A002828 -id:A002828 - cp's OEIS frontend

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

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

Offset: 1

T. D. Noe, Jan 19 2011

nonn

By Lagrange's four square theorem, the minimal number of squares required to represent a number is 4 or less. See A141490 for the numbers k that have n minimal representations.

			27 has the following representations as the sum of 4 or fewer squares: 1+1+25, 9+9+9, and 1+1+9+16. The minimal number of squares is 3 and there are 2 such representations.  Hence a(27)=2.

T. D. Noe, Table of n, a(n) for n = 1..1000

Table[r=PowersRepresentations[n,4,2]; Sort[Tally[4-Count[#,0]& /@ r]][[1,2]], {n,100}]

A277488 a(n) = number of integers one less than a prime 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

0, 1, 0, 2, 1, 1, 0, 3, 2, 3, 2, 1, 1, 2, 1, 3, 2, 3, 1, 2, 4, 4, 1, 2, 3, 2, 2, 3, 4, 3, 3, 4, 3, 4, 3, 2, 4, 5, 5, 4, 4, 3, 4, 5, 4, 4, 3, 4, 6, 4, 6, 8, 4, 4, 3, 5, 7, 3, 7, 2, 6, 6, 4, 7, 7, 4, 5, 7, 4, 8, 6, 4, 4, 3, 6, 11, 4, 5, 5, 9, 6, 3, 6, 7, 6, 9, 9, 8, 11, 6, 5, 5, 7, 8, 7, 7, 5, 8, 9, 5, 7, 6, 5, 6, 7, 6, 8, 9, 6, 9, 6, 15, 8, 10, 9, 7, 10, 6, 6, 10

Offset: 1

Antti Karttunen, Nov 08 2016

nonn

Only 325 cases in range n=1..10000 where a(n) >= A277486(n). See also comments in A277487.

			For n=6, we start iterating from k = ((6+1)^2)-1 = 48, and then 48 - A002828(48) = 45, 45 - A002828(45) = 43, 43 - A002828(43) = 40, 40 - A002828(40) = 38, and 38 - A002828(38) = 35 (which is 6^2 - 1), and when we add one to each, only 41 is prime, thus a(6) = 1.

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

Cf. A000290, A010051, A010052, A276574, A277486, A277487, A277890, A278168 (partial sums).

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
A277488(n) = { my(orgk = ((n+1)^2)-1); my(k = orgk, s = 0); while(((k == orgk) || !issquare(1+k)), s = s + if(isprime(1+k),1,0); k = k - A002828(k)); s; };
for(n=1, 10000, write("b277488.txt", n, " ", A277488(n)));

(define (A277488 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 (A010051 (+ 1 k))))))))

For all n >= 1, a(n) <= A277890(n).

A278167 a(n) = number of primes 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, 1, 2, 2, 2, 3, 5, 6, 7, 7, 9, 10, 12, 12, 15, 17, 17, 20, 20, 22, 22, 23, 27, 29, 32, 34, 38, 40, 40, 43, 46, 48, 53, 56, 60, 63, 66, 69, 71, 75, 77, 79, 83, 86, 89, 92, 98, 101, 102, 105, 109, 111, 117, 120, 123, 125, 130, 135, 140, 145, 149, 152, 159, 163, 167, 173, 177, 179, 183, 189, 194, 199, 204, 208, 215, 219, 223, 230, 234

Offset: 1

Antti Karttunen, Nov 13 2016

nonn

a(n) = number of terms of A277888 less than (n+1)^2.

			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. Of these numbers only 11 and 3 are primes, thus a(4) = 2.

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

Partial sums of A277487.

Cf. A002828, A255131, A278166, A278168, A277888.

(definec (A278167 n) (if (= 1 n) (A277487 n) (+ (A277487 n) (A278167 (- n 1)))))

a(1) = A277487(1); for n > 1, a(n) = A277487(n) + a(n-1).

A159629 Slowest increasing sequence beginning with a(1)=4 such that A002828(a(n)) = A002828(n).

Original entry on oeis.org

4, 5, 6, 9, 10, 11, 15, 17, 25, 26, 27, 30, 32, 33, 39, 49, 50, 52, 54, 58, 59, 62, 63, 66, 81, 82, 83, 87, 89, 91, 92, 97, 99, 101, 102, 121, 122, 123, 124, 125, 128, 129, 131, 132, 136, 138, 143, 147, 169, 170, 171, 173, 178, 179, 183, 184, 186, 193, 195, 199, 200, 201, 207

Offset: 1

Vladimir Shevelev, Apr 17 2009, May 04 2009

nonn

Conjecture: For every m>2 there exists a minimum index N(m) such that the minimal increasing recursive sequence S_m(n) beginning with m^2 with the condition A002828(S_m(n)) = A002828(n) coincides with a(n) for all n>N.

V. Shevelev, Several results on sequences which are similar to the positive integers arXiv:0904.2101 [math.NT], 2009.

Cf. A002828, A159619, A159615, A007814, A004760, A159559, A159560.

a2828[n_] := Which[SquaresR[1, n]>0, 1, SquaresR[2, n]>0, 2, SquaresR[3, n] > 0, 3, True, 4];
a[1] = 4; a[n_] := a[n] = For[k = a[n-1]+1, True, k++, If[a2828[k] == a2828[n], Return[k]]];
Array[a, 63] (* Jean-François Alcover, Jul 28 2018 *)

a(n+1) = min { l > a(n) : A002828(l) = A002828(n+1) }.

137 replaced by 136, extended by R. J. Mathar, Sep 17 2009

A277192 Number of integers k in range [n^2, ((n+1)^2)-1] for which the least number of squares that add up to k (A002828) is even.

Original entry on oeis.org

1, 1, 3, 3, 4, 6, 6, 8, 8, 9, 11, 10, 13, 12, 13, 16, 15, 16, 17, 19, 17, 22, 20, 21, 23, 24, 23, 25, 26, 26, 28, 30, 28, 30, 31, 31, 31, 34, 34, 35, 35, 37, 37, 40, 38, 41, 42, 40, 44, 43, 44, 45, 44, 46, 45, 50, 50, 49, 48, 54, 53, 52, 52, 57, 55, 56, 57, 58, 58, 60, 60, 58, 65, 61, 64, 66, 64, 65, 66, 68, 69, 68, 70, 69

Offset: 0

Antti Karttunen, Oct 04 2016

nonn

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

Cf. A000035, A000290, A002828, A077773, A277191, A277194.

(define (A277192 n) (add (lambda (i) (- 1 (A000035 (A002828 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)))))))

a(n) = Sum_{i=n^2 .. ((n+1)^2)-1} (1-A000035(A002828(i))).
For all n >= 0, A277191(n) + a(n) = 2n+1.
For n >= 1, a(n) = A077773(n) + A277194(n).

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).

A278491 After a(0)=0, numbers n such that (A002828(1+n) = 1) and (A002828(4+n) = 4).

Original entry on oeis.org

0, 3, 24, 35, 99, 120, 195, 323, 440, 483, 675, 728, 899, 1155, 1368, 1443, 1763, 1848, 2115, 2499, 2808, 2915, 3363, 3480, 3843, 4355, 4760, 4899, 5475, 5624, 6083, 6723, 7224, 7395, 8099, 8280, 8835, 9603, 10200, 10403, 11235, 11448, 12099, 12995, 13688, 13923, 14883, 15128, 15875, 16899, 17688, 17955, 19043, 19320, 20163

Offset: 0

Antti Karttunen, Nov 26 2016

nonn

The definition implies that after 0 these are also all numbers n such that (A002828(1+n) = 1), (A002828(2+n) = 2), (A002828(3+n) = 3) and (A002828(4+n) = 4).
Because A002828 obtains value 1 only at squares, every term must be one less than a square.
In the terms of tree defined by edge relation A255131(child) = parent, ("the least squares beanstalk"), these numbers are the nodes with four children (maximum possible).
Either of the above facts implies that this is a subsequence of A276573.
Indexing starts from zero, because a(0)=0 is a special case in this sequence, as it is only number which is its own child in the least squares beanstalk tree.

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

Cf. A002828, A273324, A278216.

Subsequence of A005563, A276573 and A278489.

\\ (For a more intelligent way to generate the terms, check Altug Alkan's PARI-code for A273324).
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
isA278491(n) = (!n || ((A002828(1+n) == 1) && (A002828(4+n) == 4)));
i=0; n=0; while(i <= 10000, if(isA278491(n), write("b278491.txt", i, " ", n); i++); n++ );

;; With Antti Karttunen's IntSeq-library.
(define A278491 (MATCHING-POS 0 0 (lambda (n) (= 4 (A278216 n)))))

a(0) = 0, and for n >= 1, a(n) = A273324(n)^2 - 1.

A278494 Primes p for which there does not exist any such integer k that k - A002828(k) = p.

Original entry on oeis.org

2, 5, 7, 13, 17, 23, 29, 31, 37, 47, 61, 79, 89, 97, 101, 103, 109, 113, 127, 157, 167, 193, 197, 199, 223, 229, 241, 257, 269, 271, 281, 293, 313, 317, 337, 353, 359, 383, 389, 397, 401, 409, 421, 433, 439, 449, 461, 463, 487, 509, 541, 569, 577, 593, 601, 607, 631, 647, 653, 673, 677, 709, 719, 727, 751, 761, 769, 773, 797

Offset: 1

Antti Karttunen, Nov 25 2016, with additional comments Nov 28 2016

nonn

Primes that are leaves in the tree defined by edge relation parent = A255131(child), "the least squares beanstalk".
Primes p such that (A002828(1+p) <> 1), (A002828(2+p) <> 2), (A002828(3+p) <> 3) and (A002828(4+p) <> 4).
See comments in A278495 which gives the count of these primes in each range [n^2, (n+1)^2].
This is a subsequence of A045352 as no prime of the form 8n+3 ever occurs in this sequence. This stems from a more general fact that A278490 contains no numbers of the form 8n+3, because A002828(8n+7) = 4 for all n. (See A004215.)

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

Intersection of A000040 and A278490.
No common terms with A277888, some common terms with A278487.
Subsequence of A045352.
Cf. A002828, A004215, A072401, A255131, A278495, A278496.
Cf. also A263091.

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

Original entry on oeis.org

0, 1, 1, 1, 4, 4, 4, 4, 4, 9, 9, 9, 4, 9, 9, 9, 16, 16, 9, 9, 16, 16, 9, 9, 16, 25, 25, 25, 25, 25, 25, 25, 16, 25, 25, 25, 36, 36, 36, 36, 36, 25, 25, 25, 36, 36, 36, 36, 16, 49, 49, 49, 36, 49, 49, 49, 36, 49, 49, 49, 49, 36, 49, 49, 64, 64, 64, 49, 64, 64, 36, 49, 36, 64, 49, 49, 36, 64, 49, 49, 64, 81, 81, 81, 64, 81, 81, 81, 36, 64, 81, 81, 81, 64, 81, 81, 64, 81, 49, 81, 100

Offset: 0

Antti Karttunen, Oct 03 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65536
A. Karttunen, Ratio a(n)/A048760(n) drawn with the help of OEIS Plot2-script

Cf. A000290, A002828, A262689, A262678.

Cf. also A048760.

(define (A262690 n) (A000290 (A262689 n)))

a(n) = A000290(A262689(n)).
Other identities. For all n >= 0:
A262678(n) = n - a(n).

A277191 Number of integers k in range [n^2, ((n+1)^2)-1] for which the least number of squares that add up to k (A002828) is odd.

Original entry on oeis.org

0, 2, 2, 4, 5, 5, 7, 7, 9, 10, 10, 13, 12, 15, 16, 15, 18, 19, 20, 20, 24, 21, 25, 26, 26, 27, 30, 30, 31, 33, 33, 33, 37, 37, 38, 40, 42, 41, 43, 44, 46, 46, 48, 47, 51, 50, 51, 55, 53, 56, 57, 58, 61, 61, 64, 61, 63, 66, 69, 65, 68, 71, 73, 70, 74, 75, 76, 77, 79, 79, 81, 85, 80, 86, 85, 85, 89, 90, 91, 91, 92, 95, 95, 98, 95, 100

Offset: 0

Antti Karttunen, Oct 04 2016

nonn

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

Cf. A000035, A000290, A002828, A277192.

After the initial zero, one more than A277193.

(define (A277191 n) (add (lambda (i) (A000035 (A002828 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)))))))

a(n) = Sum_{i=n^2 .. ((n+1)^2)-1} A000035(A002828(i)).
For all n >= 0, a(n) + A277192(n) = 2n+1.
For all n >= 1, a(n) = 1 + A277193(n).

cp's OEIS Frontend

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

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A277488 a(n) = number of integers one less than a prime 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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A278167 a(n) = number of primes encountered before reaching 0 when starting from k = ((n+1)^2)-1 and iterating map k -> k - A002828(k).

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A159629 Slowest increasing sequence beginning with a(1)=4 such that A002828(a(n)) = A002828(n).

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A277192 Number of integers k in range [n^2, ((n+1)^2)-1] for which the least number of squares that add up to k (A002828) is even.

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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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A278491 After a(0)=0, numbers n such that (A002828(1+n) = 1) and (A002828(4+n) = 4).

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A278494 Primes p for which there does not exist any such integer k that k - A002828(k) = p.

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