A260740 -id:A260740 - cp's OEIS frontend

A255131 n minus the least number of squares that add up to n: a(n) = n - A002828(n).

0, 0, 0, 0, 3, 3, 3, 3, 6, 8, 8, 8, 9, 11, 11, 11, 15, 15, 16, 16, 18, 18, 19, 19, 21, 24, 24, 24, 24, 27, 27, 27, 30, 30, 32, 32, 35, 35, 35, 35, 38, 39, 39, 40, 41, 43, 43, 43, 45, 48, 48, 48, 50, 51, 51, 51, 53, 54, 56, 56, 56, 59, 59, 59, 63, 63, 63, 64, 66, 66, 67, 67, 70, 71, 72, 72, 73, 74, 75, 75, 78, 80, 80, 80, 81

Offset: 0

Antti Karttunen, Feb 24 2015

nonn

The associated beanstalk-sequence starts from a(0) as: 0, 3, 6, 8, 11, 15, 16, 18, 21, ... (A276573).

			a(0) = 0, because no squares are needed for an empty sum, and 0 - 0 = 0.
a(3) = 0, because 3 cannot be represented as a sum of less than three squares (1+1+1), and 3 - 3 = 0.
a(4) = 3, because 4 can be represented as a sum of just one square (namely 4 itself), and 4 - 1 = 3.

Robert Israel, Table of n, a(n) for n = 0..10000
Discussion on the SeqFan mailing list

Subsequence: A005563.
Cf. A000290, A002828, A062535, A260731, A260732, A260733, A260734, A262689, A276573.
Cf. also A011371, A236840, A260740.

f:= proc(n) local F, x;
   if issqr(n) then return n-1 fi;
   if nops(select(t -> t[1] mod 4 = 3 and t[2]::odd, ifactors(n)[2])) = 0 then return n-2 fi;
   x:= n/4^floor(padic:-ordp(n, 2)/2);
   if x mod 8 = 7 then n-4 else n-3 fi
end proc:
f(0):= 0:
map(f, [$0..100]); # Robert Israel, Mar 27 2018

{0}~Join~Table[n - (If[First@ # > 0, 1, Length[First@ Split@ #] + 1] &@ SquaresR[Range@ 4, n]), {n, 84}] (* Michael De Vlieger, Sep 08 2016, after Harvey P. Dale at A002828 *)

a(n) = n - A002828(n).

a(n) = A260740(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.

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).

A261225 n minus the number of positive cubes needed to sum to n using the greedy algorithm: a(n) = n - A055401(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 7, 7, 7, 7, 7, 7, 7, 7, 14, 14, 14, 14, 14, 14, 14, 14, 21, 21, 21, 26, 26, 26, 26, 26, 26, 26, 26, 33, 33, 33, 33, 33, 33, 33, 33, 40, 40, 40, 40, 40, 40, 40, 40, 47, 47, 47, 52, 52, 52, 52, 52, 52, 52, 52, 59, 59, 63, 63, 63, 63, 63, 63, 63, 63, 70, 70, 70, 70, 70, 70, 70, 70, 77, 77, 77, 77, 77, 77, 77, 77, 84, 84, 84, 89

Offset: 0

Antti Karttunen, Aug 16 2015

nonn

			a(8) = 7, because when the greedy algorithm partitions 8 into cubes, it first finds 8 (= 2*2*2), thus A055401(8) = 1, and 8-1 = 7.

Antti Karttunen, Table of n, a(n) for n = 0..12167

Cf. A048762, A055401, A261226, A261227, A261228, A261229.

Cf. also A260740.

a(n) = n - A055401(n).
As a recurrence:
a(0) = 0; for n >= 1, a(n) = -1 + A048762(n) + a(n-A048762(n)). [Where A048762(n) gives the largest cube <= n.]

A276583 The infinite trunk of greedy squares beanstalk: The only infinite sequence such that a(n-1) = a(n) - number of squares that sum to a(n) with greedy algorithm (A053610).

Original entry on oeis.org

0, 3, 6, 8, 11, 15, 18, 21, 24, 27, 32, 35, 38, 43, 48, 51, 56, 59, 63, 66, 71, 74, 78, 80, 83, 88, 91, 95, 99, 102, 107, 110, 114, 117, 120, 123, 128, 131, 135, 138, 143, 146, 151, 154, 158, 161, 164, 168, 171, 176, 179, 183, 186, 192, 195, 198, 203, 206, 210, 213, 219, 224, 227, 232, 235, 239, 242, 248, 251, 255

Offset: 0

Antti Karttunen, Sep 07 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10095

Cf. A053610, A260740, A276582, A276584, A276585 (first differences).

Cf. also A179016, A259934, A276573, A276613, A276623 for similar constructions.

(define (A276583 n) (A276584 (A276582 n)))

a(n) = A276584(A276582(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)).

A276584 The infinite trunk of greedy squares beanstalk with reversed subsections.

Original entry on oeis.org

0, 3, 8, 6, 15, 11, 24, 21, 18, 35, 32, 27, 48, 43, 38, 63, 59, 56, 51, 80, 78, 74, 71, 66, 99, 95, 91, 88, 83, 120, 117, 114, 110, 107, 102, 143, 138, 135, 131, 128, 123, 168, 164, 161, 158, 154, 151, 146, 195, 192, 186, 183, 179, 176, 171, 224, 219, 213, 210, 206, 203, 198, 255, 251, 248, 242, 239, 235, 232, 227

Offset: 0

Antti Karttunen, Sep 07 2016 and Sep 09 2016

nonn

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

Cf. A000196, A000290, A010052, A053610, A260740, A276582, A276583.

(definec (A276584 n) (cond ((zero? n) n) ((= 1 n) 3) (else (let ((maybe_next (A260740 (A276584 (- n 1))))) (if (zero? (A010052 (+ 1 maybe_next))) maybe_next (+ -1 (A000290 (+ 2 (A000196 (+ 1 maybe_next))))))))))

a(0) = 0; a(1) = 3; for n > 1, if A260740(a(n-1))+1 is not a square, then a(n) = A260740(a(n-1)), otherwise a(n) = A000290(2+A000196(A260740(a(n-1)))) - 1.

cp's OEIS Frontend

A255131 n minus the least number of squares that add up to n: a(n) = n - A002828(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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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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A261225 n minus the number of positive cubes needed to sum to n using the greedy algorithm: a(n) = n - A055401(n).

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A276583 The infinite trunk of greedy squares beanstalk: The only infinite sequence such that a(n-1) = a(n) - number of squares that sum to a(n) with greedy algorithm (A053610).

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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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A276584 The infinite trunk of greedy squares beanstalk with reversed subsections.

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