A260733 -id:A260733 - cp's OEIS frontend

A276572 Simple self-inverse permutation of natural numbers: after a(0)=0, list each block of A260734(n) numbers in reverse order, from A260732(n) to A260733(1+n).

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

Offset: 0

Antti Karttunen, Sep 07 2016

nonn

Maps between A276573 and A276574.

Antti Karttunen, Table of n, a(n) for n = 0..10028
Index entries for sequences that are permutations of the natural numbers

Cf. A260732, A260733, A260734, A276573, A276574.

f[n_] := NestWhileList[# - (If[First@ # > 0, 1, Length[First@ Split@ #] + 1] &@ SquaresR[Range@ 4, #]) &, n^2, # != 0 &]; t = Table[Table[n, {Length[#] - 1 &@ NestWhileList[# - (If[First@ # > 0, 1, Length[First@ Split@ #] + 1] &@ SquaresR[Range@ 4, #]) &, ((n + 1)^2) - 1, # != (n^2) - 1 &]}], {n, 20}] // Flatten ; {0}~Join~Table[Length@ f@ t[[n]] - 1 + Length@ f[t[[n]] + 1] - n - 2, {n, 81}] (* Michael De Vlieger, Sep 08 2016 *)

(define (A276572 n) (if (zero? n) n (+ (- (A260733 (+ 1 (A276571 n))) n) (A260732 (A276571 n)))))

a(0) = 0; for n >= 1, a(n) = (A260733(1+A276571(n))-n)+A260732(A276571(n)).

A276573 The infinite trunk of least squares beanstalk: The only infinite sequence such that a(0) = 0 and a(n-1) = a(n) - least number of squares (A002828) that sum to a(n).

Original entry on oeis.org

0, 3, 6, 8, 11, 15, 16, 18, 21, 24, 27, 30, 32, 35, 38, 40, 43, 45, 48, 51, 53, 56, 59, 63, 64, 67, 70, 72, 75, 78, 80, 83, 85, 88, 90, 93, 96, 99, 102, 105, 108, 112, 115, 117, 120, 123, 126, 128, 131, 134, 136, 139, 143, 144, 147, 149, 152, 155, 158, 160, 162, 165, 168, 171, 173, 176, 179, 183, 186, 189, 192, 195

Offset: 0

Antti Karttunen, Sep 07 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10028

Cf. A002828, A005563, A255131, A260731, A260733, A262689, A276572, A276574, A276575 (first differences), A277016 (squares present), A277015 (their square roots), A277888 (primes), A278486 (numbers one more than a prime), A278265, A278487, A278488, A278491 (another subsequence), A278497, A278498, A278499, A278513, A278516, A278517, A278518, A278519, A278521, A278522.
Cf. A277890 & A277891 (number of even and odd terms in each range. The latter seem to be slightly more numerous), A277889.
Positions of nonzero terms in A278515.
Subsequence of A278489, no common terms with A278490.
Cf. also A179016, A259934, A276583, A276613, A276623 for similar constructions.

(define (A276573 n) (A276574 (A276572 n)))

a(n) = A276574(A276572(n)).
Other identities and observations. For all n >= 0:
A260731(a(n)) = n.
a(A260733(n+1)) = A005563(n).
A278517(n) <= a(n) <= A278519(n).
A010873(a(n)) = A278499(n). [Terms reduced modulo 4.]
A010877(a(n)) = A278488(n). [modulo 8.]
A046523(a(n)) = A278497(n). [Least number with the same prime signature.]
A008683(a(n)) = A278513(n).
A065338(a(n)) = A278498(n).
A278509(a(n)) = A278265(n).
A278216(a(n)) = A278516(n). [Number of children the n-th node of the trunk has.]

Definition clarified and more identities added to the Formula section by Antti Karttunen, Nov 28 2016

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

Original entry on oeis.org

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).

A260731 a(n) = Number of steps to reach 0 starting from x=n and using the iterated process: x -> x - A002828(x), where A002828(x) = the least number of squares that add up to x.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 8, 8, 8, 8, 9, 10, 10, 10, 10, 11, 11, 11, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16, 17, 17, 17, 18, 19, 19, 19, 20, 20, 20, 20, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 27, 27, 28, 28, 28, 29, 29, 29, 30, 31, 31, 31, 32, 32, 32, 32, 33, 33, 34, 34, 34, 35, 35, 35, 36, 36, 37, 37, 38

Offset: 0

Antti Karttunen, Aug 12 2015

nonn

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

Cf. A002828, A255131, A260732, A260733, A260734.
Left inverse of A276573, A278517 and A278519. A278518(n) gives the number of times n occurs (run lengths).
Cf. also A261221.

A002828[n_] := Which[n == 0, 0, SquaresR[1, n] > 0, 1, SquaresR[2, n] > 0, 2, SquaresR[3, n] > 0, 3, True, 4]; a[0] = 0; a[n_] := a[n] = 1 + a[n - A002828[n]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 14 2016 *)

a(0) = 0; for >= 1, a(n) = 1 + A260731(A255131(n)).
From Antti Karttunen, Nov 28 2016: (Start)
For all n >= 0, a(A278517(n)) = a(A278519(n)) = a(A276573(n)) = n.
(End)

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).

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).

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.

A276575 After a(0)=0, the first differences of A276573.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Sep 07 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10028
Burkard Polster, Root 2 and the deadly Marching Squares, Mathologer video (2015)

Cf. A001541, A002828, A132592, A260731, A260733, A276573.

a(n) = A002828(A276573(n)).
a(0) = 0; for n >= 1, a(n) = A276573(n) - A276573(n-1).
Other identities.
For all n >= 1, a(A260731(A132592(n))) = a(A260733(A001541(n))) = 2. [This is implied by the fact observed in the Polster video. Of course 2's occur at other points too.]

A277891 a(n) = number of odd numbers encountered before reaching (n^2)-1 when starting from k = ((n+1)^2)-1 and iterating map k -> k - A002828(k).

Original entry on oeis.org

1, 0, 2, 1, 2, 2, 4, 2, 4, 3, 4, 4, 6, 5, 7, 6, 5, 6, 7, 7, 6, 6, 11, 9, 9, 9, 12, 9, 10, 9, 11, 11, 12, 11, 14, 13, 15, 12, 14, 14, 16, 14, 15, 13, 15, 17, 18, 17, 14, 17, 19, 18, 20, 17, 22, 19, 22, 20, 20, 22, 20, 22, 23, 22, 24, 25, 22, 22, 25, 26, 26, 25, 28, 24, 30, 26, 28, 29, 27, 27, 28, 32, 29, 28, 32, 32, 29, 31, 30, 29, 35, 33, 32, 32, 35, 34, 35, 36

Offset: 1

Antti Karttunen, Nov 08 2016

nonn

The starting point ((n+1)^2)-1 of the iteration is included if it is odd, but the ending point (n^2)-1 is never included in the count.
a(n) = number of odd numbers on row n of A276574, after the initial zero-row.
On the average, the odd terms in A276573 (A276574) seem to occur more frequently than the even terms. (The last point in range 1..10000 where a(n) <= A277890(n) is n=862). See also comments in A277487 and the plot of ratio a(n)/A277890(n), also the plot of A277889.

			For n=6, we start iterating from k = ((6+1)^2)-1 = 48, with k -> k - A002828(k), to obtain 48 -> 45 -> 43 -> 40 -> 38 before reaching 35 (which is 6^2 - 1, an ending point and thus not included in the count), and the only odd numbers before that were 45 and 43, thus a(6) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000
A. Karttunen, Ratio a(n)/A277890(n) drawn with OEIS Plot 2 tool

Cf. A000035, A000290, A002828, A010052, A077773, A260733, A260734, A276574, A277191, A277192, A277487, A277889, A277890.

istwo(n:int)=my(f); if(n<3, return(n>=0); ); f=factor(n>>valuation(n, 2)); for(i=1, #f[, 1], if(bitand(f[i, 2], 1)==1&&bitand(f[i, 1], 3)==3, return(0))); 1
isthree(n:int)=my(tmp=valuation(n, 2)); bitand(tmp, 1)||bitand(n>>tmp, 7)!=7
A002828(n)=if(issquare(n), !!n, if(istwo(n), 2, 4-isthree(n))) \\ From Charles R Greathouse IV, Jul 19 2011
A277891(n) = { my(orgk = ((n+1)^2)-1); my(k = orgk, s = 0); while(((k == orgk) || !issquare(1+k)), s = s + (k%2); k = k - A002828(k)); s; };
for(n=1, 10000, write("b277891.txt", n, " ", A277891(n)));

(define (A277891 n) (let ((org_k (- (A000290 (+ 1 n)) 1))) (let loop ((k org_k) (s 0)) (if (and (< k org_k) (= 1 (A010052 (+ 1 k)))) s (loop (- k (A002828 k)) (+ s (A000035 k)))))))

a(n) + A277890(n) = A260734(n).

a(n) >= A277487(n).

A278497 a(n) = Least number with the prime signature of A276573(n).

Original entry on oeis.org

2, 6, 8, 2, 6, 16, 12, 6, 24, 8, 30, 32, 6, 6, 24, 2, 12, 48, 6, 2, 24, 2, 12, 64, 2, 30, 72, 12, 30, 48, 2, 6, 24, 60, 6, 96, 12, 30, 30, 72, 48, 6, 12, 120, 6, 60, 128, 2, 6, 24, 2, 6, 144, 12, 2, 24, 6, 6, 96, 48, 30, 120, 12, 2, 48, 2, 6, 30, 24, 192, 30, 60, 72, 6, 12, 210, 6, 216, 6, 6, 96, 2, 30, 2, 12, 240, 32, 12, 24, 2, 30, 256, 6, 12, 120, 6, 120

Offset: 1

Antti Karttunen, Nov 25 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10028

Cf. A002828, A046523, A260733, A276573, A278160, A278498, A278499.
Cf. A277014 (gives the indices of squares).
Cf. also A278232.

(define (A278497 n) (A046523 (A276573 n)))

a(n) = A046523(A276573(n)).

For all n >= 1, a(A260733(1+n)) = A278160(n).

cp's OEIS Frontend

A276572 Simple self-inverse permutation of natural numbers: after a(0)=0, list each block of A260734(n) numbers in reverse order, from A260732(n) to A260733(1+n).

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A276573 The infinite trunk of least squares beanstalk: The only infinite sequence such that a(0) = 0 and a(n-1) = a(n) - least number of squares (A002828) that sum to a(n).

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

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A260731 a(n) = Number of steps to reach 0 starting from x=n and using the iterated process: x -> x - A002828(x), where A002828(x) = the least number of squares that add up to x.

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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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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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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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A276575 After a(0)=0, the first differences of A276573.

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A277891 a(n) = number of odd numbers encountered before reaching (n^2)-1 when starting from k = ((n+1)^2)-1 and iterating map k -> k - A002828(k).

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