A278491 -id:A278491

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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).

+10
40

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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

FORMULA

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.]

PROG

(Scheme) (define (A276573 n) (A276574 (A276572 n)))

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Sep 07 2016

EXTENSIONS

Definition clarified and more identities added to the Formula section by Antti Karttunen, Nov 28 2016

STATUS

approved

A278216

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

+10
8

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

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

FORMULA

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.)

EXAMPLE

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.

PROG

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

CROSSREFS

Cf. A002828, A255131, A276573.

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

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 25 2016

STATUS

approved

A278489

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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.

LINKS

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

PROG

(Scheme, with Antti Karttunen's IntSeq-library)

(define A278489 (NONZERO-POS 0 0 A278216))

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 25 2016

STATUS

approved

A273324

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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.

LINKS

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

FORMULA

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

EXAMPLE

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

PROG

(PARI) isA004215(n) = {n\4^valuation(n, 4)%8==7}

lista(nn) = for(n=1, nn, if(isA004215(n^2+3), print1(n, ", ")));

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Altug Alkan, May 20 2016

STATUS

approved

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