A155043 -id:A155043 - cp's OEIS frontend

A263266 Inverse permutation to A263265: a(0) = 0; for n >= 1, a(n) = A263259(n) + A263260(A155043(n)-1) - 1.

0, 1, 2, 3, 4, 6, 5, 11, 7, 8, 9, 12, 10, 15, 13, 16, 17, 20, 14, 26, 18, 27, 19, 30, 21, 22, 23, 34, 24, 38, 25, 46, 28, 47, 29, 50, 39, 54, 31, 55, 32, 59, 33, 67, 35, 60, 36, 68, 37, 40, 41, 74, 42, 81, 43, 82, 44, 88, 48, 95, 45, 103, 51, 96, 97, 108, 52, 114, 56, 115, 57, 120, 49, 128, 61, 121, 62, 138, 63, 145, 64, 69, 70, 75, 53

Offset: 0

Antti Karttunen, Nov 24 2015

Antti Karttunen, Table of n, a(n) for n = 0..131071
A. Karttunen, Graph plotted with OEIS Plot script up to n=10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A263265.
Cf. A263281 (fixed points).
Cf. A155043, A263259, A263260.
Differs from A263268 for the first time at n=38, where a(38) = 31, while A263268(38) = 32.

(define (A263266 n) (if (zero? n) n (+ -1 (A263259 n) (A263260 (- (A155043 n) 1)))))

a(0) = 0; for n >= 1, a(n) = A263259(n) + A263260(A155043(n)-1) - 1.

A262505 Distance between the positions of the first and the last occurrence of n in A155043: a(n) = A262503(n) - A261089(n).

Original entry on oeis.org

0, 1, 3, 7, 11, 9, 13, 15, 19, 21, 31, 41, 49, 59, 67, 77, 81, 87, 55, 57, 65, 59, 63, 67, 77, 79, 89, 37, 27, 33, 37, 47, 55, 65, 67, 71, 85, 99, 65, 67, 77, 89, 93, 103, 95, 105, 109, 117, 121, 131, 135, 115, 117, 119, 133, 133, 153, 133, 143, 141, 147, 159, 175, 181, 201, 205, 223, 239, 245, 249, 259, 275, 269, 271, 273, 271, 289, 303, 77

Offset: 0

Antti Karttunen, Sep 25 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..110880

Cf. A155043, A259934, A261089, A261103, A262503, A262506.
Cf. A262508 (positions of zeros).
Cf. A262507 (the number of occurrences of n in A155043).

(define (A262505 n) (- (A262503 n) (A261089 n)))

a(n) = A262503(n) - A261089(n).

a(n) = A261103(n) + A262506(n).

A263078 a(n) = greatest k for which A155043(n+k) < A155043(n); a(n) = A263077(n)-n.

Original entry on oeis.org

-1, -2, -1, -2, 1, -4, 5, -2, -3, -4, 1, -6, 5, -2, 3, 2, 5, -6, 11, -2, 9, -4, 11, -2, -3, -4, 15, -6, 19, -8, 29, -2, 27, -4, 37, 12, 47, -4, 45, -6, 55, -8, 65, -2, 51, -4, 61, -6, -1, -2, 69, -4, 79, -6, 77, -8, 83, 2, 81, -12, 79, 10, 77, 76, 75, 6, 73, 16, 71, 14, 69, -12, 67, 22, 65, 20, 73, 18, 77, 16, 27, 26, 37, -12, 35, 34, 45, 20, 51, 18, 49, 40, 47, 26, 45, -12, 43, 42, 41, 40, 39, 30

Offset: 1

Antti Karttunen, Oct 09 2015

sign

			For n=1 we have A049820(1) = 0, thus A155043(1) = 1, and 0 is the only (and thus the largest) number from which zero can be reached with less steps (namely in zero steps, A155043(0) = 0), thus a(1) = 0 - 1 = -1.
For n=7, we have A155043(7) = 4 [as A049820(7) = 5, A049820(5) = 3, A049820(3) = 1, A049820(1) = 0], but there exists x=12 for which we have A049820(12) = 6, A049820(6) = 2, A049820(2) = 0, and this is the largest x such that A155043(x) < A155043(7), thus a(7) = 12 - 7 = 5.

Antti Karttunen, Table of n, a(n) for n = 1..124340

Cf. A155043, A261089, A262503, A262909, A263077.

Cf. A263079 (indices of the negative terms), A263080 (of the positive terms).

a[0] = 0; a[n_] := a[n] = 1 + a[n - DivisorSigma[0, n]]; Table[k = 3 n;
While[a@ k >= a@ n, k--]; k - n, {n, 102}] (* Michael De Vlieger, Oct 13 2015 *)

A263078 = n -> A263077(n) - n;
for(n=1,124340,write("b263078.txt",n," ",A263078(n)));
\\ Other code as in A263077

a(n) = A263077(n)-n.

A263079 Numbers n for which there does not exist any x > n such that A155043(x) < A155043(n).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 18, 20, 22, 24, 25, 26, 28, 30, 32, 34, 38, 40, 42, 44, 46, 48, 49, 50, 52, 54, 56, 60, 72, 84, 96, 104, 108, 120, 128, 132, 136, 140, 142, 144, 150, 152, 156, 160, 162, 168, 170, 180, 182, 184, 186, 188, 190, 192, 194, 198, 200, 204, 208, 210, 216, 220, 225, 228, 240, 248, 252, 260, 264, 276, 280, 288, 296, 300, 308, 312, 320, 328, 340, 352, 360

Offset: 1

Antti Karttunen, Oct 09 2015

nonn

Numbers n for which A263077(n) < n.
Numbers n for which A263078(n) is negative.
Numbers n at which point A155043(n) is the greatest lower bound for the rest of its terms from A155043(n) onward.

			1 is present because A049820(1) = 0, thus A155043(1) = 1, while all the larger numbers require at least the same number of steps to reach zero.

Antti Karttunen, Table of n, a(n) for n = 1..9165

Cf. A155043, A261089, A262503, A263077, A263078.

Complement: A263080.

a[0] = 0; a[n_] := a[n] = 1 + a[n - DivisorSigma[0, n]]; Position[Table[k = 3 n; While[a@ k >= a@ n, k--]; k - n, {n, 200}], Integer?Negative] // Flatten (* _Michael De Vlieger, Oct 13 2015 *)

n=0; i=0; while(n < 124340, n++; if((A263077(n) < n), i++; write("b263079.txt",i," ",n)));
\\ Other code as in A263077.

;; With Antti Karttunen's IntSeq-library.
(define A263079 (MATCHING-POS 1 1 (COMPOSE negative? A263078)))

A263259 a(n) = number of integers k <= n for which A155043(k) = A155043(n).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 1, 2, 3, 4, 2, 5, 1, 3, 2, 3, 1, 4, 1, 4, 2, 5, 1, 2, 3, 4, 1, 5, 1, 6, 1, 3, 2, 4, 1, 2, 1, 2, 2, 3, 1, 4, 1, 2, 2, 3, 2, 4, 3, 4, 1, 5, 1, 6, 2, 7, 1, 3, 1, 8, 1, 2, 2, 3, 1, 3, 1, 3, 2, 4, 1, 4, 1, 3, 2, 4, 1, 5, 1, 6, 3, 4, 2, 4, 3, 4, 3, 5, 2, 6, 3, 4, 4, 5, 5, 5, 2, 4, 3, 6, 2, 5, 3, 7, 3, 5, 4, 8, 3, 6, 4, 7, 2, 7, 3, 8, 4, 4, 2, 7

Offset: 0

Antti Karttunen, Nov 24 2015

nonn

a(n) = one-based index of n in row A155043(n) of table A263265.

Antti Karttunen, Table of n, a(n) for n = 0..12333

Cf. A000005, A155043, A261089, A262503, A262507, A263089, A263265, A263279.

Other identities. For all n >= 0:
a(A261089(n)) = 1.
a(A262503(n)) = A262507(n).
A263265(A155043(n), a(n)) = n.

A263080 Numbers n for which there exists x > n such that A155043(x) < A155043(n); numbers n for which A263078(n) is positive.

Original entry on oeis.org

5, 7, 11, 13, 15, 16, 17, 19, 21, 23, 27, 29, 31, 33, 35, 36, 37, 39, 41, 43, 45, 47, 51, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 102, 103, 105, 106, 107, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121

Offset: 1

Antti Karttunen, Oct 09 2015

nonn

			5 is present, because if we start iterating A049820 from it as: A049820(5) = 3, A049820(3) = 1, A049820(1) = 0, we get a path of length 3 to reach zero, thus A155043(5) = 3. On the other hand, if we start from 6, the path is one step shorter: A049820(6) = 2, A049820(2) = 0 [i.e., A155043(6) = 2], thus there exists a number larger than 5 which results a shorter path to zero.

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

Cf. A155043, A261089, A262503, A263077, A263078.

Complement: A263079.

a[0] = 0; a[n_] := a[n] = 1 + a[n - DivisorSigma[0, n]]; Position[Table[k = 3 n; While[a@ k >= a@ n, k--]; k - n, {n, 121}], Integer?Positive] // Flatten (* _Michael De Vlieger, Oct 13 2015 *)

n=0; i=0; while(i < 10000, n++; if((A263077(n) > n), i++; write("b263080.txt",i," ",n)));
\\ Other code as in A263077.

;; With Antti Karttunen's IntSeq-library.
(define A263080 (MATCHING-POS 1 1 (COMPOSE positive? A263078)))

A263081 a(n) = largest k for which A155043(k) < A262508(n); a(n) = A262509(n) + A262909(n).

Original entry on oeis.org

124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 24684000, 24684000, 24684000, 24684000, 24684000, 24684000, 24684000

Offset: 1

Antti Karttunen, Oct 09 2015

nonn

a(n) = largest k for which A155043(k) < A155043(A262509(n)).
If a(n) > A262509(n) then it must be a leaf (see comments in A262909 for why). Particularly, we have A045765(40722) = 124340, A045765(8191770) = 24684000.
Terms of sequence (together with the corresponding values in A262508) give particularly clean values for the boundaries that are used for example in the C++-program which computes A262896.

Antti Karttunen, Table of n, a(n) for n = 1..68

Cf. A045765, A155043, A262508, A262509, A262896, A262909, A263077, A263083.

(define (A263081 n) (+ (A262509 n) (A262909 n)))

a(n) = A263077(A262509(n)).

a(n) = A262509(n) + A262909(n).

A263280 a(n) = A262507(n) - A263279(n); number of integers k > A259934(n) for which A155043(k) = n = A155043(A259934(n)).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Nov 24 2015

nonn

a(n) gives the distance of position of A259934(n) at row n of table A263265 from its right edge.

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A155043, A259934, A262507, A263265, A263279.

Cf. also A262506.

(define (A263280 n) (- (A262507 n) (A263279 n)))

a(n) = A262507(n) - A263279(n).

A262521 Numbers where A262520 takes a negative value; numbers n for which A155043(2n) > A155043(2n + 1).

Original entry on oeis.org

112, 113, 544, 545, 684, 760, 930, 1306, 1514, 1522, 1624, 1625, 2048, 2386, 2389, 2393, 2399, 2402, 2818, 2827, 2966, 2969, 3280, 3281, 3791, 3797, 3878, 4324, 4326, 4328, 4330, 4331, 4333, 4334, 4336, 4340, 4342, 4346, 4352, 5513, 5515, 5519, 5521, 5527, 5531, 5724, 5726, 6050, 6161, 6165, 6167, 6168, 6169, 6172, 6173, 6176, 6177, 6179, 6181, 6183, 6184, 6185, 6272

Offset: 1

Antti Karttunen, Oct 02 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1004

Cf. A000005, A049820, A155043, A262518, A262519, A262520.

A262909 a(n) = greatest k such that A155043(k+A262509(n)) < A155043(A262509(n)).

Original entry on oeis.org

5197, 5193, 5177, 5115, 5113, 4419, 4417, 4259, 4245, 4243, 4239, 4059, 4047, 3991, 3941, 3633, 3593, 3449, 3445, 3437, 3423, 3421, 2897, 2789, 2517, 2261, 2079, 2077, 2067, 2063, 1527, 1379, 1135, 1127, 1117, 1103, 1083, 575, 23457, 23451, 21689, 21671, 20241, 19003, 18977, 18649, 18063, 18019, 14853, 14159, 13659, 12707, 11681, 10993, 10991, 10297, 10281, 9151, 9149, 9145, 9111, 8897, 8535, 8147, 6835, 6813, 5539, 5537

Offset: 1

Antti Karttunen, Oct 09 2015

nonn

a(n) = largest k such that A155043(k+A262509(n)) < A262508(n).
There might occur also negative terms, but no zeros.
For all terms a(n) > 0, a(n)+A262509(n) = A263081(n) is by necessity one of the leaves (A045765) in the tree generated by edge-relation A049820(child) = parent. See also comments in A262908.

Cf. A000005, A049820, A045765, A262508, A262509, A262908, A263078, A263081.

a(n) = A263078(A262509(n)).
a(n) = A263081(n) - A262509(n).
Other identities. For all n >= 1:
a(n) >= A262908(n).

cp's OEIS Frontend

A263266 Inverse permutation to A263265: a(0) = 0; for n >= 1, a(n) = A263259(n) + A263260(A155043(n)-1) - 1.

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A262505 Distance between the positions of the first and the last occurrence of n in A155043: a(n) = A262503(n) - A261089(n).

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A263078 a(n) = greatest k for which A155043(n+k) < A155043(n); a(n) = A263077(n)-n.

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A263079 Numbers n for which there does not exist any x > n such that A155043(x) < A155043(n).

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A263259 a(n) = number of integers k <= n for which A155043(k) = A155043(n).

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A263080 Numbers n for which there exists x > n such that A155043(x) < A155043(n); numbers n for which A263078(n) is positive.

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A263081 a(n) = largest k for which A155043(k) < A262508(n); a(n) = A262509(n) + A262909(n).

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A263280 a(n) = A262507(n) - A263279(n); number of integers k > A259934(n) for which A155043(k) = n = A155043(A259934(n)).

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