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 1-5 of 5 results.

A010881 Simple periodic sequence: n mod 12.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

The value of the rightmost digit in the base-12 representation of n. - Hieronymus Fischer, Jun 11 2007

Examples

			a(27) = 3 since 27 = 12*2+3.

Links

Crossrefs

Partial sums: A130490. Other related sequences A130481, A130482, A130483, A130484, A130485, A130486, A130487, A130488, A130489.

Programs

Formula

From Hieronymus Fischer, May 31 2007: (Start)
a(n) = n mod 12.
Complex representation: a(n) = (1/12)*(1-r^n)*Sum_{k=1..11} k*Product_{m=1..11, m<>k} (1-r^(n-m)) where r = exp(Pi/6*i) = (sqrt(3)+i)/2 and i = sqrt(-1).
Trigonometric representation: a(n) = (512/3)^2*(sin(n*Pi/12))^2*Sum_{k=1..11} k*Product_{m=1..11, m<>k} (sin((n-m)*Pi/12))^2.
G.f.: (Sum_{k=1..11} k*x^k)/(1-x^12).
G.f.: x*(11*x^12-12*x^11+1)/((1-x^12)*(1-x)^2). (End)
From Hieronymus Fischer, Jun 11 2007: (Start)
a(n) = (n mod 2)+2*(floor(n/2) mod 6) = A000035(n)+2*A010875(A004526(n)).
a(n) = (n mod 3)+3*(floor(n/3) mod 4) = A010872(n)+3*A010873(A002264(n)).
a(n) = (n mod 4)+4*(floor(n/4) mod 3) = A010873(n)+4*A010872(A002265(n)).
a(n) = (n mod 6)+6*(floor(n/6) mod 2) = A010875(n)+6*A000035(A152467(n)).
a(n) = (n mod 2)+2*(floor(n/2) mod 2)+4*(floor(n/4) mod 3) = A000035(n)+2*A000035(A004526(n))+4*A010872(A002265(n)). (End)
a(A001248(k) + 17) = 6 for k>2. - Reinhard Zumkeller, May 12 2010
a(n) = A034326(n+1)-1. - M. F. Hasler, Sep 25 2014

A036342 Prime concatenated analog clock numbers read clockwise.

Original entry on oeis.org

2, 3, 5, 7, 11, 23, 67, 89, 4567, 23456789, 23456789101112123, 891011121234567891011, 23456789101112123456789101112123, 567891011121234567891011121234567891011, 121234567891011121234567891011121234567891011121, 91011121234567891011121234567891011121234567891011121234567
Offset: 1

Views

Author

Patrick De Geest, Dec 15 1998

Keywords

Comments

The hours 10, 11 and 12 are taken 'complete and unreversed'.
a(28) has 1325 digits. - Michael S. Branicky, May 20 2024

Links

Crossrefs

Cf. A034326, A036343, A036344.

Programs

  • Python
    import heapq
from sympy import isprime
from itertools import islice
def A036342_gen(): # generator of terms
    h = [(i, i) for i in range(1, 13)]
    while True:
        v, last = heapq.heappop(h)
        if isprime(v):
            yield v
        nxt = 1 if last == 12 else last+1
        shift = 10 if nxt < 10 else 100
        heapq.heappush(h, (v*shift+nxt, nxt))
print(list(islice(A036342_gen(), 16))) # Michael S. Branicky, May 20 2024

Extensions

a(14)-a(15) from Jean-Marie Hachey, Oct 05 2016
Offset and data corrected by Sean A. Irvine, Oct 26 2020

A036343 Prime concatenated analog clock numbers read counterclockwise.

Original entry on oeis.org

2, 3, 5, 7, 11, 43, 109, 10987, 76543, 6543211211, 4321121110987, 3211211109876543211211, 43211211109876543211211109876543, 9876543211211109876543211211109876543211211109876543211211
Offset: 1

Views

Author

Patrick De Geest, Dec 15 1998

Keywords

Comments

The hours 10, 11 and 12 are taken 'complete and unreversed'.
a(19) has 1139 digits. - Michael S. Branicky, May 20 2024

Links

Crossrefs

Cf. A034326, A036342, A036344.

Programs

  • Python
    import heapq
from sympy import isprime
from itertools import islice
def A036343_gen(): # generator of terms
    h = [(i, i) for i in range(1, 13)]
    while True:
        v, last = heapq.heappop(h)
        if isprime(v):
            yield v
        nxt = 12 if last == 1 else last-1
        shift = 10 if nxt < 10 else 100
        heapq.heappush(h, (v*shift+nxt, nxt))
print(list(islice(A036343_gen(), 16))) # Michael S. Branicky, May 20 2024

Extensions

Offset corrected by Sean A. Irvine, Oct 26 2020

A036344 Prime concatenated analog clock numbers (clockwise and counterclockwise).

Original entry on oeis.org

2, 3, 5, 7, 11, 23, 43, 67, 89, 109, 4567, 10987, 76543, 23456789, 6543211211, 4321121110987, 23456789101112123, 891011121234567891011, 3211211109876543211211, 23456789101112123456789101112123, 43211211109876543211211109876543, 567891011121234567891011121234567891011
Offset: 1

Views

Author

Patrick De Geest, Dec 15 1998

Keywords

Comments

The hours 10, 11 and 12 are taken 'complete and unreversed'.

Links

Crossrefs

Cf. A034326, A036342, A036343.

Formula

Union of A036342 and A036343. - Sean A. Irvine, Oct 26 2020

Extensions

Offset corrected and a(21) from Sean A. Irvine, Oct 26 2020
a(22) and beyond from Michael S. Branicky, May 20 2024

A036218 Hours recorded by a 24-hour clock.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Period 24: repeat [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. - Omar E. Pol, Apr 26 2016

Links

Crossrefs

Cf. A034326.

Programs

Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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