A278491 -id:A278491 - 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.)

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.

A273324 Integers n such that n^2 + 3 is the sum of 4 but no fewer nonzero squares.

Original entry on oeis.org

2, 5, 6, 10, 11, 14, 18, 21, 22, 26, 27, 30, 34, 37, 38, 42, 43, 46, 50, 53, 54, 58, 59, 62, 66, 69, 70, 74, 75, 78, 82, 85, 86, 90, 91, 94, 98, 101, 102, 106, 107, 110, 114, 117, 118, 122, 123, 126, 130, 133, 134, 138, 139, 142, 146, 149, 150, 154, 155, 158, 162, 165, 166, 170

Offset: 1

Altug Alkan, May 20 2016

If n^2 + k is a term of A004215, then the minimum positive value of k is 3, obviously.

See also the first differences (A278536) of this sequence.

			2 is in the sequence because 2^2 + 3 = 7 is a term of A004215.

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

Cf. A000196, A004215, A117950, A278491, A278536.

isA004215(n) = {n\4^valuation(n, 4)%8==7}
lista(nn) = for(n=1, nn, if(isA004215(n^2+3), print1(n, ", ")));

a(n) = A000196(1+A278491(n)). - Antti Karttunen, Nov 26 2016

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

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A273324 Integers n such that n^2 + 3 is the sum of 4 but no fewer nonzero squares.

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