cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Antti Karttunen, Sep 07 2016

Keywords

Links

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.

Programs

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.]

Extensions

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

Views

Author

Antti Karttunen, Nov 25 2016

Keywords

Examples

			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.

Links

Crossrefs

Cf. A002828, A255131, A276573.
Cf. A278490 (positions of zeros), A278489 (positions of nonzeros), A278491 (positions of 4's).

Programs

  • 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)))))

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.)

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

Views

Author

Antti Karttunen, Nov 25 2016

Keywords

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

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.

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

Views

Author

Antti Karttunen, Nov 25 2016, with additional comments Nov 28 2016

Keywords

Comments

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.)

Links

Crossrefs

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.
Showing 1-4 of 4 results.

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