A278489 -id:A278489 - cp's OEIS frontend

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).

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

A278216 Number of children that node n has in the tree defined by the edge relation A255131(child) = parent, "the least squares beanstalk".

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Nov 25 2016

nonn

			a(0) = 4 as 0 - A002828(0) = 0, 1 - A002828(1) = 0, 2 - A002828(2) = 0 and 3 - A002828(3) = 0. (But 4 - A002828(4) = 3.) Note that 0 is the only number which is its own child as 0 - A002828(0) = 0.

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

Cf. A002828, A255131, A276573.

Cf. A278490 (positions of zeros), A278489 (positions of nonzeros), A278491 (positions of 4's).

(define (A278216 n) (let loop ((s 0) (k (+ 4 n))) (if (< k n) s (loop (+ s (if (= n (A255131 k)) 1 0)) (- k 1)))))

a(n) = Sum_{i=0..4} [A002828(n+i) = i]. (Here [ ] is the Iverson bracket, giving as its result 1 only if A002828(n+i) is i, otherwise zero.)

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.

cp's OEIS Frontend

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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A278216 Number of children that node n has in the tree defined by the 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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