
From Antti Karttunen, Jan 04 2015: (Start)
a(n) = 2 if and only if n is a square of a prime.
For all n >= 1: a(A251728(n)) = A243055(A251728(n)) + 2. That is, if n is a semiprime of the form prime(i)*prime(j), prime(i) <= prime(j) < prime(i)^2, then a(n) = (j-i)+2.
(End)
a(A000040(n)^2) = 2; a(A000040(n)*A000040(n+1)) = 3. - Reinhard Zumkeller, Apr 06 2015
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Sum_{k>=1} (A038110(k)/A038111(k))^2 = 0.2847976823663... . - Amiram Eldar, Oct 26 2024

a(0) = 0 prepended for recurrence's sake by Antti Karttunen, Dec 06 2014

A083221 Sieve of Eratosthenes arranged as an array and read by antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

Original entry on oeis.org

2, 4, 3, 6, 9, 5, 8, 15, 25, 7, 10, 21, 35, 49, 11, 12, 27, 55, 77, 121, 13, 14, 33, 65, 91, 143, 169, 17, 16, 39, 85, 119, 187, 221, 289, 19, 18, 45, 95, 133, 209, 247, 323, 361, 23, 20, 51, 115, 161, 253, 299, 391, 437, 529, 29, 22, 57, 125, 203, 319, 377, 493, 551, 667

Offset: 2

Yasutoshi Kohmoto, Jun 05 2003

This is permutation of natural numbers larger than 1.
From Antti Karttunen, Dec 19 2014: (Start)
If we assume here that a(1) = 1 (but which is not explicitly included because outside of the array), then A252460 gives an inverse permutation. See also A249741.
For navigating in this array:
A055396(n) gives the row number of row where n occurs, and A078898(n) gives its column number, both starting their indexing from 1.
A250469(n) gives the number immediately below n, and when n is an odd number >= 3, A250470(n) gives the number immediately above n. If n is a composite, A249744(n) gives the number immediately left of n.
First cube of each row, which is {the initial prime of the row}^3 and also the first number neither a prime or semiprime, occurs on row n at position A250474(n).
(End)
The n-th row contains the numbers whose least prime factor is the n-th prime: A020639(T(n,k)) = A000040(n). - Franklin T. Adams-Watters, Aug 07 2015

			The top left corner of the array:
   2,   4,   6,    8,   10,   12,   14,   16,   18,   20,   22,   24,   26
   3,   9,  15,   21,   27,   33,   39,   45,   51,   57,   63,   69,   75
   5,  25,  35,   55,   65,   85,   95,  115,  125,  145,  155,  175,  185
   7,  49,  77,   91,  119,  133,  161,  203,  217,  259,  287,  301,  329
  11, 121, 143,  187,  209,  253,  319,  341,  407,  451,  473,  517,  583
  13, 169, 221,  247,  299,  377,  403,  481,  533,  559,  611,  689,  767
  17, 289, 323,  391,  493,  527,  629,  697,  731,  799,  901, 1003, 1037
  19, 361, 437,  551,  589,  703,  779,  817,  893, 1007, 1121, 1159, 1273
  23, 529, 667,  713,  851,  943,  989, 1081, 1219, 1357, 1403, 1541, 1633
  29, 841, 899, 1073, 1189, 1247, 1363, 1537, 1711, 1769, 1943, 2059, 2117
  ...

Transpose of A083140.
One more than A249741.
Inverse permutation: A252460.
Column 1: A000040, Column 2: A001248.
Row 1: A005843, Row 2: A016945, Row 3: A084967, Row 4: A084968, Row 5: A084969, Row 6: A084970.
Main diagonal: A083141.
First semiprime in each column occurs at A251717; A251718 & A251719 with additional criteria. A251724 gives the corresponding semiprimes for the latter. See also A251728.
Permutations based on mapping numbers between this array and A246278: A249817, A249818, A250244, A250245, A250247, A250249. See also: A249811, A249814, A249815.
Also used in the definition of the following arrays of permutations: A249821, A251721, A251722.
Cf. A002260, A004736, A004280, A020639, A038179, A055396, A078898, A138511, A249820, A249730, A249735, A249744, A250469, A250470, A250472, A250474.

lim = 11; a = Table[Take[Prime[n] Select[Range[lim^2], GCD[# Prime@ n, Product[Prime@ i, {i, 1, n - 1}]] == 1 &], lim], {n, lim}]; Flatten[Table[a[[i, n - i + 1]], {n, lim}, {i, n}]] (* Michael De Vlieger, Jan 04 2016, after Yasutoshi Kohmoto at A083140 *)

More terms from Hugo Pfoertner, Jun 13 2003

A250469 a(1) = 1; and for n > 1, a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)+1, where A055396(n) is the index of smallest prime dividing n.

Original entry on oeis.org

1, 3, 5, 9, 7, 15, 11, 21, 25, 27, 13, 33, 17, 39, 35, 45, 19, 51, 23, 57, 55, 63, 29, 69, 49, 75, 65, 81, 31, 87, 37, 93, 85, 99, 77, 105, 41, 111, 95, 117, 43, 123, 47, 129, 115, 135, 53, 141, 121, 147, 125, 153, 59, 159, 91, 165, 145, 171, 61, 177, 67, 183, 155, 189, 119, 195, 71, 201, 175, 207, 73, 213, 79, 219, 185, 225, 143, 231, 83, 237, 205, 243, 89, 249, 133, 255

Offset: 1

Antti Karttunen, Dec 06 2014

nonn

Permutation of odd numbers.
For n >= 2, a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)+1. In other words, a(n) tells which number is located immediately below n in the sieve of Eratosthenes (see A083140, A083221) in the same column of the sieve that contains n.
A250471(n) = (a(n)+1)/2 is a permutation of natural numbers.
Coincides with A003961 in all terms which are primes. - M. F. Hasler, Sep 17 2016. Note: primes are a proper subset of A280693 which gives all n such that a(n) = A003961(n). - Antti Karttunen, Mar 08 2017

Antti Karttunen, Table of n, a(n) for n = 1..5002

Cf. A000040, A003961, A016945, A046523, A055396, A078898, A083140, A083221, A084967, A249744, A249810, A249820, A249817, A249818, A250471, A266645, A280692, A280693, A283465.

Cf. A250470, A268674 (left inverses, the latter is simpler).

a[1] = 1; a[n_] := If[PrimeQ[n], NextPrime[n], m1 = p1 = FactorInteger[n][[ 1, 1]]; For[k1 = 1, m1 <= n, m1 += p1; If[m1 == n, Break[]]; If[ FactorInteger[m1][[1, 1]] == p1, k1++]]; m2 = p2 = NextPrime[p1]; For[k2 = 1, True, m2 += p2, If[FactorInteger[m2][[1, 1]] == p2, k2++]; If[k1+2 == k2, Return[m2]]]]; Array[a, 100] (* Jean-François Alcover, Mar 08 2016 *)
g[n_] := If[n == 1, 0, PrimePi@ FactorInteger[n][[1, 1]]]; Function[s, MapIndexed[Lookup[s, g[First@ #2] + 1][[#1]] - Boole[First@ #2 == 1] &, #] &@ Map[Position[Lookup[s, g@#], #][[1, 1]] &, Range@ 120]]@ PositionIndex@ Array[g, 10^4] (* Michael De Vlieger, Mar 08 2017, Version 10 *)

a(1) = 1, a(n) = A083221(A055396(n)+1, A078898(n)).
a(n) = A249817(A003961(A249818(n))).
Other identities. For all n >= 1:
A250470(a(n)) = A268674(a(n)) = n. [A250470 and A268674 provide left inverses for this function.]
a(2n) = A016945(n-1). [Maps even numbers to the numbers of form 6n+3, in monotone order.]
a(A016945(n-1)) = A084967(n). [Which themselves are mapped to the terms of A084967, etc. Cf. the Example section of A083140.]
a(A000040(n)) = A000040(n+1). [Each prime is mapped to the next prime.]
For all n >= 2, A055396(a(n)) = A055396(n)+1. [A more general rule.]
A046523(a(n)) = A283465(n). - Antti Karttunen, Mar 08 2017

A250246 Permutation of natural numbers: a(1) = 1, a(n) = A246278(A055396(n), a(A078898(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 45, 34, 35, 36, 37, 38, 33, 40, 41, 54, 43, 44, 81, 46, 47, 48, 49, 50, 75, 52, 53, 42, 125, 56, 63, 58, 59, 60, 61, 62, 39, 64, 55, 90, 67, 68, 135, 70, 71, 72, 73, 74, 51, 76, 77, 66, 79, 80, 99, 82, 83

Offset: 1

Antti Karttunen, Nov 17 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences that are permutations of the natural numbers

Inverse: A250245.
Other similar permutations: A250243, A250248, A250250, A163511, A252756.
Cf. A003961, A005843, A020639, A055396, A078898, A246278, A250470, A268674, A278524, A302042, A302046.
Differs from the "vanilla version" A249818 for the first time at n=42, where a(42) = 54, while A249818(42) = 42.
Differs from A250250 for the first time at n=73, where a(73) = 73, while A250250(73) = 103.

up_to = 16384;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
A055396(n) = if(1==n,0,primepi(A020639(n)));
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A250246(n) = if(1==n,n,my(k = 2*A250246(A078898(n)), r = A055396(n)); if(1==r, k, while(r>1, k = A003961(k); r--); (k))); \\ Antti Karttunen, Apr 01 2018
(Scheme, with memoizing-macro definec from Antti Karttunen's IntSeq-library, three alternative definitions)
(definec (A250246 n) (cond ((<= n 1) n) (else (A246278bi (A055396 n) (A250246 (A078898 n)))))) ;; Code for A246278bi given in A246278
(definec (A250246 n) (cond ((<= n 1) n) ((even? n) (* 2 (A250246 (/ n 2)))) (else (A003961 (A250246 (A250470 n))))))
(define (A250246 n) (A163511 (A252756 n)))

a(1) = 1, a(n) = A246278(A055396(n), a(A078898(n))).
a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A003961(a(A250470(2n+1))). - Antti Karttunen, Jan 18 2015  - Instead of A250470, one may use A268674 in above formula. - Antti Karttunen, Apr 01 2018
As a composition of related permutations:
a(n) = A163511(A252756(n)).
Other identities. For all n >= 1:
a(n) = a(2n)/2. [The even bisection halved gives the sequence back.]
A020639(a(n)) = A020639(n) and A055396(a(n)) = A055396(n). [Preserves the smallest prime factor of n].
A001221(a(n)) = A302041(n).
A001222(a(n)) = A253557(n).
A008683(a(n)) = A302050(n).
A000005(a(n)) = A302051(n)
A010052(a(n)) = A302052(n), for n >= 1.
A056239(a(n)) = A302039(n).

A268674 a(1) = 1, after which, for odd numbers: a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)-1, and for even numbers: a(n) = a(A000265(n)).

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

Offset: 1

Antti Karttunen, Feb 11 2016

nonn

For odd numbers n > 1, a(n) tells which term is on the immediately preceding row of A083221, in the same column where n itself is.

The sequence offers a left inverse for A250469 that is slightly easier to compute than A250470.

Antti Karttunen, Table of n, a(n) for n = 1..32769
Index entries for sequences generated by sieves

Left inverse of A250469.
Cf. A000040, A000265, A001248, A055396, A078898, A083221.
Cf. also A064989.
Differs from A250470 for the first time at n=42, where a(42)=8, while A250470(42) = 10.

(* b = A250469 *) b[1] = 1; b[n_] := If[PrimeQ[n], NextPrime[n], m1 = p1 = FactorInteger[n][[1, 1]]; For[ k1 = 1, m1 <= n, m1 += p1; If[m1 == n, Break[]]; If[ FactorInteger[m1][[1, 1]] == p1, k1++]]; m2 = p2 = NextPrime[p1]; For[k2 = 1, True, m2 += p2, If[ FactorInteger[m2][[1, 1]] == p2, k2++]; If[k1 + 2 == k2, Return[m2]]]];
a[1] = a[2] = 1; a[n_?EvenQ] := a[n] = a[n/2]; a[n_] := a[n] = For[k = 1, True, k++, If[b[k] == n, Return[k]]];
Array[a, 100] (* Jean-François Alcover, Mar 14 2016 *)

a(1) = 1, after which, a(n) = a(A000265(n)) if n is even, otherwise for odd n, a(n) = A083221(A055396(n)-1, A078898(n)).
Other identities. For all n >= 1:
a(A250469(n)) = n. [This works as a left inverse for sequence A250469.]
a(2n) = a(n). [The even bisection gives the whole sequence back.]
a(2n-1) = A250470(2n-1). [Matches with A250470 on odd numbers.]
a(A000040(n+1)) = A000040(n). [Maps each odd prime to the preceding prime.]
a(A001248(n+1)) = A001248(n). [Maps each square of an odd prime to the square of the preceding prime.]

A253557 a(1) = 0; after which, a(2n) = 1 + a(n), a(2n+1) = a(A268674(2n+1)).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 4, 2, 2, 2, 3, 1, 3, 1, 5, 3, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 3, 4, 2, 1, 5, 2, 3, 3, 3, 1, 3, 3, 4, 3, 2, 1, 4, 1, 2, 2, 6, 2, 4, 1, 3, 4, 3, 1, 5, 1, 2, 2, 3, 2, 3, 1, 5, 3, 2, 1, 5, 3, 2, 3, 4, 1, 5, 3, 3, 5, 2, 2, 6, 1, 3, 2, 4, 1, 4, 1, 4, 4, 2, 1, 4, 1, 4, 2, 5, 1

Offset: 1

Antti Karttunen, Jan 12 2015

nonn

Consider the binary trees illustrated in A252753 and A252755: If we start from any n, computing successive iterations of A253554 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), a(n) gives the number of even numbers encountered on the path (i.e., including both 2 and the starting n if it was even).

This is bigomega (A001222) analog for nonstandard factorization based on the sieve of Eratosthenes (A083221). See A302041 for an omega-analog. - Antti Karttunen, Mar 31 2018

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

Essentially, one more than A253559.
Primes, A000040, gives the positions of ones.
Cf. A000079, A000120, A080791, A252753, A252754, A252755, A252756, A253554, A253555, A253556, A253558, A302037, A302041, A302042.
Differs from A001222 for the first time at n=21, where a(21) = 3, while A001222(21) = 2.

(definec (A253557 n) (cond ((= 1 n) 0) ((odd? n) (A253557 (A250470 n))) (else (+ 1 (A253557 (/ n 2))))))

a(1) = 0; after which, a(2n) = 1 + a(n), a(2n+1) = a(A268674(2n+1)).
a(n) = A253555(n) - A253556(n).
a(n) = A000120(A252754(n)). [Binary weight of A252754(n).]
Other identities.
For all n >= 0:
a(2^n) = n.
For all n >= 2:
a(n) = A080791(A252756(n)) + 1. [One more than the number of nonleading 0-bits in A252756(n).]
From Antti Karttunen, Apr 01 2018: (Start)
a(1) = 0; for n > 1, a(n) = 1 + a(A302042(n)).
a(n) = A001222(A250246(n)).
(End)

Definition (formula) corrected by Antti Karttunen, Mar 31 2018

A252754 Inverse of "Tree of Eratosthenes" permutation: a(1) = 0, after which, a(2n) = 1 + 2a(n), a(2n+1) = 2 a(A268674(2n+1)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 11, 32, 17, 10, 15, 64, 13, 128, 19, 14, 33, 256, 23, 12, 65, 18, 35, 512, 21, 1024, 31, 22, 129, 20, 27, 2048, 257, 34, 39, 4096, 29, 8192, 67, 30, 513, 16384, 47, 24, 25, 26, 131, 32768, 37, 28, 71, 38, 1025, 65536, 43, 131072, 2049, 66, 63, 36, 45, 262144, 259, 46, 41, 524288, 55, 1048576, 4097, 130, 515, 40

Offset: 1

Antti Karttunen, Jan 02 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A252753.
Fixed points of a(n)+1: A253789.
Cf. A250470, A253556, A253557, A253558, A253559, A268674, A302041.
Similar permutations: A156552, A252756, A054429, A250246, A269388.
Differs from A156552 for the first time at n=21, where a(21) = 14, while A156552(21) = 18.

A252754(n) = if(1==n,0,if(!(n%2),1 + 2*A252754(n/2),2*A252754(A268674(n)))); \\ Antti Karttunen, Mar 31 2018

;; With memoization-macro definec.
(definec (A252754 n) (cond ((= 1 n) (- n 1)) ((even? n) (+ 1 (* 2 (A252754 (/ n 2))))) (else (* 2 (A252754 (A250470 n))))))

a(1) = 0, after which, a(2n) = 1 + 2*a(n), a(2n+1) = 2 * a(A268674(2n+1)).
As a composition of related permutations:
a(n) = A054429(A252756(n)).
a(n) = A156552(A250246(n)).
From Antti Karttunen, Mar 31 2018: (Start)
A000120(a(n)) = A253557(n).
A069010(a(n)) = A302041(n).
A132971(a(n)) = A302050(n).
A106737(a(n)) = A302051(n).
(End)

Name edited and formula corrected by Antti Karttunen, Mar 31 2018

A249744 a(n) = 0 if n is 1 or a prime, otherwise, when n = A020639(n) * A032742(n), a(n) = the largest m < n such that A020639(m) = A020639(n), where A020639(n) and A032742(n) are the smallest prime and the largest proper divisor dividing n.

Original entry on oeis.org

0, 0, 0, 2, 0, 4, 0, 6, 3, 8, 0, 10, 0, 12, 9, 14, 0, 16, 0, 18, 15, 20, 0, 22, 5, 24, 21, 26, 0, 28, 0, 30, 27, 32, 25, 34, 0, 36, 33, 38, 0, 40, 0, 42, 39, 44, 0, 46, 7, 48, 45, 50, 0, 52, 35, 54, 51, 56, 0, 58, 0, 60, 57, 62, 55, 64, 0, 66, 63, 68, 0, 70, 0, 72, 69, 74, 49, 76, 0, 78, 75, 80, 0, 82, 65, 84, 81, 86, 0, 88, 77, 90, 87, 92, 85, 94, 0, 96, 93, 98, 0, 100

Offset: 1

Antti Karttunen, Dec 06 2014

nonn

For all composite numbers, a(n) tells what is the previous number processed by the sieve of Eratosthenes, i.e., number which is immediately left of n on the same row where n is in arrays like A083140, A083221.

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

Can be used to compute A078898.

Cf. A000040, A001248, A020639, A083140, A083221, A249738, A250469, A250470.

a[1] = 0; a[?PrimeQ] = 0; a[n] := For[p = FactorInteger[n][[1, 1]]; m = n - p, True, m = m - p, If[FactorInteger[m][[1, 1]] == p, Return[m]]]; Array[a, 102] (* Jean-François Alcover, Mar 08 2016 *)

(define (A249744 n) (* (A020639 n) (A249738 n)))

a(n) = A020639(n) * A249738(n).
Other identities. For all n >= 1 it holds:
a(2n) = 2n-2.
a(A001248(n)) = A000040(n). [I.e., a(p^2) = p for primes p.]

A249810 a(1) = 0, a(n) = A078898(A003961(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 5, 2, 4, 1, 8, 1, 6, 3, 14, 1, 13, 1, 11, 4, 7, 1, 23, 2, 9, 9, 17, 1, 18, 1, 41, 5, 10, 3, 38, 1, 12, 6, 32, 1, 28, 1, 20, 12, 15, 1, 68, 2, 25, 7, 26, 1, 63, 4, 50, 8, 16, 1, 53, 1, 19, 19, 122, 5, 33, 1, 29, 10, 39, 1, 113, 1, 21, 17, 35, 3, 43, 1, 95, 42, 22, 1, 83, 6, 24, 11, 59, 1, 88, 4, 44, 13, 27, 7, 203

Offset: 1

Antti Karttunen, Dec 08 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..3071

Cf. A003961, A078898, A246277, A249817, A249818, A249820, A249821, A249822, A251721, A251722, A250469, A250470.

(define (A249810 n) (if (= 1 n) 0 (A078898 (A003961 n))))

a(1) = 0, a(n) = A078898(A003961(n)).

a(1) = 0, a(n) = A078898(n) + A249820(n).

cp's OEIS Frontend

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

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A078898 Number of times the smallest prime factor of n is the smallest prime factor for numbers <= n; a(0)=0, a(1)=1.

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A083221 Sieve of Eratosthenes arranged as an array and read by antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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A250469 a(1) = 1; and for n > 1, a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)+1, where A055396(n) is the index of smallest prime dividing n.

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A250246 Permutation of natural numbers: a(1) = 1, a(n) = A246278(A055396(n), a(A078898(n))).

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A268674 a(1) = 1, after which, for odd numbers: a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)-1, and for even numbers: a(n) = a(A000265(n)).

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A253557 a(1) = 0; after which, a(2n) = 1 + a(n), a(2n+1) = a(A268674(2n+1)).

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A252754 Inverse of "Tree of Eratosthenes" permutation: a(1) = 0, after which, a(2n) = 1 + 2a(n), a(2n+1) = 2 a(A268674(2n+1)).