A261224 -id:A261224 - cp's OEIS frontend

A276582 Simple self-inverse permutation of natural numbers: after a(0)=0, list each block of A261224(n) numbers in reverse order, from A261222(n) to A261223(1+n).

0, 1, 3, 2, 5, 4, 8, 7, 6, 11, 10, 9, 14, 13, 12, 18, 17, 16, 15, 23, 22, 21, 20, 19, 28, 27, 26, 25, 24, 34, 33, 32, 31, 30, 29, 40, 39, 38, 37, 36, 35, 47, 46, 45, 44, 43, 42, 41, 54, 53, 52, 51, 50, 49, 48, 61, 60, 59, 58, 57, 56, 55, 69, 68, 67, 66, 65, 64, 63, 62, 77, 76, 75, 74, 73, 72, 71, 70, 86, 85, 84, 83, 82, 81, 80, 79, 78

Offset: 0

Antti Karttunen, Sep 07 2016

nonn

Maps between A276583 and A276584.

Antti Karttunen, Table of n, a(n) for n = 0..9996
Index entries for sequences that are permutations of the natural numbers

Cf. A261222, A261223, A261224, A276581, A276583, A276584.

(define (A276582 n) (if (zero? n) n (+ (- (A261223 (+ 1 (A276581 n))) n) (A261222 (A276581 n)))))

a(0)=0; for n >= 1, a(n) = (A261223(1+A276581(n))- n)+A261222(A276581(n)).

A276581 After a(0)=0, each n occurs A261224(n) times.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Sep 07 2016

nonn

Auxiliary function for computing A276582 & A276583.

Antti Karttunen, Table of n, a(n) for n = 0..9996

Cf. A261223, A261224, A276582, A276583.

Table[Table[n, {-1 + Length@ NestWhileList[# - Block[{m = #, c = 1}, While[a = (# - Floor[Sqrt@ #]^2) &@ m; a != 0, c++; m = a]; c] &, ((n + 1)^2) - 1, # != n^2 - 1 &]}], {n, 0, 18}] // Flatten (* Michael De Vlieger, Sep 08 2016 *)

(define (A276581 n) (let loop ((k 0)) (if (>= (A261223 (+ 1 k)) n) k (loop (+ 1 k)))))

A260734 a(n) = number of steps needed to reach (n^2)-1 when starting from k = ((n+1)^2)-1 and repeatedly applying the map that replaces k with k - A002828(k), where A002828(k) = the least number of squares that add up to k.

Original entry on oeis.org

1, 2, 2, 4, 4, 5, 5, 7, 7, 7, 8, 10, 9, 10, 10, 13, 13, 14, 13, 15, 15, 16, 17, 17, 19, 19, 19, 20, 20, 22, 22, 23, 24, 25, 24, 26, 27, 25, 28, 29, 29, 29, 30, 31, 33, 33, 33, 34, 35, 35, 37, 36, 39, 37, 38, 40, 42, 40, 42, 42, 43, 42, 45, 45, 45, 48, 45, 49, 50, 50, 48, 53, 50, 51, 54, 52, 53, 54, 56, 56, 56, 58, 59, 59, 60, 60, 60, 61, 62, 62, 62, 65, 66, 66, 65

Offset: 1

Antti Karttunen, Aug 12 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024

First differences of both A260732 and A260733.
Cf. A000290, A002828, A255131.
Cf. also A261224.

Table[Length[#] - 1 &@ NestWhileList[# - (If[First@ # > 0, 1, Length[ First@ Split@ #] + 1] &@ SquaresR[Range@ 4, #]) &, ((n + 1)^2) - 1, # != (n^2) - 1 &], {n, 95}] (* Michael De Vlieger, Sep 08 2016, after Harvey P. Dale at A002828 *)

a(n) = A260731(((n+1)^2)-1) - A260731((n^2)-1). [The definition.]
Equally, for all n >= 1:
a(n) = A260731((n+1)^2) - A260731(n^2).
a(n) = A260732(n+1) - A260732(n).
a(n) = A260733(n+1) - A260733(n).

A260740 a(n) = n minus the number of positive squares needed to sum to n using the greedy algorithm: a(n) = n - A053610(n).

Original entry on oeis.org

0, 0, 0, 0, 3, 3, 3, 3, 6, 8, 8, 8, 8, 11, 11, 11, 15, 15, 15, 15, 18, 18, 18, 18, 21, 24, 24, 24, 24, 27, 27, 27, 27, 30, 32, 32, 35, 35, 35, 35, 38, 38, 38, 38, 41, 43, 43, 43, 43, 48, 48, 48, 48, 51, 51, 51, 51, 54, 56, 56, 56, 56, 59, 59, 63, 63, 63, 63, 66, 66, 66, 66, 69, 71, 71, 71, 71, 74, 74, 74, 78, 80

Offset: 0

Antti Karttunen, Aug 12 2015

nonn

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

Cf. A048760, A053610, A062535, A255131, A261221, A261222, A261223, A261224.

Cf. also A261225.

a(n) = n - A053610(n).
As a recurrence:
a(0) = 0; for n >= 1, a(n) = -1 + A048760(n) + a(n-A048760(n)). [Where A048760(n) gives the largest square <= n.]
Other identities. For all n >= 1:
a(n) = A255131(n) - A062535(n).

A261222 a(n) = number of steps to reach 0 when starting from k = n*n and repeatedly applying the map that replaces k with k - A053610(k), where A053610(k) = the number of positive squares that sum to k using the greedy algorithm.

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 12, 15, 19, 24, 29, 35, 41, 48, 55, 62, 70, 78, 87, 97, 107, 118, 129, 141, 153, 165, 178, 191, 205, 219, 234, 249, 265, 282, 299, 317, 335, 354, 373, 392, 412, 433, 454, 475, 497, 519, 542, 565, 589, 613, 638, 664, 690, 717, 744, 772, 800, 828, 857, 887, 917, 948, 979, 1010, 1042, 1074, 1107, 1140, 1174, 1208, 1243, 1278, 1314, 1351, 1388, 1426, 1464, 1503

Offset: 0

Antti Karttunen, Aug 12 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..6000

Essentially one more than A261223.
First differences: A261224.
Cf. A000290, A053610, A260740, A261221.
Cf. also A260732, A261227.

Table[-1 + Length@ NestWhileList[# - Block[{m = #, c = 1}, While[a = (# - Floor[Sqrt@ #]^2) &@ m; a != 0, c++; m = a]; c] &, n^2, # != 0 &], {n, 0, 77}] (* Michael De Vlieger, Sep 08 2016, after Jud McCranie at A053610 *)

a(n) = A261221(n^2).
Other identities. For all n >= 1:
a(n) = 1 + A261223(n).

A261223 a(n) = number of steps to reach 0 when starting from k = (n*n)-1 and repeatedly applying the map that replaces k with k - A053610(k), where A053610(k) = the number of positive squares that sum to k using the greedy algorithm.

Original entry on oeis.org

0, 1, 3, 5, 8, 11, 14, 18, 23, 28, 34, 40, 47, 54, 61, 69, 77, 86, 96, 106, 117, 128, 140, 152, 164, 177, 190, 204, 218, 233, 248, 264, 281, 298, 316, 334, 353, 372, 391, 411, 432, 453, 474, 496, 518, 541, 564, 588, 612, 637, 663, 689, 716, 743, 771, 799, 827, 856, 886, 916, 947, 978, 1009, 1041, 1073, 1106, 1139, 1173, 1207, 1242, 1277, 1313, 1350, 1387, 1425, 1463, 1502, 1541

Offset: 1

Antti Karttunen, Aug 12 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..6000

One less than A261222.
Cf. A053610, A260740, A261221, A261224.
Cf. also A260733, A261228.

Table[-2 + Length@ NestWhileList[# - Block[{m = #, c = 1}, While[a = (# - Floor[Sqrt@ #]^2) &@ m; a != 0, c++; m = a]; c] &, (n + 1)^2, # != 0 &], {n, 0, 77}] (* Michael De Vlieger, Sep 08 2016, after Jud McCranie at A053610 *)

a(n) = A261221((n^2)-1).

a(n) = A261222(n)-1.

A261229 a(n) = number of steps to reach (n^3)-1 when starting from k = ((n+1)^3)-1 and repeatedly applying the map that replaces k with k - A055401(k), where A055401(k) = the number of positive cubes needed to sum to k using the greedy algorithm.

Original entry on oeis.org

1, 3, 6, 9, 14, 19, 25, 31, 38, 44, 54, 62, 71, 81, 91, 103, 115, 129, 142, 157, 172, 187, 203, 221, 238, 256, 274, 294, 313, 335, 357, 378, 400, 424, 449, 473, 499, 525, 552, 579, 609, 637, 666, 698, 729, 760, 792, 827, 860, 895, 929, 967, 1002, 1039, 1077, 1117, 1155, 1195, 1235, 1278, 1318, 1361, 1404, 1448, 1492, 1538, 1583, 1631, 1677, 1725

Offset: 1

Antti Karttunen, Aug 16 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..512

First differences of both A261227 and A261228.
Cf. A000578, A055401, A261225, A261226.
Cf. also A261224.

a(n) = A261226(((n+1)^3)-1) - A261226((n^3)-1). [The definition.]
Equally, for all n >= 1:
a(n) = A261226((n+1)^3) - A261226(n^3).
a(n) = A261227(n+1) - A261227(n).
a(n) = A261228(n+1) - A261228(n).

A261221 a(n) = number of steps to reach 0 when starting from k = n and repeatedly applying the map that replaces k with k - A053610(k), where A053610(k) = the number of positive squares needed to sum to k using the greedy algorithm.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 23, 24, 24, 24, 24, 25, 25, 25, 25, 26, 26, 26, 26, 26, 27, 27, 27, 28, 28, 28, 29

Offset: 0

Antti Karttunen, Aug 12 2015

nonn

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

Cf. A260740, A261222, A261223, A261224.

Cf. also A260731, A261226.

a(0) = 0; for n >= 1, a(n) = 1 + a(A260740(n)).

cp's OEIS Frontend

A276582 Simple self-inverse permutation of natural numbers: after a(0)=0, list each block of A261224(n) numbers in reverse order, from A261222(n) to A261223(1+n).

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A276581 After a(0)=0, each n occurs A261224(n) times.

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A260734 a(n) = number of steps needed to reach (n^2)-1 when starting from k = ((n+1)^2)-1 and repeatedly applying the map that replaces k with k - A002828(k), where A002828(k) = the least number of squares that add up to k.

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A260740 a(n) = n minus the number of positive squares needed to sum to n using the greedy algorithm: a(n) = n - A053610(n).

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A261222 a(n) = number of steps to reach 0 when starting from k = n*n and repeatedly applying the map that replaces k with k - A053610(k), where A053610(k) = the number of positive squares that sum to k using the greedy algorithm.

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A261223 a(n) = number of steps to reach 0 when starting from k = (n*n)-1 and repeatedly applying the map that replaces k with k - A053610(k), where A053610(k) = the number of positive squares that sum to k using the greedy algorithm.

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Programs

Mathematica

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A261229 a(n) = number of steps to reach (n^3)-1 when starting from k = ((n+1)^3)-1 and repeatedly applying the map that replaces k with k - A055401(k), where A055401(k) = the number of positive cubes needed to sum to k using the greedy algorithm.

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A261221 a(n) = number of steps to reach 0 when starting from k = n and repeatedly applying the map that replaces k with k - A053610(k), where A053610(k) = the number of positive squares needed to sum to k using the greedy algorithm.

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