A002808 -id:A002808 - cp's OEIS frontend

A099862 a(n) = (2*k)-th composite number; a bisection of A002808.

6, 9, 12, 15, 18, 21, 24, 26, 28, 32, 34, 36, 39, 42, 45, 48, 50, 52, 55, 57, 60, 63, 65, 68, 70, 74, 76, 78, 81, 84, 86, 88, 91, 93, 95, 98, 100, 104, 106, 110, 112, 115, 117, 119, 121, 123, 125, 128, 130, 133, 135, 138, 141, 143, 145, 147, 150, 153, 155, 158, 160

Offset: 1

N. J. A. Sloane, Nov 19 2004

			a(1) = 6 is the second composite number.

Cf. A002808, A099861.

Complement of A377898.

c:=proc(n) if isprime(n) then else n fi end: B:=[seq(c(n),n=2..250)]: seq(B[2*m],m=1..75); # Emeric Deutsch, Dec 09 2004

Partition[Select[Range[200], CompositeQ], 2][[All, 2]] (* Jean-François Alcover, Mar 22 2023 *)

More terms from Emeric Deutsch, Dec 09 2004

A130882 a(n) = smallest k such that A002808(n+1) = A002808(n) + (A002808(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 4, 7, 2, 4, 5, 13, 2, 7, 4, 19, 2, 4, 23, 2, 5, 2, 13, 4, 31, 2, 3, 2, 17, 37, 2, 19, 4, 43, 2, 4, 47, 2, 7, 2, 5, 53, 2, 5, 2, 4, 29, 61, 2, 3, 2, 4, 67, 2, 4, 5, 73, 2, 3, 2, 4, 79, 2, 4, 83, 2, 5, 2, 43, 89, 2, 7, 2, 3, 2, 47, 97

Offset: 1

Rémi Eismann, Aug 21 2007 - Jan 09 2011

nonn

a(n) is the "weight" of composite numbers.

The decomposition of composite numbers into weight * level + gap is A002808(n) = a(n) * A179621(n) + A073783(n) if a(n) > 0.

			For n = 1 we have A002808(n) = 4, A002808(n+1) = 6; there is no k such that 6 - 4 = 2 = (4 mod k), hence a(1) = 0.
For n = 3 we have A002808(n) = 8, A002808(n+1) = 9; 7 is the smallest k such that 9 - 8 = 1 = (8 mod k), hence a(3) = 7.
For n = 24 we have A002808(n) = 36, A002808(n+1) = 38; 17 is the smallest k such that 38 - 36 = 2 = (36 mod k), hence a(24) = 17.

Remi Eismann, Table of n, a(n) for n=1..9999

Cf. A020639, A117078, A117563, A001223, A118534, A090369, A090368, A130533, A130650, A130703.

A246370 a(1)=0, a(p_n) = 1 + a(n), a(c_n) = a(n), where p_n = n-th prime = A000040(n), c_n = n-th composite number = A002808(n); Also number of nonleading 0-bits in the binary representation of A135141(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 27 2014

nonn

			Consider n=30. It is the 19th composite number in A002808: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, ...
Thus we consider next n=19, which is the 8th prime in A000040: 2, 3, 5, 7, 11, 13, 17, 19, ...
So we proceed with n=8, which is the 3rd composite number, and then with n=3, which is the 2nd prime, and then with n=2 which is the 1st prime, and we have finished.
All in all, it took us 5 steps (A246348(30) = 6 = 5+1) to reach 1, and on the journey, we encountered three primes, 19, 3 and 2, thus a(30) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..32998

Cf. A000040, A002808, A000720, A065855, A080791, A135141, A246348, A246369, A246377.

a(1) = 1, and for n >= 1, if A010051(n) = 1 [i.e. when n is prime], a(n) = 1 + a(A000720(n)), otherwise a(n) = a(A065855(n)). [A000720(n) and A065855(n) tell the number of primes, and respectively, composites <= n].
a(n) = A080791(A135141(n)). [a(n) tells also the number of nonleading zeros in binary representation of A135141(n)].
a(n) = A000120(A246377(n))-1. [Respectively, one less than the number of 1-bits in 0/1-swapped version of that sequence].
a(n) = A246348(n) - A246369(n) - 1.

A246380 Permutation of natural numbers: a(1) = 1, a(2n) = nthcomposite(a(n)), a(2n-1) = nthprime(a(A064989(2n-1)-1)), where nthprime = A000040, nthcomposite = A002808, and A064989(n) shifts the prime factorization of n one step towards smaller primes.

Original entry on oeis.org

1, 4, 2, 9, 7, 6, 23, 16, 3, 14, 13, 12, 43, 35, 17, 26, 37, 8, 101, 24, 5, 22, 19, 21, 53, 62, 83, 51, 79, 27, 233, 39, 191, 54, 149, 15, 103, 134, 11, 36, 47, 10, 151, 34, 41, 30, 29, 33, 73, 75, 241, 86, 113, 114, 89, 72, 1153, 108, 443, 40, 593, 296, 547, 56, 167, 245, 173, 76, 563, 194, 1553, 25

Offset: 1

Antti Karttunen, Aug 29 2014

nonn

Has an infinite number of infinite cycles. See comments in A246379.

Antti Karttunen, Table of n, a(n) for n = 1..3098
Index entries for sequences that are permutations of the natural numbers

Inverse: A246379.
Similar or related permutations: A246376, A246378, A246363, A246364, A246366, A246368, A064216, A246682.
Cf. A000035, A000040, A002808, A010051, A064989.

default(primelimit,(2^31)+(2^30));
A002808(n) = { my(k=-1); while( -n + n += -k + k=primepi(n), ); n }; \\ This function from M. F. Hasler
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A246380(n) = if(1==n, 1, if(!(n%2), A002808(A246380(n/2)), prime(A246380(A064989(n)-1))));
for(n=1, 3098, write("b246380.txt", n, " ", A246380(n)));
(Scheme, with memoization-macro definec)
(definec (A246380 n) (cond ((< n 2) n) ((even? n) (A002808 (A246380 (/ n 2)))) (else (A000040 (A246380 (- (A064989 n) 1))))))

a(1) = 1, a(2n) = nthcomposite(a(n)), a(2n-1) = nthprime(a(A064989(2n-1)-1)), where nthprime = A000040, nthcomposite = A002808, and A064989(n) shifts the prime factorization of n one step towards smaller primes.
As a composition of related permutations:
a(n) = A246378(A246376(n)).
Other identities. For all n > 1 the following holds:
A010051(a(n)) = A000035(n). [Maps odd numbers larger than one to primes, and even numbers to composites, in some order. Permutations A246378 & A246682 have the same property].

A246682 Permutation of natural numbers: a(1) = 0, a(2) = 1, and for n > 1, a(2n) = nthcomposite(a(n)), a(2n-1) = nthprime(a(A064989(2n-1))), where nthprime = A000040, nthcomposite = A002808, and A064989(n) shifts the prime factorization of n one step towards smaller primes.

Original entry on oeis.org

0, 1, 2, 4, 3, 6, 5, 9, 7, 8, 11, 12, 31, 10, 13, 16, 127, 14, 709, 15, 19, 20, 5381, 21, 17, 46, 23, 18, 52711, 22, 648391, 26, 29, 166, 41, 24, 9737333, 858, 71, 25, 174440041, 30, 3657500101, 32, 37, 6186

Offset: 1

Antti Karttunen, Sep 01 2014

nonn

Note the indexing: the domain starts from 1, while the range includes also zero.

Has an infinite number of infinite cycles. See comments at A246681.

Index entries for sequences that are permutations of the natural numbers

Inverse: A246681.
Similar or related permutations: A246376, A246378, A243071, A246368, A064216, A246380.
Cf. A000035, A000040, A002808, A007097, A010051, A064989, A049076, A078442.

default(primelimit,(2^31)+(2^30));
A002808(n) = { my(k=-1); while( -n + n += -k + k=primepi(n), ); n }; \\ This function from M. F. Hasler
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A246682(n) = if(n < 3, n-1, if(!(n%2), A002808(A246682(n/2)), prime(A246682(A064989(n)))));
for(n=1, 46, write("b246682.txt", n, " ", A246682(n)));
(Scheme, two variants)
(definec (A246682 n) (cond ((<= n 2) (- n 1)) ((even? n) (A002808 (A246682 (/ n 2)))) (else (A000040 (A246682 (A064989 n))))))
(define (A246682 n) (A246378 (A243071 n)))

a(1) = 0, a(2) = 1, and for n > 1, a(2n) = nthcomposite(a(n)), a(2n-1) = nthprime(a(A064989(2n-1))), where nthprime = A000040, nthcomposite = A002808, and A064989(n) shifts the prime factorization of n one step towards smaller primes.
As a composition of related permutations:
a(n) = A246378(A243071(n)).
Other identities.
For all n >= 1 the following holds:
a(A000040(n)) = A007097(n-1). [Maps primes to the iterates of primes].
A049076(a(A000040(n))) = n. [Follows from above].
For all n > 1 the following holds:
A010051(a(n)) = A000035(n). [Maps odd numbers larger than one to primes, and even numbers to composites, in some order. Permutations A246378 & A246380 have the same property].

A257731 Permutation of natural numbers: a(1) = 1, a(prime(n)) = lucky(1+a(n)), a(composite(n)) = unlucky(a(n)), where prime(n) = n-th prime number A000040, composite(n) = n-th composite number A002808 and lucky = A000959, unlucky = A050505.

Original entry on oeis.org

1, 3, 9, 2, 33, 5, 7, 14, 4, 45, 163, 8, 15, 11, 20, 6, 25, 59, 63, 203, 12, 22, 13, 17, 28, 10, 35, 78, 235, 83, 1093, 251, 18, 30, 19, 24, 31, 39, 16, 47, 67, 101, 43, 290, 107, 1283, 87, 309, 26, 41, 27, 34, 21, 42, 53, 23, 61, 88, 115, 128, 321, 57, 354, 137, 1499, 112, 349, 376, 36, 55, 1401, 38, 49, 46, 29, 56, 70, 32, 99, 81

Offset: 1

Antti Karttunen, May 06 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..6185
Index entries for sequences that are permutations of the natural numbers

Inverse: A257732.
Cf. A000040, A000720, A000959, A002808, A010051, A050505, A065855.
Related or similar permutations: A246377, A255421, A257726, A257733.
Cf. also A032600, A255553, A255554.
Differs from A257733 for the first time at n=19, where a(19) = 63, while A257733(19) = 203.

a(1) = 1; for n > 1: if A010051(n) = 1 [i.e., if n is a prime], then a(n) = A000959(1+a(A000720(n))), otherwise a(n) = A050505(a(A065855(n))).
As a composition of other permutations:
a(n) = A257726(A246377(n)).
a(n) = A257733(A255421(n)).

A257732 Permutation of natural numbers: a(1) = 1, a(lucky(n)) = prime(a(n-1)), a(unlucky(n)) = composite(a(n)), where lucky(n) = n-th lucky number A000959, unlucky(n) = n-th unlucky number A050505, and prime = A000040, composite = A002808.

Original entry on oeis.org

1, 4, 2, 9, 6, 16, 7, 12, 3, 26, 14, 21, 23, 8, 13, 39, 24, 33, 35, 15, 53, 22, 56, 36, 17, 49, 51, 25, 75, 34, 37, 78, 5, 52, 27, 69, 101, 72, 38, 102, 50, 54, 43, 106, 10, 74, 40, 94, 73, 134, 83, 98, 55, 135, 70, 76, 62, 141, 18, 100, 57, 125, 19, 99, 175, 114, 41, 130, 167, 77, 176, 95, 89, 104, 137, 86, 184, 28, 149, 133, 80, 164, 30

Offset: 1

Antti Karttunen, May 06 2015

In other words, a(1) = 1 and for n > 1, if n is the k-th lucky number larger than 1 [i.e., n = A000959(k+1)] then a(n) = nthprime(a(k)), otherwise, when n is the k-th unlucky number [i.e., n = A050505(k)], then a(n) = nthcomposite(a(k)).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A257731.
Cf. A000040, A000959, A002808, A050505, A109497, A145649.
Related or similar permutations: A246378, A255422, A257725, A257734.
Cf. also A032600, A255553, A255554.

a(1) = 1; for n > 1: if A145649(n) = 1 [i.e., if n is lucky], then a(n) = A000040(a(A109497(n)-1)), otherwise a(n) = A002808(a(n-A109497(n))).
As a composition of other permutations:
a(n) = A246378(A257725(n)).
a(n) = A255422(A257734(n)).

A378373 Number of composite numbers (A002808) between consecutive nonsquarefree numbers (A013929), exclusive.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 02 2024

nonn

All terms are 0, 1, 2, or 3 (cf. A078147).
The inclusive version is a(n) + 2.
The nonsquarefree numbers begin: 4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, ...

			The composite numbers counted by a(n) form the following set partition of A120944:
{6}, {}, {10}, {14,15}, {}, {}, {21,22}, {}, {26}, {}, {30}, {33,34,35}, {38,39}, ...

For prime (instead of nonsquarefree) we have A046933.
For squarefree (instead of nonsquarefree) we have A076259(n)-1.
For prime power (instead of nonsquarefree) we have A093555.
For prime instead of composite we have A236575.
For nonprime prime power (instead of nonsquarefree) we have A378456.
For perfect power (instead of nonsquarefree) we have A378614, primes A080769.
A002808 lists the composite numbers.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147.
A073247 lists squarefree numbers with nonsquarefree neighbors.
A120944 lists squarefree composite numbers.
A377432 counts perfect-powers between primes, zeros A377436.
A378369 gives distance to the next nonsquarefree number (A120327).
Cf. A065890, A067535, A067871, A080101, A081221, A151800, A243348, A366833, A377046, A378033, A378039, A378086.

v=Select[Range[100],!SquareFreeQ[#]&];
Table[Length[Select[Range[v[[i]]+1,v[[i+1]]-1],CompositeQ]],{i,Length[v]-1}]

A073169 a(n)=A002808(n)-n, difference between n-th composite and n.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Jul 19 2002

nonn

a(n) = the number of numbers of set {1, prime} (A008578(n)) less than n-th composite numbers (A002808(n)). a(n) = inverse (frequency distribution) sequence of A162177(n), i.e. number of terms of sequence A162177(n) less than n for n >= 1. a(n) = A002808(n) + A162177(n) - A158611(n+1) for n >= 1. a(n) = A002808(n) + A162177(n) - A008578(n) for n >= 1. [From Jaroslav Krizek, Jul 23 2009]

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A007821, A022449, A000040, A002808, A073168.

c[n_Integer] := FixedPoint[n+PrimePi[ # ]+1&, n] Table[c[w]-w, {w, 1, 128}]
With[{c=Select[Range[100],CompositeQ]},#[[1]]-#[[2]]&/@Thread[ {c,Range[ Length[ c]]}]] (* Harvey P. Dale, Feb 03 2015 *)

a(n)=1+A073425(n). [From R. J. Mathar, Jul 31 2009]

Correction for change of offset in A158611 and A008578 in Aug 2009 Jaroslav Krizek, Jan 27 2010

A203418 Vandermonde determinant of the first n composite numbers (A002808).

Original entry on oeis.org

1, 2, 16, 240, 11520, 13271040, 254803968000, 15892123484160000, 5126163351050649600000, 89288743527804466888704000000, 50689719717698351557731837542400000000, 125765178831579421305165126665125232640000000000

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

Each term divides its successor, as in A203419, and each term is divisible by the corresponding superfactorial, A000178(n), as in A203420.

G. C. Greubel, Table of n, a(n) for n = 1..42

Cf. A000040, A000178, A002808, A203419, A203420.

A002808:=[n: n in [2..250] | not IsPrime(n)];
a:= func< n | n eq 0 select 1 else (&*[(&*[A002808[k+2] - A002808[j+1]: j in [0..k]]): k in [0..n-1]]) >;
[a(n): n in [0..20]]; // G. C. Greubel, Feb 24 2024

composite = Select[Range[100], CompositeQ]; (* A002808 *)
z = 20;
f[j_]:= composite[[j]];
v[n_]:= Product[Product[f[k] - f[j], {j, 1, k-1}], {k, 2, n}];
d[n_]:= Product[(i - 1)!, {i, 1, n}];
Table[v[n], {n,z}]             (* this sequence *)
Table[v[n+1]/v[n], {n,z}]      (* A203419 *)
Table[v[n]/d[n], {n,z}]        (* A203420 *)

A002808=[n for n in (2..250) if not is_prime(n)]
def a(n): return product(product( A002808[k+1] - A002808[j] for j in range(k+1)) for k in range(n))
[a(n) for n in range(15)] # G. C. Greubel, Feb 24 2024

cp's OEIS Frontend

A099862 a(n) = (2*k)-th composite number; a bisection of A002808.

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A130882 a(n) = smallest k such that A002808(n+1) = A002808(n) + (A002808(n) mod k), or 0 if no such k exists.

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A246370 a(1)=0, a(p_n) = 1 + a(n), a(c_n) = a(n), where p_n = n-th prime = A000040(n), c_n = n-th composite number = A002808(n); Also number of nonleading 0-bits in the binary representation of A135141(n).

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A246380 Permutation of natural numbers: a(1) = 1, a(2n) = nthcomposite(a(n)), a(2n-1) = nthprime(a(A064989(2n-1)-1)), where nthprime = A000040, nthcomposite = A002808, and A064989(n) shifts the prime factorization of n one step towards smaller primes.

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A246682 Permutation of natural numbers: a(1) = 0, a(2) = 1, and for n > 1, a(2n) = nthcomposite(a(n)), a(2n-1) = nthprime(a(A064989(2n-1))), where nthprime = A000040, nthcomposite = A002808, and A064989(n) shifts the prime factorization of n one step towards smaller primes.

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A257731 Permutation of natural numbers: a(1) = 1, a(prime(n)) = lucky(1+a(n)), a(composite(n)) = unlucky(a(n)), where prime(n) = n-th prime number A000040, composite(n) = n-th composite number A002808 and lucky = A000959, unlucky = A050505.

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A257732 Permutation of natural numbers: a(1) = 1, a(lucky(n)) = prime(a(n-1)), a(unlucky(n)) = composite(a(n)), where lucky(n) = n-th lucky number A000959, unlucky(n) = n-th unlucky number A050505, and prime = A000040, composite = A002808.

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A378373 Number of composite numbers (A002808) between consecutive nonsquarefree numbers (A013929), exclusive.

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A073169 a(n)=A002808(n)-n, difference between n-th composite and n.

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