A261222 -id:A261222 - 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)).

A260732 a(n) = number of steps needed to reach zero when starting from k = n^2 and repeatedly applying the map that replaces k with k - {the least number of squares (A002828) that add up to k}.

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 14, 19, 24, 31, 38, 45, 53, 63, 72, 82, 92, 105, 118, 132, 145, 160, 175, 191, 208, 225, 244, 263, 282, 302, 322, 344, 366, 389, 413, 438, 462, 488, 515, 540, 568, 597, 626, 655, 685, 716, 749, 782, 815, 849, 884, 919, 956, 992, 1031, 1068, 1106, 1146, 1188, 1228, 1270, 1312, 1355, 1397, 1442, 1487, 1532, 1580, 1625

Offset: 0

Antti Karttunen, Aug 12 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1024

Partial sums of A260734.
Essentially one more than A260733.
Cf. A002828, A260731.
Cf. also A261222.

Table[Length[#] - 1 &@ NestWhileList[# - (If[First@ # > 0, 1, Length[ First@ Split@ #] + 1] &@ SquaresR[Range@ 4, #]) &, n^2, # != 0 &], {n, 0, 68}] (* Michael De Vlieger, Sep 08 2016, after Harvey P. Dale at A002828 *)

a(n) = A260731(n^2).

For all n >= 1: a(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).

A261088 Number of steps needed to reach zero when starting from k = n^2 and repeatedly applying the map that replaces k with k - d(k), where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 10, 10, 19, 15, 19, 21, 24, 28, 39, 33, 53, 44, 49, 53, 60, 61, 69, 72, 79, 82, 92, 93, 117, 108, 115, 115, 140, 121, 174, 146, 205, 155, 233, 217, 267, 192, 295, 209, 225, 222, 238, 249, 267, 270, 299, 290, 336, 313, 373, 328, 411, 347, 451, 380, 486, 400, 534, 422, 447, 441, 460, 460, 511, 479, 496, 504, 545, 529, 602, 553, 579, 577, 626, 612, 681, 632, 747, 665, 796, 695

Offset: 0

Antti Karttunen, Sep 23 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..5792

Cf. A000005, A000290, A155043, A049820, A259934, A261085.

Cf. also A260732, A261222.

f[n_]:=Length[NestWhileList[#-DivisorSigma[0,#]&,n^2,#!= 0&]]-1;f/@Range[0,85] (* Ivan N. Ianakiev, Sep 25 2015 *)

allocatemem((2^31)+(2^30));
uplim = 2^25;
v155043 = vector(uplim);
v155043[1] = 1; v155043[2] = 1;
for(i=3, uplim, v155043[i] = 1 + v155043[i-numdiv(i)]; if(!(i%65536),print1(i,", ")););
A155043 = n -> if(!n,n,v155043[n]);
A261088 = n -> A155043(n^2);
for(n=0, 5792, write("b261088.txt", n, " ", A261088(n)));

(define (A261088 n) (A155043 (A000290 n)))

a(n) = A155043(A000290(n)) = A155043(n^2).

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.

A261224 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 - A053610(k), where A053610(k) = the number of positive squares that sum to k using the greedy algorithm.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 5, 5, 6, 6, 7, 7, 7, 8, 8, 9, 10, 10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 15, 15, 16, 17, 17, 18, 18, 19, 19, 19, 20, 21, 21, 21, 22, 22, 23, 23, 24, 24, 25, 26, 26, 27, 27, 28, 28, 28, 29, 30, 30, 31, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 37, 37, 38, 38, 39, 39, 39, 40, 41, 41, 42, 42, 42, 43, 43, 44, 44, 45, 45, 46

Offset: 1

Antti Karttunen, Aug 12 2015

nonn

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

First differences of both A261222 and A261223.
Cf. A000290, A053610, A260740, A261221.
Cf. also A260734, A261229.

Table[-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, 91}] (* Michael De Vlieger, Sep 08 2016, after Jud McCranie at A053610 *)

a(n) = A261221(((n+1)^2)-1) - A261221((n^2)-1). [The definition.]
Equally, for all n >= 1:
a(n) = A261221((n+1)^2) - A261221(n^2).
a(n) = A261222(n+1) - A261222(n).
a(n) = A261223(n+1) - A261223(n).

A261227 a(n) = number of steps to reach 0 when starting from k = n^3 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

0, 1, 2, 5, 11, 20, 34, 53, 78, 109, 147, 191, 245, 307, 378, 459, 550, 653, 768, 897, 1039, 1196, 1368, 1555, 1758, 1979, 2217, 2473, 2747, 3041, 3354, 3689, 4046, 4424, 4824, 5248, 5697, 6170, 6669, 7194, 7746, 8325, 8934, 9571, 10237, 10935, 11664, 12424, 13216, 14043, 14903, 15798, 16727, 17694, 18696, 19735, 20812, 21929, 23084, 24279, 25514

Offset: 0

Antti Karttunen, Aug 16 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..512

Essentially one more than A261228.
First differences: A261229.
Cf. A000578, A055401, A261225, A261226.
Cf. also A261222.

a(0) = 0, a(1) = 1; for n >= 2, a(n) = A261229(n-1) + a(n-1).

a(n) = A261226(n^3).

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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A260732 a(n) = number of steps needed to reach zero when starting from k = n^2 and repeatedly applying the map that replaces k with k - {the least number of squares (A002828) 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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A261088 Number of steps needed to reach zero when starting from k = n^2 and repeatedly applying the map that replaces k with k - d(k), where d(k) is the number of divisors of k (A000005).

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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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A261224 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 - 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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A261227 a(n) = number of steps to reach 0 when starting from k = n^3 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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