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: Brian Kehrig

Brian Kehrig's wiki page.

Brian Kehrig has authored 4 sequences.

A382681 Conjecturally, the numbers k (not multiples of 5) such that for all x >= 0, k*2^x has a '0' in its decimal expansion.

Original entry on oeis.org

7501221, 7508793, 10006109, 10625334, 12970254, 15002442, 15017586, 15685077, 17975049, 20012218, 20752359, 21250668, 22500771, 23501007, 24625029, 24875024, 25033207, 25034183, 25034771, 25940508, 29003907, 29057504, 29450021, 29590047, 29625044, 29850293, 30004884, 30035172, 30175941
Offset: 1

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Author

Brian Kehrig, Jun 02 2025

Keywords

Comments

If k is a multiple of 5, then k*2^x always ends with '0' for x >= 1. We exclude these trivial cases.
If k is in this sequence, then so is k*2^x for all x >= 1.
All terms up to 30175941 have been tested up to x=10^6.

Examples

			7501221*2^0 = 7501221 contains a '0'
7501221*2^1 = 15002442 contains a '0'
7501221*2^2 = 30004884 contains a '0'
7501221*2^3 = 60009768 contains a '0'
7501221*2^4 = 120019536 contains a '0'
7501221*2^5 = 240039072 contains a '0'
...
Conjecturally, all further numbers of the form 7501221*2^k also contain '0'. Thus, 7501221 is in the sequence.

Crossrefs

Cf. A007377, A027870, A130694.

A383618 Smallest prime gap whose first occurrence is >= 2^n.

Original entry on oeis.org

1, 4, 6, 6, 8, 8, 10, 16, 16, 16, 26, 30, 36, 38, 46, 46, 66, 74, 80, 94, 108, 116, 142, 156, 158, 166, 186, 200, 228, 254, 264, 294, 298, 334, 362, 388, 388, 422, 466, 488, 510, 536, 576, 590, 632, 676, 708, 764, 782, 796, 848, 926, 928, 968, 1006, 1048
Offset: 1

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Author

Brian Kehrig, May 02 2025

Keywords

Comments

In the case of a prime gap that straddles a power of 2, we take the lower end of the gap, as listed in A000230.

Examples

			a(3) = 6, because prime gaps of 1, 2, and 4 all appear before 2^3 = 8, but a prime gap of 6 does not occur until later.

Links

Crossrefs

Cf. A000230.

A364678 Maximum number of primes between consecutive multiples of n, as permitted by divisibility considerations.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 6, 6, 6, 5, 7, 7, 6, 7, 7, 7, 7, 8, 7, 8, 9, 8, 10, 8, 10, 10, 10, 11, 11, 11, 10, 11, 11, 11, 12, 12, 12, 12, 13, 12, 13, 14, 13, 13, 14, 14, 15, 15, 14, 15, 15, 15, 16, 15, 15, 16, 16, 17, 16, 17, 18, 18, 18, 18, 18, 17, 19, 19, 19, 19, 20, 20, 19, 19, 20, 21, 21
Offset: 1

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Author

Brian Kehrig, Aug 24 2023

Keywords

Comments

Alternatively: a(n) = the maximum number of elements of an admissible k-tuple strictly contained in (0,n) such that all elements are relatively prime to n. Recall that an admissible tuple is defined as a tuple of integers with the property that all primes p have at least one residue class that has no intersection with the tuple.
For n > 1, we have a(n) <= A023193(n-1), with equality if (but not only if) n is prime or a power of 2. The smallest n for which it is not an equality is n=14.
Conjecture 1: Every nonnegative integer appears in this sequence.
Conjecture 2: For all n, there is an infinitude of k's such that there are a(n) primes between n*k and n*(k+1).
Conjecture 2 resembles the k-tuples conjecture a.k.a. the first Hardy-Littlewood conjecture, although it is not the same.
A notable value is a(35) = 8. Compare with A000010(210) = 48. This says that between any two consecutive multiples of 210 the 48 values that are not divisible by 2, 3, 5 or 7 are equally distributed between 6 equal divisions of 210; that is, 8 are in the interval [0, 34], 8 in the interval [35, 69], etc. - Peter Munn, Feb 16 2024

Examples

			Between two multiples of 15 (n and n+15), only n+1, n+2, n+4, n+7, n+8, n+11, n+13, and n+14 could possibly be prime based on divisibility by 3 and 5. However, 4 of these are even and 4 are odd, so at most 4 of them can be prime. Thus, a(15)=4.

Links

Crossrefs

Cf. A000010, A023193.
Multiples of n following which the maximum number of primes occur for particular n: A005097 (2), A144769 (3), A123986 (4), A056956 (6), A007811 (10), A123985 (12), A309871 (18).

Programs

  • Python
    # see Links section

A361723 Numbers k such that there are 18 primes between 100*k and 100*k + 99.

Original entry on oeis.org

1228537713709, 23352869714018, 28703237474266, 144785865481702, 161394923966449, 168975708209638, 174748809066898, 207552241231357, 278215179205531, 312303328909720, 592248982143877, 812939886634531, 939100782752014, 983930290209021, 1111161494544274
Offset: 1

Views

Author

Brian Kehrig, Mar 21 2023

Keywords

Comments

There are A261571(18) = 948729 possible patterns for centuries having 18 primes.

Examples

			1228537713709 is in the sequence because there are 18 primes between 122853771370900 and 122853771370999: 122853771370900 + x, where x is one of (1, 3, 7, 19, 21, 27, 31, 33, 37, 49, 51, 61, 69, 73, 87, 91, 97, or 99).

Links

  • Brian Kehrig, Table of n, a(n) for n = 1..39 (terms up to 10^16)
  • Note (Apr 24 2024): An older version of the b-file missed a(33) and a(38). The present b-file is correct.

Crossrefs

Cf. A038822 (number of primes between 100n and 100n+99), A186311 (first occurrences).
Cf. A181098 (no primes), A186393-A186408 (1 to 16 primes), A186509 (17 primes).
Cf. A261571 (number of patterns for centuries with n primes).

Programs

  • PARI
    isok(k) = sum(i=0, 99, isprime(100*k + i)) == 18; \\ Michel Marcus, Mar 23 2023

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