A268674 -id:A268674 - cp's OEIS frontend

A268675 Self-inverse permutation of natural numbers: a(1) = 1; a(n) = A000079(A193231(A007814(n))) * A250469(a(A268674(n))).

1, 2, 3, 8, 5, 6, 7, 4, 21, 10, 11, 24, 13, 14, 15, 32, 17, 42, 19, 40, 9, 22, 23, 12, 55, 26, 27, 56, 29, 30, 31, 16, 69, 34, 35, 168, 37, 38, 39, 20, 41, 18, 43, 88, 93, 46, 47, 96, 91, 110, 123, 104, 53, 54, 25, 28, 117, 58, 59, 120, 61, 62, 63, 64, 65, 138, 67, 136, 33, 70, 71, 84, 73, 74, 75, 152, 77, 78, 79, 160

Offset: 1

Antti Karttunen, Feb 11 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1914
Index entries for sequences that are permutations of the natural numbers

Cf. A000079, A000265, A007814, A193231, A250469, A268674.

Cf. also A268385.

a(1) = 1, and for n > 1, a(n) = A000079(A193231(A007814(n))) * A250469(a(A268674(n))).
Other identities. For all n >= 1:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]
A020639(a(n)) = A020639(n). [More generally, it preserves the smallest prime dividing n.]
A055396(a(n)) = A055396(n).

A279349 a(1) = 1, for n > 1, if n is even, a(n) = A055938(a(n/2)), otherwise a(n) = A005187(a(A268674(n))).

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 7, 12, 8, 9, 11, 13, 19, 14, 10, 27, 35, 17, 67, 20, 22, 24, 131, 28, 15, 40, 16, 29, 259, 21, 515, 58, 23, 72, 18, 36, 1027, 136, 25, 43, 2051, 45, 4099, 51, 50, 264, 8195, 59, 26, 30, 32, 83, 16387, 33, 41, 60, 38, 520, 32771, 44, 65539, 1032, 46, 121, 31, 48, 131075, 147, 53, 37, 262147, 75, 524291, 2056, 78, 275, 34, 52

Offset: 1

Antti Karttunen, Dec 12 2016

nonn

Index entries for sequences that are permutations of the natural numbers

Inverse: A279348.
Cf. A005187, A055938, A268674.
Related or similar permutations: A250246, A252754, A252756, A279339, A279342, A279344.

(definec (A279349 n) (cond ((= 1 n) n) ((even? n) (A055938 (A279349 (/ n 2)))) (else (A005187 (A279349 (A268674 n))))))

a(1) = 1, for n > 1, if n is even, a(n) = A055938(a(n/2)), otherwise a(n) = A005187(a(A268674(n))).
As a composition of other permutations:
a(n) = A279339(A250246(n)).
a(n) = A279344(A252754(n)).
a(n) = A279342(A252756(n)).

A319679 Number of distinct prime signatures that occur when map x -> A268674(x) is iterated, starting from x = n, until an even number is reached; a(1) = 0 by convention.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2

Offset: 1

Antti Karttunen, Nov 20 2018

nonn

Number of distinct values that A046523 obtains when applied to n and all the terms above it in that column where it is located in array A083221.

Antti Karttunen, Table of n, a(n) for n = 1..32769

Cf. A046523, A083221, A268674.

Cf. also A319689.

up_to = 32769;
v268674 = readvec("./b268674_terms_only-32769.txt"); \\ Prepared from the b-file of A268674.
A268674(n) = v268674[n];
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A319679(n) = if(1==n,0,my(m=Map(),s,k=0); while(1,if(!mapisdefined(m,s=A046523(n)), mapput(m,s,s); k++); if(!(n%2), return(k)); n = A268674(n)));

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).

A302042 A032742 analog for a nonstandard factorization process based on the sieve of Eratosthenes (A083221).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 6, 1, 7, 5, 8, 1, 9, 1, 10, 9, 11, 1, 12, 5, 13, 7, 14, 1, 15, 1, 16, 15, 17, 7, 18, 1, 19, 11, 20, 1, 21, 1, 22, 21, 23, 1, 24, 7, 25, 25, 26, 1, 27, 25, 28, 27, 29, 1, 30, 1, 31, 13, 32, 11, 33, 1, 34, 33, 35, 1, 36, 1, 37, 17, 38, 11, 39, 1, 40, 39, 41, 1, 42, 35, 43, 35, 44, 1, 45, 49, 46, 45, 47, 13, 48

Offset: 1

Antti Karttunen, Mar 31 2018

nonn

Like [A020639(n), A032742(n)] also ordered pair [A020639(n), a(n)] is unique for each n. Iterating n, a(n), a(a(n)), a(a(a(n))), ..., until 1 is reached, and taking the smallest prime factor (A020639) of each term gives a multiset of primes in ascending order, unique for each natural number n >= 1. Permutation pair A250245/A250246 maps between this non-standard prime factorization of n and the ordinary prime factorization of n.

			For n = 66, A020639(66) [its smallest prime factor] is 2. Because A055396(66) = A000720(2) = 1, a(66) is just A078898(66) = 66/2 = 33.
For n = 33, A020639(33) = 3 and A055396(33) = 2, so a(33) = A250469(A078898(33)) = A250469(6) = 15. [15 is under 6 in array A083221].
For n = 15, A020639(15) = 3 and A055396(15) = 2, so a(15) = A250469(A078898(15)) = A250469(3) = 5. [5 is under 3 is array A083221].
For n = 5, A020639(5) = 5 and A055396(5) = 3, so a(5) = A250469(A250469(A078898(5))) = A250469(A250469(1)) = 1.
Collecting the primes given by A020639 we get a multiset of factors: [2, 3, 3, 5]. Note that 2*3*3*5 = 90 = A250246(66).
If we start from n = 66, iterating the map n -> A302044(n) [instead of n -> A302042(n)] and apply A020639 to each term obtained we get just a single instance of each prime: [2, 3, 5]. Then by applying A302045 to the same terms we get the corresponding exponents (multiplicities) of those primes: [1, 2, 1].

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences generated by sieves

Cf. A020639, A032742, A055396, A078898, A083221, A250245, A250246, A250469, A268674, A276151, A280496, A302032.

Cf. also following analogs: A302041 (omega), A253557 (bigomega), A302043, A302044, A302045 (exponent of the least prime present), A302046 (prime signature filter), A302050 (Moebius mu), A302051 (tau), A302052 (char.fun of squares), A302039, A302055 (Arith. derivative).

\\ Assuming A250469 and its inverse A268674 have been precomputed, then the following is fast enough:
A302042(n) = if(1==n,n,my(k=0); while((n%2), n = A268674(n); k++); n = n/2; while(k>0, n = A250469(n); k--); (n));

A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
A078898(n) = if(n<=1,n, my(spf=A020639(n),k=1,m=n/spf); while(m>1,if(A020639(m)>=spf,k++); m--); (k));
\\ Faster if we precompute A078898 as an ordinal transform of A020639:
up_to = 65537;
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; };
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A302042(n) = if((1==n)||isprime(n),1,my(c = A078898(n), p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); (k*p));

For n > 1, a(n) = A250469^(r)(A078898(n)), where r = A055396(n)-1 and A250469^(r)(n) stands for applying r times the map x -> A250469(x), starting from x = n.
a(n) = A250245(A032742(A250246(n))) = A250245(A280496(n)) = A250245(A250246(n) / A020639(n)).
a(n) = n - A302043(n).

A269380 a(1) = 1, after which, for odd numbers: a(n) = A260739(n)-th number k for which A260738(k) = A260738(n)-1, and for even numbers: a(n) = a(n/2).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 1, 4, 3, 7, 2, 11, 5, 6, 1, 13, 4, 9, 3, 8, 7, 17, 2, 23, 11, 10, 5, 25, 6, 19, 1, 12, 13, 15, 4, 29, 9, 14, 3, 37, 8, 41, 7, 16, 17, 43, 2, 21, 23, 18, 11, 47, 10, 31, 5, 20, 25, 35, 6, 53, 19, 22, 1, 27, 12, 61, 13, 24, 15, 67, 4, 55, 29, 26, 9, 71, 14, 33, 3, 28, 37, 77, 8, 49, 41, 30, 7, 83, 16, 89, 17, 32, 43, 39, 2

Offset: 1

Antti Karttunen, Mar 01 2016

nonn

For odd numbers n > 1, a(n) tells which term is on the immediately preceding row of A255127 (square array generated by Ludic sieve), in the same column where n itself is.

Antti Karttunen, Table of n, a(n) for n = 1..33011

Cf. A255127, A260738, A260739.
Cf. A269172, A269355, A269357, A269382, A269386, A269388 (sequences that use this function).
Cf. also A268674, A269370.

a(1) = 1; after which, for even numbers a(n) = a(n/2), and for odd numbers a(n) = A255127(A260738(n)-1, A260739(n)).
Other identities. For all n >= 1:
a(A269379(n)) = n.

A302041 An omega analog for a nonstandard factorization based on the sieve of Eratosthenes (A083221).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2

Offset: 1

Antti Karttunen, Mar 31 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences generated by sieves

Cf. A001221, A069010, A250246, A252754, A302044.
Cf. A302040 (positions of terms < 2).
Cf. A253557 (a similar analog for bigomega), A302050, A302051, A302052, A302039, A302055 (other analogs).
Differs from A302031 for the first time at n=59, where a(59) = 1, while A302031(59) = 2.

\\ Assuming A250469 and its inverse A268674 have been precomputed, then the following is reasonably fast:
A302044(n) = if(1==n,n,my(k=0); while((n%2), n = A268674(n); k++); n = (n/2^valuation(n, 2)); while(k>0, n = A250469(n); k--); (n));
A302041(n) = if(1==n, 0,1+A302041(A302044(n)));

up_to = 65537;
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
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A000265(n) = (n/2^valuation(n, 2));
A302044(n) = { my(c = A000265(A078898(n))); if(1==c,1,my(p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); (k*p)); };
A302041(n) = if(1==n, 0,1+A302041(A302044(n)));

\\ Or, using also some of the code from above:
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A055396(n) = if(1==n,0,primepi(A020639(n)));
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)));
A302041(n) = omega(A250246(n));

a(1) = 0; for n > 1, a(n) = 1 + a(A302044(n)).
a(n) = A001221(A250246(n)).
a(n) = A069010(A252754(n)).

A302044 A028234 analog for factorization process based on the sieve of Eratosthenes (A083221).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 7, 5, 1, 1, 9, 1, 5, 1, 11, 1, 3, 1, 13, 7, 7, 1, 15, 1, 1, 5, 17, 7, 9, 1, 19, 11, 5, 1, 21, 1, 11, 1, 23, 1, 3, 1, 25, 25, 13, 1, 27, 1, 7, 7, 29, 1, 15, 1, 31, 13, 1, 11, 33, 1, 17, 5, 35, 1, 9, 1, 37, 17, 19, 11, 39, 1, 5, 11, 41, 1, 21, 7, 43, 35, 11, 1, 45, 1, 23, 1, 47, 13, 3, 1, 49, 19, 25, 1, 51, 1, 13, 25

Offset: 1

Antti Karttunen, Mar 31 2018

nonn

Iterating n, a(n), a(a(n)), a(a(a(n))), ..., until 1 is reached, and taking the smallest prime factor (A020639) of each term gives a sequence of distinct primes in ascending order, while applying A302045 to the same terms gives the corresponding exponents (multiplicities) of those primes. Permutation pair A250245/A250246 maps between this non-standard prime factorization and the ordinary factorization of n. See also comments and examples in A302042.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences generated by sieves

Cf. A000265, A020639, A055396, A078898, A028234, A083221, A250245, A250246, A250469, A302034, A302042, A302045.

Cf. A302040 (positions of 1's).

\\ Assuming A250469 and its inverse A268674 have been precomputed, then the following is fast enough:
A302044(n) = if(1==n,n,my(k=0); while((n%2), n = A268674(n); k++); n = (n/2^valuation(n, 2)); while(k>0, n = A250469(n); k--); (n));

up_to = 65537;
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; };
A000265(n) = (n/2^valuation(n, 2));
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A302044(n) = { my(c = A000265(A078898(n))); if(1==c,1,my(p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); (k*p)); };

For n > 1, a(n) = A250469^(r)(A000265(A078898(n))), where r = A055396(n)-1 and A250469^(r)(n) stands for applying r times the map x -> A250469(x), starting from x = n.

a(n) = A250245(A028234(A250246(n))).

A302045 a(1) = 0, for n > 1, a(n) = A001511(A078898(n)); Number of instances of the smallest prime factor A020639(n) in nonstandard factorization of n that is based on the sieve of Eratosthenes (A083221).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 31 2018

nonn

Iterating the map n -> A302044(n) until 1 is reached, and taking the smallest prime factor (A020639) of each term gives a sequence of distinct primes in ascending order, while applying this function (A302045) to those terms gives the corresponding "exponents" of those primes, that is, the count of consecutive occurrences of each prime when iterating the map n -> A302042(n), which gives the same primes with repetitions. Permutation pair A250245/A250246 maps between this non-standard prime factorization of n and the ordinary factorization of n. See also comments and examples in A302042.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences generated by sieves

Cf. A001511, A067029, A078898, A250245, A250246, A268674, A302042, A302044.

Cf. also A302035, A302050, A302051.

A020639(n) = { if(1==n,n,vecmin(factor(n)[, 1])); };
A078898(n) = { if(n<=1,n, my(spf=A020639(n),k=1,m=n/spf); while(m>1,if(A020639(m)>=spf,k++); m--); (k)); };
A001511(n) = 1+valuation(n,2);
A302045(n) = if(1==n,0,A001511(A078898(n)));

a(1) = 0, for n > 1, a(n) = A001511(A078898(n)).

For n > 1, a(n) = A250245(A067029(A250246(n))).

cp's OEIS Frontend

A268675 Self-inverse permutation of natural numbers: a(1) = 1; a(n) = A000079(A193231(A007814(n))) * A250469(a(A268674(n))).

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A279349 a(1) = 1, for n > 1, if n is even, a(n) = A055938(a(n/2)), otherwise a(n) = A005187(a(A268674(n))).

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A319679 Number of distinct prime signatures that occur when map x -> A268674(x) is iterated, starting from x = n, until an even number is reached; a(1) = 0 by convention.

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PARI

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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Mathematica

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

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A302042 A032742 analog for a nonstandard factorization process based on the sieve of Eratosthenes (A083221).

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A269380 a(1) = 1, after which, for odd numbers: a(n) = A260739(n)-th number k for which A260738(k) = A260738(n)-1, and for even numbers: a(n) = a(n/2).

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A302041 An omega analog for a nonstandard factorization based on the sieve of Eratosthenes (A083221).

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PARI

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A302044 A028234 analog for factorization process based on the sieve of Eratosthenes (A083221).

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