A250469 -id:A250469 - cp's OEIS frontend

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)).

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.]

A250245 Permutation of natural numbers: a(1) = 1, a(n) = A083221(A055396(n),a(A246277(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, 39, 34, 35, 36, 37, 38, 63, 40, 41, 54, 43, 44, 33, 46, 47, 48, 49, 50, 75, 52, 53, 42, 65, 56, 99, 58, 59, 60, 61, 62, 57, 64, 95, 78, 67, 68, 111, 70, 71, 72, 73, 74, 51, 76, 77, 126, 79, 80, 45, 82

Offset: 1

Antti Karttunen, Nov 17 2014

nonn

The first 7-cycle occurs at: (33 39 63 57 99 81 45) which is mirrored by the cycle (66 78 126 114 198 162 90) with double-size terms.
The cycle which contains 55 as its smallest term, goes as: 55, 65, 95, 185, 425, 325, 205, 455, 395, 1055, 2945, 6035, 30845, ...
while to the other direction (A250246) it goes as: 55, 125, 245, 115, 625, 8575, 40375, ...
The cycle which contains 69 as its smallest term, goes as: 69, 111, 183, 351, 261, 273, 387, 489, 939, 1863, 909, 1161, 981, 1281, 4167, ...
while to the other direction (A250246) it goes as: 69, 135, 87, 105, 225, 207, 231, 195, 525, 1053, 3159, 24909, ...

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

Inverse: A250246.
Other similar permutations: A250244, A250247, A250249, A243071, A252755.
Cf. A005843, A020639, A055396, A064989, A083221, A246277, A250469.
Differs from the "vanilla version" A249817 for the first time at n=42, where a(42) = 54, while A249817(42) = 42.
Differs from A250246 for the first time at n = 33, where a(33) = 39, while A250246(33) = 45.
Differs from A250249 for the first time at n=73, where a(73) = 73, while A250249(73) = 103.

a(1) = 1, a(n) = A083221(A055396(n), a(A246277(n))).
a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A250469(a(A064989(2n+1))). - Antti Karttunen, Jan 18 2015
As a composition of related permutations:
a(n) = A252755(A243071(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].

A250470 a(n) = A249817(A064989(A249818(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, 10, 41, 7, 16, 19, 43, 2, 25, 9, 18, 11, 47, 8, 21, 5, 20, 23, 53, 6, 59, 29, 22, 1, 27, 14, 61, 13, 24, 15, 67, 4, 71, 31, 26, 17, 35, 22, 73, 3, 28, 37, 79, 10, 33, 41, 30, 7, 83, 12, 55, 19, 32, 43, 39, 2, 89, 25, 34, 9, 97, 26, 101

Offset: 1

Antti Karttunen, Dec 06 2014

nonn

Odd bisection, A250472, is a permutation of natural numbers. A250479 gives the even bisection.

For odd numbers n >= 3, 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 above n in the sieve of Eratosthenes (see A083140, A083221) in the same column of the sieve that contains n.

Antti Karttunen, Table of n, a(n) for n = 1..9048

Cf. A055396, A064989, A078898, A083140, A083221, A249744, A249810, A249820, A249817, A249818, A250469.
Odd bisection: A250472.
Even bisection: A250479.
Differs from A064989 for the first time at n=21, where a(21) = 8, while
A064989(21) = 10.

(define (A250470 n) (A249817 (A064989 (A249818 n))))

a(n) = A249817(A064989(A249818(n))).
Other identities. For all n >= 1:
a(A250469(n)) = n. [This is an inverse function for injection A250469.]
For all odd numbers n >= 3: A055396(a(n)) = A055396(n)-1.

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).

A269379 a(1) = 1; for n > 1, a(n) = A255127(A260738(n)+1, A260739(n)).

Original entry on oeis.org

1, 3, 5, 9, 7, 15, 11, 21, 19, 27, 13, 33, 17, 39, 35, 45, 23, 51, 31, 57, 49, 63, 25, 69, 29, 75, 65, 81, 37, 87, 55, 93, 79, 99, 59, 105, 41, 111, 95, 117, 43, 123, 47, 129, 109, 135, 53, 141, 85, 147, 125, 153, 61, 159, 73, 165, 139, 171, 103, 177, 67, 183, 155, 189, 113, 195, 71, 201, 169, 207, 77, 213, 101, 219, 185, 225, 83

Offset: 1

Antti Karttunen, Mar 01 2016

nonn

a(n) = the number located immediately below n in A255127 (square array generated by Ludic sieve) in the same column where n itself is, or in other words, the number removed in the next filtering stage at the same step as when n was removed in the A260738(n)-th stage.

Permutation of odd numbers.

Antti Karttunen, Table of n, a(n) for n = 1..32999

Cf. A255127, A260738, A260739.
Cf. A269171, A269356, A269358, A269382, A269385, A269387 (sequences that use this function).
Cf. A269380 (left inverse).
Cf. also A250469, A269369.

(define (A269379 n) (if (= 1 n) n (A255127bi (+ (A260738 n) 1) (A260739 n)))) ;; Code for A255127bi given in A255127.

a(1) = 1; for n > 1, a(n) = A255127(A260738(n)+1, A260739(n)).
Other identities. For all n >= 1:
A269380(a(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))).

A249814 "Mountains of Eratosthenes" permutation: a(1) = 1, a(n) = A249741(A001511(n), a(A003602(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 14, 15, 24, 13, 20, 11, 10, 17, 26, 27, 34, 29, 44, 47, 48, 25, 38, 39, 54, 21, 32, 19, 12, 33, 50, 51, 64, 53, 80, 67, 76, 57, 86, 87, 114, 93, 140, 95, 120, 49, 74, 75, 94, 77, 116, 107, 90, 41, 62, 63, 84, 37, 56, 23, 16, 65, 98, 99, 124, 101, 152, 127, 118, 105, 158, 159, 204, 133, 200, 151, 142

Offset: 1

Antti Karttunen, Nov 06 2014

This sequence is a "recursed variant" of A249811.
From Antti Karttunen, Jan 18 2015: (Start)
This can be viewed as an entanglement or encoding permutation where the complementary pairs of sequences to be interwoven together are even and odd numbers (A005843/A005408) which are entangled with another complementary pair: even numbers in the order they appear in A253886 and odd numbers in their usual order: (A253886/A005408).
From the above follows also that this sequence can be represented as a binary tree. Each child to the left is obtained by doubling the parent and subtracting one, and each child to the right is obtained by applying A253886 to the parent:
                                    1
                                    |
                 ...................2...................
                3                                       4
      5......../ \........8                   7......../ \........6
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  9       14         15       24         13       20         11       10
17 26   27  34     29  44   47  48     25  38   39  54     21  32   19  12
(End)
For listening I recommend some (mostly) percussive MIDI-instrument and the pitch offset set to at least 29 and the tempo (rate) to about 60. - Antti Karttunen, Feb 17 2015

Antti Karttunen, Table of n, a(n) for n = 1..8191
Index entries for sequences that are permutations of the natural numbers

Inverse: A249813.
Similar or related permutations: A246684, A249811, A250244, A252755.
Cf. A000079, A000265, A001511, A003602, A006093, A083140, A083221, A249741, A250469, A253886.
Compare also the scatterplot of this sequence to the graphs of A252755 and A246684.
Differs from A246684 for the first time at n=14, where a(14) = 20, while A246684(14) = 26.

In the following formulas, A083221 and A249741 are interpreted as bivariate functions:
a(1) = 1, for n>1: a(n) = A083221(A001511(n), a(A003602(n))) - 1 = A249741(A001511(n), a(A003602(n))).
a(1) = 1, a(2n) = A253886(a(n)), a(2n+1) = (2*a(n+1))-1. - Antti Karttunen, Jan 18 2015
As a composition of other permutations:
a(n) = A250244(A246684(n)).
Other identities. For all n >= 1, the following holds:
a(n) = (1+a((2*n)-1))/2. [The odd bisection from a(1) onward with one added and then halved gives the sequence back.]
a(A000079(n-1)) = A006093(n).

A269369 a(1) = 1, a(n) = A260439(n)-th number k for which A260438(k) = A260438(n)+1; a(n) = A255551(A260438(n)+1, A260439(n)).

Original entry on oeis.org

1, 3, 7, 5, 19, 11, 9, 17, 13, 23, 39, 29, 15, 35, 21, 41, 61, 47, 27, 53, 25, 59, 81, 65, 31, 71, 45, 77, 103, 83, 33, 89, 37, 95, 123, 101, 43, 107, 57, 113, 145, 119, 49, 125, 55, 131, 165, 137, 51, 143, 63, 149, 187, 155, 85, 161, 97, 167, 207, 173, 91, 179, 67, 185, 229, 191, 69, 197, 73, 203, 249, 209, 75

Offset: 1

Antti Karttunen, Mar 01 2016

nonn

For n > 1, a(n) = the number located immediately below n in A255551 (square array generated by Lucky sieve) in the same column where n itself is.

Permutation of odd numbers.

Antti Karttunen, Table of n, a(n) for n = 1..25850

Cf. A255551, A260438, A260439, A269372, A269375, A269377.
Cf. A269370 (left inverse).
Cf. also A250469, A269379.

(define (A269369 n) (if (= 1 n) n (A255551bi (+ (A260438 n) 1) (A260439 n)))) ;; Code for A255551bi given in A255551.

a(1) = 1; for n > 1, a(n) = A255551(A260438(n)+1, A260439(n)).
Other identities. For all n >= 1:
A269370(a(n)) = n.

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.]

cp's OEIS Frontend

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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Mathematica

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A250245 Permutation of natural numbers: a(1) = 1, a(n) = A083221(A055396(n),a(A246277(n))).

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A250470 a(n) = A249817(A064989(A249818(n))).

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

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PARI

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A269379 a(1) = 1; for n > 1, a(n) = A255127(A260738(n)+1, A260739(n)).

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

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PARI

PARI

PARI

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

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PARI

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A249814 "Mountains of Eratosthenes" permutation: a(1) = 1, a(n) = A249741(A001511(n), a(A003602(n))).

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