A277905 -id:A277905 - cp's OEIS frontend

A277899 a(n) = A097249(A260443(n)).

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

Offset: 0

Antti Karttunen, Nov 15 2016

nonn

a(n) = number of times we must iterate A097246, starting at A260443(n), before the result is squarefree.

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A097246, A097249, A260443, A277905.

Cf. A023758 (positions of zeros).

(define (A277899 n) (A097249_for_coeff_list (A260443as_coeff_list n)))
(define (A097249_for_coeff_list nums) (let loop ((nums nums) (s 0)) (if (<= (reduce max 0 nums) 1) s (loop (A097246_for_coeff_list nums) (+ 1 s)))))
(define (A097246_for_coeff_list nums) (add_two_lists (map A000035 nums) (cons 0 (map A004526 nums))))
;; For the other required functions, see A260443.

a(n) = A097249(A260443(n)).

A277896 a(n) = the least k > n for which A048675(k) = A048675(n), 0 if no such number exists (when n is a power of 2).

Original entry on oeis.org

0, 0, 4, 0, 9, 8, 25, 0, 12, 18, 49, 16, 121, 50, 20, 0, 169, 24, 289, 27, 28, 98, 361, 32, 45, 242, 36, 75, 529, 40, 841, 0, 44, 338, 63, 48, 961, 578, 52, 54, 1369, 56, 1681, 147, 60, 722, 1849, 64, 175, 90, 68, 363, 2209, 72, 99, 150, 76, 1058, 2809, 80, 3481, 1682, 84, 0, 117, 88, 3721, 507, 92, 126, 4489, 96, 5041, 1922, 100, 867, 275

Offset: 1

Antti Karttunen, Nov 15 2016

nonn

Apart from zeros, a permutation of A013929.

Antti Karttunen, Table of n, a(n) for n = 1..192

Cf. A000079, A013929, A048675, A209229, A277886, A277893, A277905.

Numbers not in this sequence: A005117 (A019565).

(define (A277896 n) (if (= 1 (A209229 n)) 0 (let ((v (A048675 n))) (let loop ((k (+ 1 n))) (if (= (A048675 k) v) k (loop (+ 1 k)))))))

a(A000079(n)) = 0.

For all n, except powers of two, a(n) >= A277893(n).

A277903 a(n) = the least k such that A000123(k) >= n.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12

Offset: 1

Antti Karttunen, Nov 14 2016

nonn

Greatest monotonic left inverse of A000123.

Each n occurs A018819(n) times.

Antti Karttunen, Table of n, a(n) for n = 1..9828

Cf. A000123, A018819, A048675, A277904, A277905, A278163.

(definec (A277903 n) (let loop ((k 0)) (if (>= (A000123 k) n) k (loop (+ 1 k)))))
(define (A000123 n) (A018819 (+ n n)))
(definec (A018819 n) (cond ((zero? n) 1) ((even? n) (+ (A018819 (- n 1)) (A018819 (/ n 2)))) (else (A018819 (- n 1)))))

a(n) = A048675(A277905(n)).

a(A000123(n)) = n.

A277904 Irregular table: row n (n >= 0) is obtained by listing numbers 0 .. A018819(n)-1.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19

Offset: 1

Antti Karttunen, Nov 14 2016

			A018819 -> range -> terms on row n
  1        [0,0]:    0;
  1        [0,0]:    0;
  2        [0,1]:    0, 1;
  2        [0,1]:    0, 1;
  4        [0,3]:    0, 1, 2, 3;
  4        [0,3]:    0, 1, 2, 3;
  6        [0,5]:    0, 1, 2, 3, 4, 5;
etc.

Antti Karttunen, Table of n, a(n) for n = 1..9828

Cf. A000123, A018819, A277903.
Used for constructing A277905.
Retaining only every second row gives A278164.

(define (A277904 n) (if (= 1 n) 0 (- n (A000123 (+ -1 (A277903 n))) 1)))

a(1) = 0; for n > 1, a(n) = n - A000123(A277903(n)-1) - 1.

A373120 Number of distinct possible binary ranks of integer partitions of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 33, 43, 55, 70, 89, 109, 136, 167, 206, 251, 306, 371, 445, 535, 639, 759, 904, 1069, 1262, 1489, 1747, 2047, 2390, 2784, 3237, 3754, 4350, 5027, 5798, 6680, 7671, 8808, 10091, 11543, 13190, 15040, 17128, 19477, 22118

Offset: 0

Gus Wiseman, May 26 2024

nonn

			The partitions of 4 are (4), (3,1), (2,2), (2,1,1), (1,1,1,1), with respective binary ranks 8, 5, 4, 4, 4, so a(4) = 3.

The strict case is A000009.
A048675 gives binary rank of prime indices, distinct A087207.
A118462 lists binary ranks of strict integer partitions, row sums A372888.
A277905 groups all positive integers by binary rank of prime indices.
A372890 adds up binary ranks of integer partitions.
Binary indices (A048793):
- length A000120, complement A023416
- min A001511, opposite A000012
- max A029837 or A070939, opposite A070940
- sum A029931, product A096111
- reverse A272020
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
- opposite A371572, sum A230877
Cf. A000041, A018819, A158704, A158705, A225620, A231204, A372688.

Table[Length[Union[Total[2^(#-1)]&/@IntegerPartitions[n]]],{n,0,15}]

cp's OEIS Frontend

A277899 a(n) = A097249(A260443(n)).

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A277896 a(n) = the least k > n for which A048675(k) = A048675(n), 0 if no such number exists (when n is a power of 2).

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A277903 a(n) = the least k such that A000123(k) >= n.

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A277904 Irregular table: row n (n >= 0) is obtained by listing numbers 0 .. A018819(n)-1.

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A373120 Number of distinct possible binary ranks of integer partitions of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1).

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Mathematica