A255131 -id:A255131 - cp's OEIS frontend

A260740 a(n) = n minus the number of positive squares needed to sum to n using the greedy algorithm: a(n) = n - A053610(n).

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).

A277487 a(n) = number of primes 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, 1, 0, 0, 1, 2, 1, 1, 0, 2, 1, 2, 0, 3, 2, 0, 3, 0, 2, 0, 1, 4, 2, 3, 2, 4, 2, 0, 3, 3, 2, 5, 3, 4, 3, 3, 3, 2, 4, 2, 2, 4, 3, 3, 3, 6, 3, 1, 3, 4, 2, 6, 3, 3, 2, 5, 5, 5, 5, 4, 3, 7, 4, 4, 6, 4, 2, 4, 6, 5, 5, 5, 4, 7, 4, 4, 7, 4, 0, 5, 6, 7, 4, 4, 9, 4, 5, 2, 6, 6, 7, 11, 3, 6, 4, 9, 5, 7, 7, 7, 6, 8, 8, 7, 6, 4, 6, 5, 7, 8, 5, 9, 8, 8, 5, 12, 7, 5, 6

Offset: 1

Antti Karttunen, Nov 08 2016

nonn

Number of primes on row n of A276574, after the initial zero-row.
Note how for the most n in range 1..10000, a(n) < A277486(n), even though for the most n in the same range A277890(n) < A277891(n). In range n=1..10000, there are only 209 cases where a(n) >= A277486(n).
On the other hand, when a(n) is compared to A277488(n), there is no such marked bias.

			For n=3, starting from k = ((3+1)^2)-1, and iterating k -> A255131(k), yields 15 -> 11 -> 8, where the iteration stops as the next lower number one less than a square has been reached. Of these numbers only 11 is a prime, thus a(3) = 1.

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

Cf. A000290, A002828, A010052, A276574, A277486, A277488, A277890, A277891, A278167 (partial sums).

Cf. A277888.

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
A277487(n) = { my(orgk = ((n+1)^2)-1); my(k = orgk, s = 0); while(((k == orgk) || !issquare(1+k)), s = s + if(isprime(k),1,0); k = k - A002828(k)); s; };
for(n=1, 10000, write("b277487.txt", n, " ", A277487(n)));

(define (A277487 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 k)))))))

a(n) <= A277891(n).

A276574 The infinite trunk of least squares beanstalk with reversed subsections.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Sep 07 2016

			This can be viewed as an irregular table, where after 0, each row has A260734(n) = 1, 2, 2, 4, 4, 5, 5, 7, ... terms:
0;
3;
8, 6;
15, 11;
24, 21, 18, 16;
35, 32, 30, 27;
48, 45, 43, 40, 38;
63, 59, 56, 53, 51;
80, 78, 75, 72, 70, 67, 64;
99, 96, 93, 90, 88, 85, 83;
120, 117, 115, 112, 108, 105, 102;
...
Each row begins with (n^2)-1 (see A005563), and each successive term is obtained by subtracting A002828(k) from the previous term k, until ((n-1)^2)-1 would be encountered, which is not listed second time (as it already occurs as the first term of the previous row), but instead, the current row is finished and the next row is started with the term ((n+1)^2)-1.

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

Cf. A005563 (left edge), A277023 (right edge).
Cf. A000196, A000290, A002828, A010052, A255131, A260734, A262689, A276572.
Used to construct A276573.
Cf. A277015 (tells which rows end with squares, listed in A277016).

(definec (A276574 n) (cond ((zero? n) n) ((= 1 n) 3) (else (let ((maybe_next (A255131 (A276574 (- 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, let k = A255131(a(n-1)). If k+1 is not a square, then a(n) = k, otherwise a(n) = A000290(2+A000196(k+1)) - 1.

Example section added and the formula rewritten to a simpler form (which is now correct) - Antti Karttunen, Oct 16 2016

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).

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.

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).

A278496 a(n) = A000196(A278494(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 25 2016

nonn

Each n occurs A278495(n) times.

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

Cf. A000196, A002828, A255131, A278494, A278495.

(define (A278496 n) (A000196 (A278494 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))).

cp's OEIS Frontend

A260740 a(n) = n minus the number of positive squares needed to sum to n using the greedy algorithm: a(n) = n - A053610(n).

Views

Author

Keywords

Links

Crossrefs

Formula

A277487 a(n) = number of primes encountered before reaching (n^2)-1 when starting from k = ((n+1)^2)-1 and iterating map k -> k - A002828(k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Scheme

Formula

A276574 The infinite trunk of least squares beanstalk with reversed subsections.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Scheme

Formula

Extensions

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Scheme

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Scheme

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

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).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Scheme

Formula

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).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Scheme

Formula