A278216 -id:A278216 - cp's OEIS frontend

A278516 Number of children the n-th node (counted from the root 0) has in the infinite trunk of least squares beanstalk tree: a(n) = A278216(A276573(n)).

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

Offset: 0

Antti Karttunen, Nov 28 2016

nonn

Naturally, all terms must be > 0 and <= 4.

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

Cf. A002828, A276573, A278216, A278518.

(define (A278516 n) (A278216 (A276573 n)))

a(n) = A278216(A276573(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

A278489 Nonleaves in the tree defined by edge relation A255131(child) = parent, the least squares beanstalk.

Original entry on oeis.org

0, 3, 6, 8, 9, 11, 15, 16, 18, 19, 21, 24, 27, 30, 32, 35, 38, 39, 40, 41, 43, 45, 48, 50, 51, 53, 54, 56, 59, 63, 64, 66, 67, 70, 71, 72, 73, 74, 75, 78, 80, 81, 83, 85, 87, 88, 90, 91, 93, 95, 96, 99, 102, 104, 105, 107, 108, 111, 112, 114, 115, 117, 120, 123, 126, 128, 129, 130, 131, 134, 135, 136, 137, 138, 139, 143, 144

Offset: 0

Antti Karttunen, Nov 25 2016

nonn

Numbers n for which there exists at least one such integer k that k - A002828(k) = n, in other words, numbers n such that either A002828(1+n) is 1 or A002828(2+n) is 2 or A002828(3+n) is 3 or A002828(4+n) is 4, as the maximum value that A002828 may obtain is 4.

Indexing starts from zero, because a(0)=0 is a special case in this sequence.

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

Complement: A278490.
Positions of nonzeros in A278216.
Cf. A276573 (the infinite trunk of the tree, is a subsequence).
Cf. A278491 (another subsequence).
Cf. A002828, A255131.

A278490 Leaves in the tree defined by edge relation A255131(child) = parent, the least squares beanstalk.

Original entry on oeis.org

1, 2, 4, 5, 7, 10, 12, 13, 14, 17, 20, 22, 23, 25, 26, 28, 29, 31, 33, 34, 36, 37, 42, 44, 46, 47, 49, 52, 55, 57, 58, 60, 61, 62, 65, 68, 69, 76, 77, 79, 82, 84, 86, 89, 92, 94, 97, 98, 100, 101, 103, 106, 109, 110, 113, 116, 118, 119, 121, 122, 124, 125, 127, 132, 133, 140, 141, 142, 145, 148, 150, 153, 154, 156, 157

Offset: 1

Antti Karttunen, Nov 25 2016

nonn

Numbers n for which there are no solutions to k - A002828(k) = n for any k, in other words, numbers n such that (A002828(1+n) <> 1) and (A002828(2+n) <> 2) and (A002828(3+n) <> 3) and (A002828(4+n) <> 4), as the maximum value that A002828 may obtain is 4.

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

Complement: A278489.
Positions of zeros in A278216.
Cf. A278494 (primes in this sequence).

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.

A278495 a(n) = number of primes in range [n^2, (n+1)^2] that are leaves in "the least squares beanstalk" tree.

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 1, 1, 2, 4, 1, 2, 1, 3, 2, 4, 3, 3, 3, 5, 3, 2, 2, 4, 4, 4, 4, 3, 4, 4, 4, 4, 2, 3, 3, 2, 4, 2, 5, 4, 6, 3, 5, 4, 5, 5, 4, 6, 3, 3, 6, 8, 4, 5, 3, 5, 5, 5, 4, 6, 6, 7, 5, 5, 7, 6, 8, 8, 8, 8, 5, 5, 5, 8, 7, 7, 7, 3, 13, 5, 8, 6, 8, 7, 8, 5, 14, 7, 8, 8, 10, 7, 5, 8, 6, 7, 6, 9, 4, 10, 4, 9, 8, 6, 8, 8, 8, 6, 10, 11, 13, 9

Offset: 1

Antti Karttunen, Nov 25 2016

nonn

Number of terms of A278494 in range [n^2, (n+1)^2], where A278494 are primes p for which there does not exist any such integer k that k - A002828(k) = p.
In other words, number of primes p in range [n^2, (n+1)^2] for which (A002828(1+p) <> 1) and (A002828(2+p) <> 2) and (A002828(3+p) <> 3) and (A002828(4+p) <> 4).
Conjecture: a(n) > 0 for all n >= 1.
Similar guesses are easy to make but hard to prove. I also conjecture that A277487(n) > 0 for all n > 80, and that both A277486(n) > 0 and A277488(n) > 0 for all n > 7. If any of these claims were proved true, it would imply the proof of Legendre's conjecture as well. See also comments in A014085 and sequences A277888 & A278487.

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

Cf. A000290, A002828, A010051, A010052, A014085 (an upper bound), A278216, A278494 (primes that are counted), A278496.

Cf. also A277486, A277487, A277488.

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
A278495(n) = { my(s = 0); for(k=(n^2),(n+1)^2, if((isprime(k) && (A002828(1+k) <> 1) && (A002828(2+k) <> 2) && (A002828(3+k) <> 3) && (A002828(4+k) <> 4)),s = s+1) ); s; };
for(n=1, 10000, write("b278495.txt", n, " ", A278495(n)));

(define (A278495 n) (let loop ((k (+ -1 (A000290 (+ 1 n)))) (s 0)) (if (= 1 (A010052 k)) s (loop (- k 1) (+ s (* (A010051 k) (if (zero? (A278216 k)) 1 0)))))))

For all n >= 1, a(n) <= A014085(n).

cp's OEIS Frontend

A278516 Number of children the n-th node (counted from the root 0) has in the infinite trunk of least squares beanstalk tree: a(n) = A278216(A276573(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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A278489 Nonleaves in the tree defined by edge relation A255131(child) = parent, the least squares beanstalk.

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A278490 Leaves in the tree defined by edge relation A255131(child) = parent, the least squares beanstalk.

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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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A278495 a(n) = number of primes in range [n^2, (n+1)^2] that are leaves in "the least squares beanstalk" tree.

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