A004001 -id:A004001 - cp's OEIS frontend

A276443 Permutation of natural numbers: a(1) = 1, a(A087686(n)) = A000069(1+a(n-1)), a(A088359(n)) = A001969(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001, and A000069 & A001969 are odious & evil numbers.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 15, 13, 14, 16, 17, 18, 20, 24, 19, 23, 30, 21, 27, 25, 22, 29, 31, 26, 28, 32, 33, 34, 36, 40, 48, 35, 39, 46, 60, 37, 43, 54, 41, 51, 49, 38, 45, 58, 47, 63, 61, 42, 53, 55, 50, 44, 57, 59, 62, 52, 56, 64, 65, 66, 68, 72, 80, 96, 67, 71, 78, 92, 120, 69, 75, 86, 108, 73, 83, 102, 81

Offset: 1

Antti Karttunen, Sep 03 2016

Inverse: A276444.
Cf. A000069, A001969, A004001, A080677, A087686, A088359, A093879.
Similar or related permutations: A003188, A276441, A276445 (compare the scatter plots).

(definec (A276443 n) (cond ((< n 2) n) ((zero? (A093879 (- n 1))) (A000069 (+ 1 (A276443 (+ -1 (A080677 n)))))) (else (A001969 (+ 1 (A276443 (+ -1 (A004001 n))))))))

a(1) = 1; for n > 1, if A093879(n-1) = 0 [when n is in A087686], a(n) = A000069(1+a(A080677(n)-1)), otherwise [when n is in A088359], a(n) = A001969(1+a(A004001(n)-1)).
As a composition of other permutations:
a(n) = A003188(A276441(n)).

A317648 a(1) = a(2) = 1; for n >= 3, a(n) = a(t(n)) + a(n-t(n)) where t = A004001.

Original entry on oeis.org

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

Offset: 1

Altug Alkan, Aug 02 2018

This sequence hits every positive integer.

Let b(1) = b(2) = b(3) = 1; for n >= 4, b(n) = b(t(n)) + b(n-t(n)) where t = A004001. Observe the symmetric relation between this sequence (a(n)) and b(n) thanks to line plots of a(n)-n/2 and b(n)-n/2 in Links section.

Altug Alkan, Table of n, a(n) for n = 1..65536
Altug Alkan, Line plot of a(n)-n/2 for n <= 2^17
Altug Alkan, Line plots of A004001(n)-n/2 and a(n)-n/2 for n <= 2^14
Altug Alkan, Line plots of a(n)-n/2 and b(n)-n/2 for n <= 2^11

Cf. A004001, A317686.

b:= proc(n) option remember; `if`(n<3, 1,
      b(b(n-1)) +b(n-b(n-1)))
    end:
a:= proc(n) option remember; `if`(n<3, 1,
      a(b(n)) +a(n-b(n)))
    end:
seq(a(n), n=1..100); # after Alois P. Heinz at A317686

t[1] = 1; t[2] = 1; t[n_] := t[n] = t[t[n-1]] + t[n - t[n-1]];
a[1] = a[2] = 1; a[n_] := a[n] = a[t[n]] + a[n - t[n]];
Array[a, 100] (* Jean-François Alcover, Nov 01 2020 *)

t=vector(99); t[1]=t[2]=1; for(n=3, #t, t[n] = t[t[n-1]]+t[n-t[n-1]]); a=vector(99); a[1]=a[2]=1; for(n=3, #a, a[n] = a[t[n]]+a[n-t[n]]); a

a(n+1) - a(n) = 0 or 1 for all n >= 1.

A147869 Expansion of Product_{k>0} (1 + A004001(k)*x^k).

Original entry on oeis.org

1, 1, 1, 3, 4, 7, 11, 17, 25, 41, 59, 86, 125, 180, 263, 382, 536, 738, 1073, 1466, 2028, 2841, 3889, 5275, 7211, 9800, 13249, 17860, 23948, 31921, 42864, 56802, 75115, 99788, 131239, 172870, 226789, 296404, 386745, 504939, 655227, 849628, 1101270

Offset: 0

Roger L. Bagula, Nov 16 2008

nonn

			From _Petros Hadjicostas_, Apr 11 2020: (Start)
Let f(m) = A004001(m). Using the strict partitions of each n (see A000009), we get
a(1) = f(1) = 1,
a(2) = f(2) = 1,
a(3) = f(3) + f(1)*f(2) = 2 + 1*1 = 3,
a(4) = f(4) + f(1)*f(3) = 2 + 1*2 = 4,
a(5) = f(5) + f(1)*f(4) + f(2)*f(3) = 3 + 1*2 + 1*2 = 7,
a(6) = f(6) + f(1)*f(5) + f(2)*f(4) + f(1)*f(2)*f(3) = 4 + 1*3 + 1*2 + 1*1*2 = 11,
a(7) = f(7) + f(1)*f(6) + f(2)*f(5) + f(3)*f(4) + f(1)*f(2)*f(4) = 4 + 1*4 + 1*3 + 2*2 + 1*1*2 = 17. (End)

Cf. A000009, A004001, A147654, A147655, A147559, A147880.

f[0] = 0; f[1] = 1; f[2] = 1; f[n_] := f[n] = f[f[n - 1]] + f[n - f[n - 1]];
P[x_, n_] := P[x, n] = Product[1 + f[m]*x^m, {m, 0, n}];
Take[CoefficientList[P[x, 45], x], 45]

a(n) = [x^n] Product_{k > 0} (1 + A004001(k)*x^k).

a(n) = Sum_{(b_1,...,b_n)} f(1)^b_1 * f(2)^b_2 * ... * f(n)^b_n, where f(m) = A004001(m), and the sum is taken over all lists (b_1,...,b_n) with b_j in {0,1} and Sum_{j=1..n} j*b_j = n. - Petros Hadjicostas, Apr 11 2020

Various sections edited by Petros Hadjicostas, Apr 11 2020

A266188 a(n) = A004001(A087686(n)).

Original entry on oeis.org

1, 1, 2, 4, 4, 7, 8, 8, 8, 12, 14, 15, 15, 16, 16, 16, 16, 21, 24, 26, 27, 27, 29, 30, 30, 31, 31, 31, 32, 32, 32, 32, 32, 38, 42, 45, 47, 48, 48, 51, 53, 54, 54, 56, 57, 57, 58, 58, 58, 60, 61, 61, 62, 62, 62, 63, 63, 63, 63, 64, 64, 64, 64, 64, 64, 71, 76, 80, 83, 85, 86, 86, 90, 93, 95, 96, 96, 99, 101, 102, 102, 104

Offset: 1

Antti Karttunen, Jan 10 2016

nonn

Discarding duplicates gives A087686 back, i.e., this set of numbers is closed with respect to A004001.

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

Cf. A004001, A080677, A087686.

(define (A266188 n) (A004001 (A087686 n)))

a(n) = A004001(A087686(n)).

A276444 Permutation of natural numbers: a(1) = 1; a(A001969(1+n)) = A088359(a(n)), a(A000069(1+n)) = A087686(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001, and A000069 & A001969 are odious & evil numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 14, 15, 13, 16, 17, 18, 21, 19, 24, 27, 22, 20, 26, 30, 25, 31, 28, 23, 29, 32, 33, 34, 38, 35, 42, 48, 39, 36, 45, 54, 43, 58, 49, 40, 51, 37, 47, 57, 46, 62, 55, 44, 56, 63, 59, 50, 60, 41, 53, 61, 52, 64, 65, 66, 71, 67, 76, 86, 72, 68, 80, 96, 77, 106, 87, 73, 90, 69, 83, 102, 81

Offset: 1

Antti Karttunen, Sep 03 2016

Inverse: A276443.
Cf. A000069, A001969, A004001, A010060, A087686, A088359, A115384, A245710.
Similar or related permutations: A006068, A276442, A276446.

(definec (A276444 n) (cond ((< n 2) n) ((zero? (A010060 n)) (A088359 (A276444 (A245710 n)))) (else (A087686 (+ 1 (A276444 (- (A115384 n) 1)))))))

a(1) = 1, and for n > 1, if A010060(n) = 0 [when n is one of the evil numbers, A001969], a(n) = A088359(a(A245710(n))), otherwise a(n) = A087686(1+a(A115384(n)-1)).
As a composition of other permutations:
a(n) = A276442(A006068(n)).

A276446 Permutation of natural numbers: a(1) = 1; a(A000069(1+n)) = A088359(a(n)), a(A001969(1+n)) = A087686(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001, and A000069 & A001969 are odious & evil numbers.

Original entry on oeis.org

1, 3, 2, 6, 7, 4, 5, 11, 14, 15, 13, 8, 9, 10, 12, 20, 26, 30, 25, 31, 28, 23, 29, 16, 17, 18, 21, 19, 24, 27, 22, 37, 47, 57, 46, 62, 55, 44, 56, 63, 59, 50, 60, 41, 53, 61, 52, 32, 33, 34, 38, 35, 42, 48, 39, 36, 45, 54, 43, 58, 49, 40, 51, 70, 85, 105, 84, 120, 103, 82, 104, 126, 117, 98, 118, 79, 101, 119, 100, 127, 122, 108, 123, 89

Offset: 1

Antti Karttunen, Sep 03 2016

Inverse: A276445.
Cf. A000069, A001969, A004001, A010060, A087686, A088359, A115384, A245710.
Similar or related permutations: A006068, A267112, A276444.

(definec (A276446 n) (cond ((< n 2) n) ((zero? (A010060 n)) (A087686 (+ 1 (A276446 (A245710 n))))) (else (A088359 (A276446 (- (A115384 n) 1))))))

a(1) = 1, and for n > 1, if A010060(n) = 1 [when n is one of the odious numbers, A000069], a(n) = A088359(a(A115384(n)-1)), otherwise a(n) = A087686(1+a(A245710(n))).
As a composition of other permutations:
a(n) = A267112(A006068(n)).

A283468 a(n) = A004001(A004001(n-1)) - A004001(n-A004001(n-1)), a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 0, 0, -1, 0, 0, 0, -1, 0, 1, 1, 0, 0, 0, 0, -1, 0, 1, 2, 2, 1, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, -1, 0, 1, 2, 3, 3, 2, 3, 4, 4, 3, 4, 4, 3, 3, 3, 2, 3, 3, 2, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, -1, 0, 1, 2, 3, 4, 4, 3, 4, 5, 6, 6, 5, 6, 7, 7, 6, 7, 7, 6, 6, 6, 5, 6, 7, 7, 6, 7, 7, 6, 6, 6, 5, 6, 6, 5, 5, 5, 4, 4, 4, 4, 3, 4, 4, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Mar 18 2017

sign

The only negative terms seem to be -1's, occurring as a(1+(2^n)), for n >= 2.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A004001, A080677, A283471 (positions of zeros), A283469, A283470, A283472.

Cf. also A283655.

a[n_] := a[n] = If[n <= 2, 1, a[a[n - 1]] + a[n - a[n - 1]]]; Table[a[#] - a[n - #] &@ a[n - 1] + Boole[n <= 2], {n, 120}] (* Michael De Vlieger, Mar 18 2017, after Robert G. Wilson v at A004001 *)

(define (A283468 n) (if (<= n 2) 1 (- (A004001 (A004001 (- n 1))) (A004001 (- n (A004001 (- n 1)))))))
;; Code for A004001 given under that entry.

a(1) = a(2) = 1; for n > 2, a(n) = A004001(A004001(n-1)) - A004001(A080677(n-1)).

A283481 Positions of odd terms in A004001.

Original entry on oeis.org

1, 2, 5, 9, 11, 12, 17, 19, 22, 25, 26, 27, 33, 35, 37, 38, 40, 43, 46, 47, 48, 50, 51, 55, 56, 57, 58, 65, 67, 69, 72, 74, 77, 79, 80, 82, 83, 87, 89, 90, 92, 93, 97, 100, 101, 102, 107, 110, 111, 112, 117, 118, 119, 120, 121, 129, 131, 133, 135, 136, 138, 140, 143, 145, 148, 150, 151, 153, 154, 158, 160, 163, 165, 166, 168, 169

Offset: 1

Antti Karttunen, Mar 18 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8121

Cf. A283482 (complement), A283480 (a left inverse).
Positions of ones in A095901.
Cf. A004001.

a[n_] := a[n] = If[n <= 2, 1, a[a[n - 1]] + a[n - a[n - 1]]]; Select[Range@ 169, OddQ@ a[#] &] (* Michael De Vlieger, Mar 18 2017, after Robert G. Wilson v at A004001 *)

Other identities. For all n >= 1:

A283480(a(n)) = n.

A286541 Compound filter (the left & right summand of Hofstadter-Conway $10000 sequence): a(n) = P(A004001(A004001(n-1)), A004001(n-A004001(n-1))), where P(n,k) is sequence A000027 used as a pairing function, with a(1) = a(2) = 0.

Original entry on oeis.org

0, 0, 1, 1, 2, 5, 5, 5, 8, 13, 19, 19, 25, 25, 25, 25, 32, 41, 51, 62, 62, 73, 86, 86, 99, 99, 99, 113, 113, 113, 113, 113, 128, 145, 163, 182, 202, 202, 222, 244, 267, 267, 290, 315, 315, 340, 340, 340, 366, 394, 394, 422, 422, 422, 451, 451, 451, 451, 481, 481, 481, 481, 481, 481, 512, 545, 579, 614, 650, 687, 687, 724, 763, 803, 844, 844, 885, 928, 972, 972

Offset: 1

Antti Karttunen, May 18 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A004001, A286559, A286560.

Cf. A283468, A283469, A283470, A283472 and also A283471, A283473.

(define (A286541 n) (if (<= n 2) 0 (* (/ 1 2) (+ (expt (+ (A004001 (A004001 (- n 1))) (A004001 (- n (A004001 (- n 1))))) 2) (- (A004001 (A004001 (- n 1)))) (- (* 3 (A004001 (- n (A004001 (- n 1)))))) 2))))

a(1) = a(2) = 0, for n > 2, a(n) = (1/2)*(2 + ((A004001(A004001(n-1))+A004001(n-A004001(n-1)))^2) - A004001(A004001(n-1)) - 3*A004001(n-A004001(n-1))).

A286569 Restricted growth sequence transform of "Hofstadter chaotic heart", A284019 (= A004001(n) - A005185(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 3, 2, 1, 1, 4, 2, 2, 2, 1, 1, 1, 3, 5, 4, 3, 3, 2, 1, 1, 1, 6, 2, 1, 4, 4, 3, 3, 2, 3, 3, 3, 3, 3, 5, 5, 7, 7, 8, 9, 9, 2, 5, 9, 1, 3, 7, 2, 3, 1, 1, 1, 1, 10, 2, 5, 6, 1, 7, 4, 4, 3, 3, 1, 5, 5, 7, 3, 9, 9, 5, 5, 9, 9, 5, 9, 7, 5, 7, 11, 7, 9, 11, 11, 12, 12, 13, 14, 9, 5, 3, 15, 7, 9, 16, 4, 12, 11, 5, 1, 16, 3, 3, 17, 1, 6, 18

Offset: 1

Antti Karttunen, May 18 2017

nonn

			We start by setting a(1) = 1 for A284019(1) = 0. Then after, whenever A284019(k) is equal to some A284019(m) with m < k, we set a(k) = a(m). Otherwise (when the value is a new one, not encountered before), we allot for a(k) the least natural number not present among a(1) .. a(k-1).
For n=2, as A284019(2) = 0, which was already present at A284019(1), we set a(2) = a(1) = 1.
For n=3, as A284019(3) = 0, which was already present at n=1, we set a(3) = a(1) = 1.
For n=4, as A284019(4) = -1, which is a new value not encountered before, we set a(4) = 1 + max(a(1),a(2),a(3)) = 2.
For n=5, as A284019(5) = 0, which was already present at n=1, we set a(5) = a(1) = 1.
For n=7, as A284019(7) = -1, which was already present at n=4, we set a(7) = a(4) = 2.
For n=11, as A284019(11) = 1, which is a new value not encountered before (sign matters here), we set a(11) = 1 + max(a(1),..,a(10)) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..47000

Cf. A004001, A005185, A284019, A286560.

cp's OEIS Frontend

A276443 Permutation of natural numbers: a(1) = 1, a(A087686(n)) = A000069(1+a(n-1)), a(A088359(n)) = A001969(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001, and A000069 & A001969 are odious & evil numbers.

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A317648 a(1) = a(2) = 1; for n >= 3, a(n) = a(t(n)) + a(n-t(n)) where t = A004001.

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A147869 Expansion of Product_{k>0} (1 + A004001(k)*x^k).

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A266188 a(n) = A004001(A087686(n)).

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A276444 Permutation of natural numbers: a(1) = 1; a(A001969(1+n)) = A088359(a(n)), a(A000069(1+n)) = A087686(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001, and A000069 & A001969 are odious & evil numbers.

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A276446 Permutation of natural numbers: a(1) = 1; a(A000069(1+n)) = A088359(a(n)), a(A001969(1+n)) = A087686(1+a(n)), where A088359 & A087686 = numbers that occur only once & more than once in A004001, and A000069 & A001969 are odious & evil numbers.

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A283468 a(n) = A004001(A004001(n-1)) - A004001(n-A004001(n-1)), a(1) = a(2) = 1.

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A283481 Positions of odd terms in A004001.

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