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: Krzysztof Ostrowski

Krzysztof Ostrowski's wiki page.

Krzysztof Ostrowski has authored 4 sequences.

A184997 Number of distinct remainders that are possible when a safe prime p is divided by n (for p > 2*n+1).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 5, 2, 3, 3, 9, 1, 11, 5, 3, 4, 15, 3, 17, 3, 5, 9, 21, 2, 15, 11, 9, 5, 27, 3, 29, 8, 9, 15, 15, 3, 35, 17, 11, 6, 39, 5, 41, 9, 9, 21, 45, 4, 35, 15, 15, 11, 51, 9, 27, 10, 17, 27, 57, 3, 59, 29, 15, 16, 33, 9, 65, 15, 21, 15, 69, 6, 71, 35, 15, 17, 45, 11, 77, 12
Offset: 1

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Author

Krzysztof Ostrowski, Apr 24 2011

Keywords

Comments

A number r could be a remainder of division p/n (for n > 0 and safe prime p > 2*n+1) if it satisfies two conditions:
1) r is coprime to n,
2) (r-1)/2 is coprime to n (assuming r-1 is even) or (n+r-1)/2 is coprime to n (assuming n+r-1 is even).
If one of these conditions isn't satisfied then either p or (p-1)/2 isn't a prime number.
If n1 and n2 are coprime then a(n1*n2) = a(n1)*a(n2), per the Chinese remainder theorem.

Examples

			a(60) = 3 as there are only three distinct remainders possible (23, 47 and 59) when dividing some safe prime p by 60. It's true for all safe primes except 5, 7 and 11.

Links

Crossrefs

Cf. A005385, A000010.

A189466 Number of superior highly composite numbers < 10^n.

Original entry on oeis.org

2, 4, 6, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 23, 24, 25, 25, 26, 27, 28, 28, 29, 30, 31, 32, 33, 33, 34, 34, 35, 36, 36, 37, 38, 38, 39, 39, 40, 41, 41, 42, 42, 43, 43, 44, 44, 46, 46, 46, 47, 47, 48, 49, 49, 50, 50, 51, 51, 52, 52, 53
Offset: 1

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Author

Krzysztof Ostrowski, Apr 22 2011

Keywords

Comments

Number of superior highly composite numbers (A002201) with at most n digits.

Examples

			a(2) = 4 since there are 4 superior highly composite numbers < 10^2 {2,6,12,60}

Links

Crossrefs

Cf. A000705, A002201, A112781.

A189394 Highly composite numbers whose number of divisors is also highly composite.

Original entry on oeis.org

1, 2, 6, 12, 60, 360, 1260, 2520, 5040, 55440, 277200, 720720, 3603600, 61261200, 2205403200, 293318625600, 6746328388800, 195643523275200
Offset: 1

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Author

Krzysztof Ostrowski, Apr 21 2011

Keywords

Comments

Both n and d(n) are highly composite numbers.
It is extremely likely that this sequence is complete. The highly composite numbers have a very special form. The number of divisors of a large HCN has a high power of 2 in its factorization -- which is not the form of an HCN. - T. D. Noe, Apr 21 2011
All but a(7) and a(12) are a multiple of the previous term: ratios a(n+1) / a(n) are (2, 3, 2, 5, 6, 7/2, 2, 2, 11, 5, 13/5, 5, 17, 36, 133, 23, 29, ...?). - M. F. Hasler, Jun 20 2022

Examples

			d(60) = 12; both 60 and 12 are highly composite numbers

Links

Crossrefs

Cf. A002182, A002183, A141320.

Programs

Extensions

Typo in a(15) corrected by Ben Beer, Jul 20 2016
Keywords fini and full, following Øverlier's thesis, added by Michel Marcus, May 25 2023

A188719 Number of safe primes < 10^n.

Original entry on oeis.org

2, 7, 25, 115, 670, 4324, 30657, 229568, 1775675, 14156112, 115500275, 960392564
Offset: 1

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Author

Krzysztof Ostrowski, Apr 09 2011

Keywords

Comments

Number of safe primes with at most n digits; or a(10^n).

Examples

			a(10^2) = a(100) = #{5, 7, 11, 23, 47, 59, 83} = 7

Links

Crossrefs

A156875, A006880, A005385

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