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 1-8 of 8 results.

A144101 Characteristic sequence for A144100.

Original entry on oeis.org

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

Views

Author

Reikku Kulon, Sep 10 2008

Keywords

Links

Crossrefs

Cf. A007947, A144100.

A144907 a(n) = 1 if n is prime, 2 * rad(n) if four divides n and rad(n) otherwise.

Original entry on oeis.org

1, 1, 1, 4, 1, 6, 1, 4, 3, 10, 1, 12, 1, 14, 15, 4, 1, 6, 1, 20, 21, 22, 1, 12, 5, 26, 3, 28, 1, 30, 1, 4, 33, 34, 35, 12, 1, 38, 39, 20, 1, 42, 1, 44, 15, 46, 1, 12, 7, 10, 51, 52, 1, 6, 55, 28, 57, 58, 1, 60, 1, 62, 21, 4, 65, 66, 1, 68, 69, 70, 1, 12, 1, 74, 15, 76, 77, 78, 1, 20, 3, 82
Offset: 1

Views

Author

Reikku Kulon, Sep 24 2008

Keywords

Links

Crossrefs

Cf. A000040, A007947, A144100, A144310.

Programs

  • Haskell
    a144907 x | a010051 x == 1 = 1
          | x `mod` 4 == 0 = 2 * rad
          | otherwise      = rad  where rad = a007947 x
-- Reinhard Zumkeller, Mar 12 2014
  • Mathematica
    rad[n_]:= Module[{aux = FactorInteger[n]},Product[aux[[i, 1]],{i, Length[aux]}]]; a[n_] := Which[PrimeQ[n], 1, IntegerQ[n/4], 2*rad[n], True, rad[n]] Table[a[n], {n, 1, 100}] (* José María Grau Ribas, Feb 16 2010 *)

A144310 Triangle read by rows: prime numbers p(n) along left edge (n, 1) and p(n - 1) along right edge (n, n), with (n, k) = rad((n - 1, k - 1) + (n - 1, k)), multiplied by two if four divides (n - 1, k - 1) + (n - 1, k), for 1 < k < n when n > 2.

Original entry on oeis.org

2, 3, 2, 5, 5, 3, 7, 10, 4, 5, 11, 17, 14, 3, 7, 13, 28, 31, 17, 10, 11, 17, 41, 59, 12, 3, 21, 13, 19, 58, 20, 71, 15, 12, 34, 17, 23, 77, 78, 91, 86, 3, 46, 51, 19, 29, 20, 155, 13, 177, 89, 7, 97, 70, 23, 31, 7, 35, 84, 190, 266, 12, 52, 167, 93, 29
Offset: 1

Views

Author

Reikku Kulon, Sep 17 2008

Keywords

Comments

Among the initial terms, 12 occurs most frequently. 3 and 17 are also common.

Examples

			(7, 2) = 41
(7, 3) = 59
41 + 59 = 100
rad(100) = A007947(100) = 10
Four divides 100, so (8, 3) = 20.

Crossrefs

Cf. A000040, A007947, A144100

A144775 Define f(n) = 2 * rad(n) if four divides n and rad(n) otherwise: then a(n) = 0 for composite n where f(n) is not less than n and otherwise equals the number of positive integers k less than n for which f(k) < f(n).

Original entry on oeis.org

0, 1, 2, 0, 4, 0, 6, 3, 2, 0, 10, 0, 12, 0, 0, 4, 16, 8, 18, 0, 0, 0, 22, 13, 7, 0, 2, 0, 28, 0, 30, 5, 0, 0, 0, 16, 36, 0, 0, 24, 40, 0, 42, 0, 21, 0, 46, 16, 13, 15, 0, 0, 52, 11, 0, 35, 0, 0, 58, 0, 60, 0, 31, 5, 0, 0, 66, 0, 0, 0, 70, 20, 72, 0, 27, 0, 0, 0, 78, 32, 2, 0, 82, 0, 0, 0, 0, 55
Offset: 1

Views

Author

Reikku Kulon, Sep 21 2008

Keywords

Comments

This sequence obtains a new maximum a(n) = n - 1 for prime n.
a(n) = 0 often, but not always, when n - 1 and n + 1 are twin primes.

Examples

			f(8) = 2 * rad(8) = 4. f(k) < 4 for 1, 2 and 3 (f(k) = k for 0 < k < 8); a(8) = 3.

Crossrefs

Cf. A000040, A001097, A007947, A144100, A144774

A144776 Define f(n) = 1 if n is prime, 2 * rad(n) if four divides n and rad(n) otherwise: then a(n) = 0 for composite n where f(n) is not less than n and otherwise equals the number of positive integers k less than n for which f(k) < f(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 5, 5, 0, 0, 0, 0, 0, 0, 8, 0, 12, 0, 0, 0, 0, 0, 17, 14, 0, 10, 0, 0, 0, 0, 14, 0, 0, 0, 22, 0, 0, 0, 28, 0, 0, 0, 0, 29, 0, 0, 26, 25, 26, 0, 0, 0, 24, 0, 42, 0, 0, 0, 0, 0, 0, 41, 21, 0, 0, 0, 0, 0, 0, 0, 35, 0, 0, 42, 0, 0, 0, 0, 46, 23, 0, 0, 0, 0, 0, 0, 64, 0, 58, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Reikku Kulon, Sep 21 2008

Keywords

Comments

For the given terms, nearly all n for which a(n) obtains a new maximum are multiples of eight. Only 18, 36 and 45 are not.

Examples

			f(8) = 4 and f(9) = 3. For 1, 2, 3, 5 and 7, f(k) = 1, so a(8) = a(9) = 5.

Links

Crossrefs

Cf. A000040, A001097, A007947, A144100, A144774, A144775

Programs

  • PARI
    A007947(n) = factorback(factorint(n)[, 1]);
f(n) = if(isprime(n),1,if(n%4,A007947(n),2*A007947(n)));
A144776(n) = if(n<2,0,my(x=f(n)); if(!isprime(n)&&(x>=n),0,sum(k=1,n-1,(f(k)Antti Karttunen, Jul 03 2018

A144102 Decimal expansion of A144101 interpreted as a binary fraction.

Original entry on oeis.org

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

Views

Author

Reikku Kulon, Sep 10 2008

Keywords

Comments

Decimal point precedes initial 4.

Crossrefs

Cf. A144100, A144101.

A144862 Composite numbers n such that sqrt(n) > f(n), where f(n) = 2 * rad(n) if 4 divides n and rad(n) otherwise.

Original entry on oeis.org

27, 32, 54, 64, 81, 125, 128, 162, 192, 216, 243, 250, 256, 288, 324, 343, 375, 384, 405, 432, 486, 500, 512, 567, 576, 625, 640, 648, 675, 686, 729, 768, 800, 864, 896, 972, 1000, 1024, 1029, 1125, 1152, 1215, 1250, 1280, 1296, 1323, 1331, 1350, 1372, 1458
Offset: 1

Views

Author

Reikku Kulon, Sep 23 2008

Keywords

Comments

Subset of A144100.

Crossrefs

Cf. A000040, A007947, A144100.

Programs

  • PARI
    rad(n) = local(p); p=factor(n)[, 1]; prod(i=1, length(p), p[i]) ;
ff(n) = if ((n % 4)==0 , 2*rad(n), rad(n)); isok(n) = (n != 1) && (! isprime(n)) && (n > ff(n)^2); \\ Michel Marcus, Aug 09 2013

A144908 Composite numbers n such that sqrt(n) > A144907(n) and absolute normalized digital mean |dm(b, n) * 2 / (b - 1)| decreases for b in [2, k] for some k > 2.

Original entry on oeis.org

32, 64, 125, 128, 192, 250, 256, 288, 343, 375, 384, 500, 512, 576, 640, 648, 768, 800, 896, 1024, 1029, 1125, 1152, 1280, 1296, 1536, 1568, 1600, 1715, 1792, 1875, 2025, 2048, 2058, 2304, 2401, 2500, 2560, 2592, 2816, 3072, 3136, 3200, 3328, 3375, 3456
Offset: 1

Views

Author

Reikku Kulon, Sep 24 2008

Keywords

Comments

Subset of A144100.
Believed to have particular importance for linear congruential pseudorandom number generators.

Examples

			125 is a member: A144907(125) is 5, which is less than 11.18, the square root of 125;
125 in base 2 is 1111101; dm(2, 125) = (6 * 1 - 1) / 14 = 5/14 ~ 0.357;
125 in base 3 is 11122; dm(3, 125) = (3 * 0 + 2 * 2) / 10 = 2/5 = 0.4;
125 in base 4 is 1331; dm(4, 125) = (2 * -1 + 2 * 3) / 8 = 1/2 = 0.5;
5/14 * 2 / 1 = 5/7 ~ 0.714;
2/5 * 2 / 2 = 2/5 = 0.4;
1/2 * 2 / 3 = 1/3 ~ 0.333;
For b in [2, 4], |dm(b, 125) * 2 / (b - 1)| is decreasing.

Crossrefs

Cf. A144100, A144907, A144777
Showing 1-8 of 8 results.

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