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.

Previous Showing 21-28 of 28 results.

A072912 Number of Fibonacci numbers F(k) <= 10^n which end in 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25
Offset: 0

Views

Author

Shyam Sunder Gupta and Benoit Cloitre, Aug 15 2002

Keywords

Examples

			a(2)=6 because there are 6 Fibonacci numbers F(k) <= 10^2 which end in 0.

Crossrefs

Different from A002280.
Cf. A008597, A214855.

Programs

  • PARI
    a(n) = (sum(i=0,ceil(n*log(10)/log((1+sqrt(5))/2)),if(fibonacci(i)%10+1+sign(fibonacci(i)-10^n),0,1)))

Formula

a(n) = ceiling(n*log(10)/(15*log(phi))) +0 or +1.

A144602 Christoffel word of slope 4/11.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1
Offset: 0

Views

Author

N. J. A. Sloane, Jan 13 2009

Keywords

Comments

The path is on the slope after 0, 15, 30, 45, 60, 75,... (A008597) steps, which gives a simple C-finite recurrence. - R. J. Mathar, May 28 2025

Crossrefs

See A144595 for further details.

Formula

a(n) = a(n-15). - R. J. Mathar, May 28 2025
G.f.: -x^3*(1+x^4+x^8+x^11) / ( (x-1)*(1+x^4+x^3+x^2+x)*(1+x+x^2)*(1-x+x^3-x^4+x^5-x^7+x^8) ). - R. J. Mathar, May 28 2025

A193109 Least k such that 2^x + k produces primes for x=1..n and composite for x=n+1.

Original entry on oeis.org

0, 1, 9, 3, 225, 15, 65835, 1605, 19425, 2397347205, 153535525935
Offset: 1

Views

Author

Arkadiusz Wesolowski, Jul 21 2011

Keywords

Comments

All terms except the first four are congruent to 15 mod 30.
a(10) was found in 2005 by T. D. Noe and a(11) was found in the same year by Don Reble.
Other known values: a(13) = 29503289812425.
a(12) > 10^13. - Tyler Busby, Feb 19 2023

Crossrefs

Another version of A110096.
Cf. A008597, A057733, A092506, A104070, A144487.

Programs

  • Mathematica
    Table[k = 0; While[i = 1; While[i <= n && PrimeQ[2^i + k], i++]; i <= n || PrimeQ[2^i + k], k++]; k, {n, 9}] (* T. D. Noe, Jul 21 2011 *)
  • PARI
    is(k, n) = for(x=1, n, if(!isprime(k+2^x), return(0))); 1;
a(n) = {my(s=2); forprime(p=3, n, if(znorder(Mod(2, p))==(p-1), s*=p)); forstep(k=s*(n>1)/2, oo, s, if(is(k, n) && !isprime(k+2^(n+1)), return(k))); } \\ Jinyuan Wang, Jul 30 2020

A212950 Amounts (in cents) of Canadian coins in denominations suggested by Shallit.

Original entry on oeis.org

1, 5, 10, 25, 83, 100, 200
Offset: 1

Views

Author

Jonathan Vos Post, May 31 2012

Keywords

Examples

			1c, 5c, 10c, 25c, 100c (a dollar coin, popularly known as a "loonie," because it bears a picture of a loon), 200c (the "toonie"), and the optimal suggested new coin in the denomination 83c.

Links

Crossrefs

Cf. A212773, A212774, A008588, A008593, A008597, A008598, A135631.
Cf. A208953 (analog for American coins).

A216364 Fermat pseudoprimes to base 2 divisible by 15.

Original entry on oeis.org

645, 1905, 18705, 55245, 62745, 72885, 215265, 451905, 831405, 1246785, 1472505, 1489665, 1608465, 1815465, 2077545, 2113665, 2882265, 4535805, 6135585, 6242685, 8322945, 9063105, 9816465, 16263105, 18137505, 19523505, 53661945, 63560685, 81612105, 81722145
Offset: 1

Views

Author

Marius Coman, Sep 05 2012

Keywords

Comments

Most of the numbers in the sequence above can be written in one of just two forms: 15*(42*n + 1) and 15*(42*n - 13):
(I) numbers of the first form and the corresponding n in the brackets: 645(1), 1905(3), 1246785(1979), 2113665(3355), 2882265(4575), 6135585(9739); 6242685(9909); 8322945(13211), 81612105(129543);
(II) numbers of the second form and the corresponding n in the brackets: 18705(30), 55245(88), 72885(116), 215265(342), 831405(1320), 1815465(2882), 2077545(3298), 4535805(7200), 9816465(15582), 18137505(28790), 19523505(30990), 53661945(85178), 81722145(129718).
But these pseudoprimes can be categorized in many ways taking, beside 42, p - 1, where p is a prime divisor common to many of them (e.g., numbers of the form 15*(46*n + 43) and the corresponding n in the brackets: 62745 (90); 451905 (654); 1489665(2158); 9063105(13134); 63560685(92116)) or p + 1 (e.g., numbers of the form 15*(90*n + 67) and the corresponding n in the brackets: 1472505(1090); 16263105(12046)).
What is also interesting about these numbers: the Fermat pseudoprimes to base 2 formed with their prime divisors, different from 3 and 5 (e.g., 645 = 15*43 and 1905 = 15*127) are Fermat pseudoprimes to base 2, but also 5461 = 43*127; 18705 = 15*29*43 and 55245 = 15*29*127 are Fermat pseudoprimes to base 2, and 158369 = 29*43*127.
Note: Fermat pseudoprimes to base 2 divisible by 5 are mostly of the form 3*k or 3*k + 1; of the first 100 numbers divisible by 5 checked, fewer than 10 are of the form 3*k + 2.

Links

Crossrefs

Intersection of A001567 and A008597.
Cf. A215672, A215944, A216023.

Programs

  • Mathematica
    Select[15*Range[10^6], PowerMod[2, # - 1, #] == 1 &] (* Amiram Eldar, Mar 07 2020 *)
  • PARI
    is_a216364(n) = {Mod(2, n)^n==2 & !isprime(n) & Mod(n, 15)==0} \\ Michael B. Porter, Jan 27 2013

A257645 a(n) = 15*n + 14.

Original entry on oeis.org

14, 29, 44, 59, 74, 89, 104, 119, 134, 149, 164, 179, 194, 209, 224, 239, 254, 269, 284, 299, 314, 329, 344, 359, 374, 389, 404, 419, 434, 449, 464, 479, 494, 509, 524, 539, 554, 569, 584, 599, 614, 629, 644, 659, 674, 689, 704, 719, 734, 749, 764, 779
Offset: 0

Views

Author

Arkadiusz Wesolowski, Nov 05 2015

Keywords

Comments

A123159(a(n)) <= 4.
This is not a subsequence of A047725 (for example, 239 is missing in A047725). - Bruno Berselli, Nov 06 2015
Equivalently, intersection of A016897 and A016789. - Bruno Berselli, Jan 24 2018

Links

Crossrefs

Cf. A008597, A016897, A175887, A123159, A204542.

Programs

  • Magma
    [15*n+14: n in [0..51]];
  • Maple
    seq(15*n+14, n=0..51);
  • Mathematica
    15 Range[50] - 1
  • PARI
    for(n=0, 51, print1(15*n+14, ", "));

Formula

G.f.: (14 + x)/(1 - x)^2.
a(n) = A008597(n+1) - 1. - Omar E. Pol, Nov 05 2015
a(n) = A016897(3n+2) = A175887(2n+2) = A204542(4n+4). - Bruno Berselli, Nov 06 2015
E.g.f.: (15*x + 14)*exp(x). - G. C. Greubel, Apr 23 2018
a(n) = 2*a(n-1)-a(n-2). - Wesley Ivan Hurt, Dec 27 2023

A198291 Least k such that 2^x - k produces primes or negative values of primes for x=1..n and (possibly in absolute value) composite for x=n+1.

Original entry on oeis.org

0, 33, 111, 285, 1455, 10275, 21, 75, 45, 13573477665, 232317867705
Offset: 1

Views

Author

Arkadiusz Wesolowski, Jan 26 2012

Keywords

Comments

All terms after the first seven are congruent to 15 mod 30.
a(n) exists for every n under Dickson's conjecture. [Charles R Greathouse IV, Jan 30 2012]

Examples

			There are some numbers (7, 9, 15, 21) for which both abs(2^1 - k) and abs(2^2 - k) are primes. Let k = 33, then 2^1 - 33 is -31, the negative of a prime. 2^2 - 33 is -29, the negative of a prime as well. The absolute value of 2^3 - 33 is composite, hence 33 is a term of the sequence.

Crossrefs

Cf. A008597.

Programs

  • Mathematica
    Table[k = 0; While[i = 1; While[i <= n && PrimeQ[2^i - k], i++]; i <= n || PrimeQ[2^i - k] || Abs[2^i - k] == 1, k++]; k, {n, 9}]
  • PARI
    /* Optimized version, starts from twin primes */
list(lim)=my(v=vector(50),least=2,k,p=2);forprime(q=3,lim,if(q-p>2,p=q;next,k=q+2;p=q);for(j=3,least,if(!isprime(abs(2^j-k)),next(2)));my(j=least+1);while(isprime(abs(2^j-k)),j++);if(abs(2^j-k)<2,next);j--;if(!v[j],v[j]=k;print("a("j") = "k);while(v[least],least++)));forstep(i=#v,1,-1,if(v[i],v=vector(i,j,v[j]);break));v \\ Charles R Greathouse IV, Jan 30 2012

Extensions

a(10) from Charles R Greathouse IV, Jan 30 2012
a(11) from Charles R Greathouse IV, Jan 31 2012

A212951 Amounts (in hundredths of a Euro) of coins in denominations suggested by Shallit.

Original entry on oeis.org

1, 2, 5, 10, 20, 50, 100, 133, 200
Offset: 1

Views

Author

Jonathan Vos Post, May 31 2012

Keywords

Comments

The European Union uses eight coins - worth 1, 2, 5, 10, 20, and 50 cents, plus 1- and 2-Euro coins - with a range of values from 0 to 499. The average cost of making change in Europe, Jeffrey Shallit calculates, is 4.6 coins. The best way to lower the cost, to 3.92, would be for Europeans to add yet another coin, worth either 1.33 or 1.37 Euros (the sequence as shown uses 133, though 137 is an equally valid solution).

Examples

			1, 2, 5, 10, 20, and 50 cents, plus 1- and 2-Euro coins (100 and 200 cents), and the proposed 1.33-Euro coin (133 cents).

Links

Crossrefs

Cf. A212773, A212774, A008588, A008593, A008597, A008598, A135631.
Cf. A208953 (analog for American coins).
Cf. A212950 (analog for Canadian coins).
Previous Showing 21-28 of 28 results.

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