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.

User: Harry Altman

Harry Altman's wiki page.

Harry Altman has authored 7 sequences.

A354024 Terms m > 1 of A351467 such that A005245(m) == 1 (mod 3).

Original entry on oeis.org

4, 10, 12, 28, 30, 32, 36, 82, 84, 90, 96, 108, 244, 246, 252, 256, 270, 288, 324, 730, 732, 738, 756, 768, 810, 864, 972, 2188, 2190, 2196, 2214, 2268, 2304, 2430, 2592, 2916, 6562, 6564, 6570, 6588, 6642, 6804, 6912, 7290, 7776, 8748
Offset: 1

Views

Author

Harry Altman, May 14 2022

Keywords

Comments

m appears in this list if and only if it can be written as 2^p*3^r for p in {2,5,8} or as (3^r+1)3^s for r > 1.

Links

Crossrefs

Cf. A351467, A350723, A005245, A354023, A354025.

Extensions

a(46) corrected by David Radcliffe, Aug 04 2025

A354025 Terms m of A351467 such that A005245(m) == 2 (mod 3).

Original entry on oeis.org

2, 5, 6, 13, 14, 15, 16, 18, 37, 38, 39, 40, 42, 45, 48, 54, 109, 110, 111, 112, 114, 117, 120, 126, 128, 135, 144, 162, 325, 326, 327, 328, 330, 333, 336, 342, 351, 360, 378, 384, 405, 432, 486, 973, 974, 975, 976, 978, 981, 984, 990, 999
Offset: 1

Views

Author

Harry Altman, May 14 2022

Keywords

Comments

m appears in this list if and only if it can be written as 2^p*3^r for p in {1,4,7,10} or as 4(3^r+1)3^s (r>0) or 2(2*3^r+1)3^s or (4*3^r+1)3^s.

Links

Crossrefs

Cf. A351467, A350723, A005245, A354023, A354024.

A354023 Terms m of A351467 such that A005245(m) == 0 (mod 3).

Original entry on oeis.org

3, 7, 8, 9, 19, 20, 21, 24, 27, 55, 56, 57, 60, 63, 64, 72, 81, 163, 164, 165, 168, 171, 180, 189, 192, 216, 243, 487, 488, 489, 492, 495, 504, 512, 513, 540, 567, 576, 648, 729, 1459, 1460, 1461, 1464, 1467, 1476, 1485, 1512, 1536, 1539
Offset: 1

Views

Author

Harry Altman, May 14 2022

Keywords

Comments

m appears in this list if and only if m>1 and it can be written as 2^p*3^r for p in {0,3,6,9} or as 2(3^r+1)3^s (r>0) or (2*3^r+1)3^s.

Links

Crossrefs

Cf. A351467, A350723, A005245, A354024, A354025.

A351467 Numbers with integer defect at most 1; m such that A350723(m) <= 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 24, 27, 28, 30, 32, 36, 37, 38, 39, 40, 42, 45, 48, 54, 55, 56, 57, 60, 63, 64, 72, 81, 82, 84, 90, 96, 108, 109, 110, 111, 112, 114, 117, 120, 126, 128, 135, 144, 162, 163, 164, 165, 168, 171
Offset: 1

Views

Author

Harry Altman, Feb 12 2022

Keywords

Comments

m appears in this list if and only if it can be written as 2^p*3^r for p <= 10 or as 2^p*(2^q*3^r+1)*3^s for p+q <= 2.

References

  • Harry Altman, Integer Complexity: The Integer Defect, Moscow Journal of Combinatorics and Number Theory 8-3 (2019), 193-217.

Links

Crossrefs

Cf. A350723, A005245, A349983. Contains A000792 as a subset.

A350723 Integer defect of n: a(n) = A005245(n) - A349983(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 2, 0, 1, 1, 1, 1, 2, 0, 1, 1, 1, 2, 3, 1, 2, 2, 0, 1, 2, 1, 2, 1, 2, 2, 2, 0, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 3, 1, 2, 2, 2, 2, 3, 0, 1, 1, 1, 2, 3, 1, 2, 2, 1, 1, 2, 2, 3, 2, 3, 2, 3, 1
Offset: 1

Views

Author

Harry Altman, Feb 06 2022

Keywords

References

  • Harry Altman, Integer Complexity: The Integer Defect, Moscow Journal of Combinatorics and Number Theory 8-3 (2019), 193-217.

Links

Crossrefs

Difference between A005245 and A349983 (the latter being almost the same as A007600).

A349983 a(n) is the largest k such A000792(k) <= n.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11
Offset: 1

Views

Author

Harry Altman, Jan 08 2022

Keywords

References

  • Harry Altman, Integer Complexity: The Integer Defect, Moscow Journal of Combinatorics and Number Theory 8-3 (2019), 193-217.

Links

Crossrefs

Cf. A005245, A000792, A007600.

Formula

a(n) = max{ 3*floor(log_3(n)), 3*floor(log_3(n/2))+2, 3*floor(log_3(n/4))+4, 1 }.
a(n) = A007600(n)-1 except when n appears in A000792, in which case a(n) = A007600(n).

A230697 Length of shortest addition-multiplication chain for n.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 3, 3, 4, 4, 4, 5, 5, 4, 3, 4, 4, 5, 4, 5, 5, 6, 4, 4, 5, 4, 5, 5, 5, 6, 4, 5, 5, 5, 4, 5, 5, 5, 5, 6, 5, 6, 6, 5, 6, 6, 5, 5, 5, 6, 6, 6, 5, 6, 6, 6, 6, 7, 5, 6, 6, 6, 4, 5, 5, 6, 5, 6, 6, 7, 5, 6, 6, 5, 6, 6, 6, 7, 5, 4, 5, 5, 5, 6, 6, 6, 6
Offset: 1

Views

Author

Harry Altman, Oct 27 2013

Keywords

Examples

			A shortest addition-multiplication chain for 16 is (1,2,4,16), of length a(16) = 3.
A shortest addition-multiplication chain for 281 is (1,2,4,5,16,25,256,281), of length a(281) = 7. This is the first case where not all terms in some shortest chain are the sum or product of the immediately preceding term and one more preceding term. In other words, 281 is the smallest of the analog of non-Brauer numbers (A349044) for addition-multiplication chains. The next ones are 913, 941, 996, 997, 998, 1012, 1077, 1079, 1542, 1572, 1575, 1589, 1706, 1792, 1795, 1816, 1864, ... . - _Pontus von Brömssen_, May 02 2025

Links

Crossrefs

Cf. A003313, A005245, A128998, A173419, A349044, A354914, A383001, A383002.

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