A002828 -id:A002828 - cp's OEIS frontend

A277193 Number of integers k in range [n^2, ((n+1)^2)-1] for which 3 = the least number of squares that add up to k (A002828).

0, 1, 1, 3, 4, 4, 6, 6, 8, 9, 9, 12, 11, 14, 15, 14, 17, 18, 19, 19, 23, 20, 24, 25, 25, 26, 29, 29, 30, 32, 32, 32, 36, 36, 37, 39, 41, 40, 42, 43, 45, 45, 47, 46, 50, 49, 50, 54, 52, 55, 56, 57, 60, 60, 63, 60, 62, 65, 68, 64, 67, 70, 72, 69, 73, 74, 75, 76, 78, 78, 80, 84, 79, 85, 84, 84, 88, 89, 90, 90, 91, 94, 94, 97, 94, 99

Offset: 0

Antti Karttunen, Oct 04 2016

nonn

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

Cf. A000290, A002828, A010052, A072401, A077773, A229062, A277194.

After the initial zero, one less than A277191.

(define (A277193 n) (add (lambda (i) (* (- 1 (A010052 i)) (- 1 (A229062 i)) (- 1 (A072401 i)))) (A000290 n) (+ -1 (A000290 (+ 1 n)))))
;; Implements sum_{i=lowlim..uplim} intfun(i)
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))

Sum_{i=n^2 .. ((n+1)^2)-1} (1-A010052(i))*(1-A229062(i))*(1-A072401(i)).
Other identities. For all n >= 0:
1 + A077773(n) + a(n) + A277194(n) = 2n+1.
For n >= 1, a(n) = A277191(n)-1.

A277194 Number of integers k in range [n^2, ((n+1)^2)-1] for which 4 = the least number of squares that add up to k (A002828).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 04 2016

nonn

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

Cf. A000290, A002828, A010052, A072401, A077773, A229062, A277192, A277193.

(define (A277194 n) (add A072401 (A000290 n) (+ -1 (A000290 (+ 1 n)))))
;; Implements sum_{i=lowlim..uplim} intfun(i)
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))

a(n) = Sum_{i=n^2 .. ((n+1)^2)-1} A072401(i).
Other identities.
For n >= 0, 1 + A077773(n) + A277193(n) + a(n) = 2n+1.
For n >= 1, A277192(n) = A077773(n) + a(n).

A278166 a(n) = number of integers one more than a prime encountered before reaching 0 when starting from k = ((n+1)^2)-1 and iterating map k -> k - A002828(k).

Original entry on oeis.org

1, 3, 3, 5, 7, 9, 9, 11, 12, 14, 15, 18, 19, 22, 23, 26, 29, 31, 34, 37, 42, 46, 47, 51, 54, 58, 60, 64, 68, 70, 74, 78, 82, 85, 88, 92, 95, 99, 104, 109, 114, 118, 122, 128, 134, 137, 140, 149, 153, 158, 164, 173, 177, 183, 187, 191, 199, 205, 210, 217, 222, 231, 236, 241, 248, 256, 262, 273, 278, 287, 291, 298, 307, 316, 322, 332

Offset: 1

Antti Karttunen, Nov 13 2016

nonn

			For n=4, starting from k = ((4+1)^2)-1, and iterating k -> A255131(k), yields 24 -> 21 -> 18 -> 16 -> 15 -> 11 -> 8 -> 6 -> 3 before 0 is reached. Subtracting one from each gives [23, 20, 17, 15, 14, 10, 7, 5, 2], of which only 23, 17, 7, 5 and 2 are primes, thus a(4) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..10000
A. Karttunen, Ratio a(n)/A278168(n) plotted with OEIS Plot 2 tool

Partial sums of A277486.

Cf. A002828, A278167, A278168.

(definec (A278166 n) (if (= 1 n) (A277486 n) (+ (A277486 n) (A278166 (- n 1)))))

a(1) = A277486(1); for n > 1, a(n) = A277486(n) + a(n-1).

A278168 a(n) = number of integers one less than a prime encountered before reaching 0 when starting from k = ((n+1)^2)-1 and iterating map k -> k - A002828(k).

Original entry on oeis.org

0, 1, 1, 3, 4, 5, 5, 8, 10, 13, 15, 16, 17, 19, 20, 23, 25, 28, 29, 31, 35, 39, 40, 42, 45, 47, 49, 52, 56, 59, 62, 66, 69, 73, 76, 78, 82, 87, 92, 96, 100, 103, 107, 112, 116, 120, 123, 127, 133, 137, 143, 151, 155, 159, 162, 167, 174, 177, 184, 186, 192, 198, 202, 209, 216, 220, 225, 232, 236, 244, 250, 254, 258, 261, 267, 278, 282, 287, 292, 301

Offset: 1

Antti Karttunen, Nov 13 2016

nonn

			For n=4, starting from k = ((4+1)^2)-1, and iterating k -> A255131(k), yields 24 -> 21 -> 18 -> 16 -> 15 -> 11 -> 8 -> 6 -> 3 before 0 is reached. Subtracting one from each gives [25, 22, 19, 17, 16, 12, 9, 7, 4], of which only 19, 17, and 7 are primes, thus a(4) = 3.

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

Partial sums of A277488.

Cf. A002828, A255131, A278166, A278167.

(definec (A278168 n) (if (= 1 n) (A277488 n) (+ (A277488 n) (A278168 (- n 1)))))

a(1) = A277488(1); for n > 1, a(n) = A277488(n) + a(n-1).

A262678 a(n) = n - A262690(n), where A262690(n) = largest square k <= n such that A002828(n-k) = A002828(n)-1.

Original entry on oeis.org

0, 0, 1, 2, 0, 1, 2, 3, 4, 0, 1, 2, 8, 4, 5, 6, 0, 1, 9, 10, 4, 5, 13, 14, 8, 0, 1, 2, 3, 4, 5, 6, 16, 8, 9, 10, 0, 1, 2, 3, 4, 16, 17, 18, 8, 9, 10, 11, 32, 0, 1, 2, 16, 4, 5, 6, 20, 8, 9, 10, 11, 25, 13, 14, 0, 1, 2, 18, 4, 5, 34, 22, 36, 9, 25, 26, 40, 13, 29, 30, 16, 0, 1, 2, 20, 4, 5, 6, 52, 25, 9, 10, 11, 29, 13, 14, 32, 16, 49, 18, 0

Offset: 0

Antti Karttunen, Oct 03 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65536

Cf. A002828, A262689, A262690.

Cf. also A053186.

Scheme
```
(define (A262678 n) (- n (A262690 n)))
```

a(n) = n - A262690(n).

A284000 a(n) = a(a(n-A002828(n))) + a(n-a(n-A002828(n))) with a(1) = a(2) = a(3) = 1, where A002828(n) = the least number of squares that add up to n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 23 2017

nonn

Does a(n)/n converge to some value near 0.6 ? See for example: a(10) = 6, a(100) = 62, a(1000) = 604, a(10000) = 6050, a(100000) = 60414.

Antti Karttunen, Table of n, a(n) for n = 1..10001

Cf. A002828, A004001, A255131, A284006, A284007.

For n <= 3 a(n) = 1, else a(n) = a(a(A255131(n))) + a(n-a(A255131(n))).

A302694 a(n) is the smallest integer k such that A002828(k*n) = 3.

Original entry on oeis.org

3, 3, 1, 3, 6, 1, 2, 3, 3, 3, 1, 1, 6, 1, 2, 3, 3, 3, 1, 6, 1, 1, 2, 1, 3, 3, 1, 2, 6, 1, 2, 3, 1, 3, 1, 3, 6, 1, 2, 3, 3, 1, 1, 1, 6, 1, 2, 1, 3, 3, 1, 6, 6, 1, 2, 1, 1, 3, 1, 2, 6, 1, 2, 3, 3, 1, 1, 3, 1, 1, 2, 3, 3, 3, 1, 1, 1, 1, 2, 6, 3, 3, 1, 1, 6, 1, 2, 1, 3, 3

Offset: 1

Juri-Stepan Gerasimov, Apr 11 2018

nonn

All terms are squarefree.

			a(2) = 3 because A002828(1*2) = 2, A002828(2*2) = 1,..., and 3 is the smallest multiplier leading to A002828(3*2) = 3.

Cf. A002828, A005117, A007521, A007913, A302690.

A302694 := proc(n)
    for k from 1 do
        if A002828(k*n) = 3 then
            return k;
        end if;
    end do:
end proc:
seq(A302694(n),n=1..100) ; # R. J. Mathar, Apr 16 2018

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))); \\ A002828
a(n) = {my(m=1); while(a002828(m*n)!=3, m++); m; } \\ Michel Marcus, Apr 12 2018

a(n^2) = 3.

Conjecture: a(n) <= 6.

Name corrected and more terms added by Michel Marcus, Apr 12 2018

A320002 a(0) = 1; for n > 0, a(n) = A002828(n) * a(n-A002828(n)), where A002828(n) is the least number of squares that add up to n.

Original entry on oeis.org

1, 1, 2, 3, 3, 6, 9, 12, 18, 18, 36, 54, 54, 108, 162, 216, 216, 432, 432, 648, 864, 1296, 1944, 2592, 3888, 3888, 7776, 11664, 15552, 23328, 34992, 46656, 69984, 104976, 139968, 209952, 209952, 419904, 629856, 839808, 1259712, 1679616, 2519424, 3779136, 5038848, 7558272, 11337408, 15116544, 22674816, 22674816, 45349632

Offset: 0

Antti Karttunen, Nov 24 2018

nonn

Product of A002828(x) computed over all x encountered when map x -> x - A002828(x) is iterated, starting from x = n, until 0 is reached.
Sequence is monotonic because A255131 is monotonic.
All terms are 3-smooth (A003586).

Antti Karttunen, Table of n, a(n) for n = 0..121

Cf. A002828, A003586, A255131, A276573.

Cf. also A320008, A320009.

Nest[Append[#1, #2 #1[[-#2]] ] & @@ {#, If[First@ # > 0, 1, Length@ First@ Split@ # + 1] &@ SquaresR[Range@ 4, Length@ #]} &, {1}, 50] (* Michael De Vlieger, Nov 25 2018, after Harvey P. Dale at A002828 *)

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 A002828
A255131(n) = (n-A002828(n));
A320002(n) = { my(m=1, v); while(n>0, v = A002828(n); m *= v; n -= v); (m); };

A320002(n) = if(0==n,1,A002828(n)*A320002(n-A002828(n)));

a(0) = 1; for n > 0, a(n) = A002828(n) * a(n-A002828(n)).

A141490 Least number k having n representations as the sum of the minimal number of squares, A002828.

Original entry on oeis.org

1, 27, 28, 63, 103, 124, 135, 175, 207, 247, 255, 252, 327, 351, 412, 375, 511, 423, 543, 679, 540, 639, 687, 495, 567, 663, 759, 775, 847, 988, 783, 1111, 735, 1327, 855, 927, 1191, 999, 1308, 975, 1143, 1383, 1263, 1071, 1463, 1359, 1495, 1375, 1479

Offset: 1

Martin Renner, Jan 15 2011

nonn

That is, a(n) is the least k such that A180466(k) = n.

			a(1) = 1 since 1 = 1^2;
a(2) = 27 since 27 = 1^2 + 1^2 + 5^2 = 3^2 + 3^2 + 3^2 (2 ways);
a(3) = 28 since 28 = 1^2 + 1^2 + 1^2 +5^2 = 1^2 + 3^2 + 3^2 + 3^2 = 2^2 + 2^2 + 2^2 + 4^2 (3 ways).

Eric W. Weisstein, MathWorld: Waring's Problem

Cf. A180466 (number of representations of n as a minimal number of squares, A002828(n))

t=Table[r=PowersRepresentations[n, 4, 2]; Sort[Tally[4-Count[#, 0] & /@ r]][[1, 2]], {n, 1000}]; u=Union[t]; c=Complement[Range[Max[u]], u]; If[c == {}, mx=u[[-1]], mx=c[[1]]-1]; Flatten[Table[Position[t, n, 1, 1], {n, mx}]]

A175270 Numbers k such that the least number of squares that add up to k equals the least number of triangular numbers that add up to k. Equivalently, A002828(k) = A061336(k).

Original entry on oeis.org

0, 1, 2, 13, 14, 18, 19, 20, 29, 33, 34, 35, 36, 37, 44, 54, 58, 59, 61, 62, 65, 72, 73, 75, 77, 86, 90, 96, 97, 101, 106, 107, 118, 129, 130, 131, 134, 137, 138, 140, 146, 147, 148, 152, 155, 157, 158, 160, 161, 164, 166, 176, 179, 181, 184, 187, 193, 195, 200

Offset: 1

Ctibor O. Zizka, Mar 19 2010

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000217, A000290, A002828, A061336.

is2s(n)=my(f=factor(n>>valuation(n, 2))); for(i=1, #f~, if(bitand(f[i, 2], 1) && bitand(f[i, 1], 3)==3, return(0))); 1
is2t(n)=my(m9=n%9,f); if(m9==5 || m9==8, return(0)); is2s(4*n+1)
is(n)=my(o2=valuation(n, 2),f); if(n==0, return(1)); if(bitand(o2, 1)==0 && bitand(n>>o2, 7)==7, return(0)); if(issquare(n), return(ispolygonal(n,3))); if(ispolygonal(n,3), return(0)); is2t(n)==is2s(n) \\ Charles R Greathouse IV, Mar 17 2022

Data corrected and extended by Mohammed Yaseen, Mar 17 2022

cp's OEIS Frontend

A277193 Number of integers k in range [n^2, ((n+1)^2)-1] for which 3 = the least number of squares that add up to k (A002828).

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A277194 Number of integers k in range [n^2, ((n+1)^2)-1] for which 4 = the least number of squares that add up to k (A002828).

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A278166 a(n) = number of integers one more than a prime encountered before reaching 0 when starting from k = ((n+1)^2)-1 and iterating map k -> k - A002828(k).

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A278168 a(n) = number of integers one less than a prime encountered before reaching 0 when starting from k = ((n+1)^2)-1 and iterating map k -> k - A002828(k).

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A262678 a(n) = n - A262690(n), where A262690(n) = largest square k <= n such that A002828(n-k) = A002828(n)-1.

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A284000 a(n) = a(a(n-A002828(n))) + a(n-a(n-A002828(n))) with a(1) = a(2) = a(3) = 1, where A002828(n) = the least number of squares that add up to n.

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A302694 a(n) is the smallest integer k such that A002828(k*n) = 3.

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A320002 a(0) = 1; for n > 0, a(n) = A002828(n) * a(n-A002828(n)), where A002828(n) is the least number of squares that add up to n.

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