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 1-4 of 4 results.

A133808 Numbers that are primally tight, have 2 as first prime and weakly ascending powers.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 18, 30, 32, 36, 54, 64, 108, 128, 150, 162, 210, 216, 256, 324, 450, 486, 512, 648, 750, 900, 972, 1024, 1296, 1458, 1470, 1944, 2048, 2250, 2310, 2916, 3750, 3888, 4096, 4374, 4500, 5832, 6750, 7350, 7776, 8192, 8748, 10290, 11250
Offset: 1

Views

Author

Olivier Gérard, Sep 23 2007

Keywords

Comments

All numbers of the form 2^k1*p_2^k2*...*p_n^k_n, where k1 <= k2 <= ... <= k_n and the p_i are the n first primes.
Subset of A073491, A133810.

Examples

			10 = 2*5 with missing prime factor 3 between 2 and 5 is not in the sequence.
12 = 2^2*3 with 2's exponent > 3's exponent is not in the sequence.

Links

Crossrefs

Cf. A025487, A087980, A073491, A133809-A133813.
Cf. A027748, A124010, A151800, A000040.

Programs

  • Haskell
    import Data.Set (singleton, deleteFindMin, insert)
a133808 n = a133808_list !! (n-1)
a133808_list = 1 : f (singleton (2, 2, 1)) where
   f s = y : f (insert (y * p, p, e + 1) $ insert (y * q^e, q, e) s')
             where q = a151800 p
                   ((y, p, e), s') = deleteFindMin s
-- Reinhard Zumkeller, Apr 13 2015
  • PARI
    isok(n) = {my(f = factor(n)); my(nbf = #f~); if (prod(i=1, nbf, prime(i)) ! = prod(i=1, nbf, f[i, 1]), return (0)); for (j=2, nbf, if (f[j,2] < f[j-1,2], return (0));); return (1);} \\ Michel Marcus, Jun 04 2014

A144100 Numbers k such that k is strictly greater than f(k), where f(k) = 1 if k is prime, 2 * rad(k) if 4 divides k and rad(k) otherwise.

Original entry on oeis.org

2, 3, 5, 7, 8, 9, 11, 13, 16, 17, 18, 19, 23, 24, 25, 27, 29, 31, 32, 36, 37, 40, 41, 43, 45, 47, 48, 49, 50, 53, 54, 56, 59, 61, 63, 64, 67, 71, 72, 73, 75, 79, 80, 81, 83, 88, 89, 90, 96, 97, 98, 99, 100, 101, 103, 104, 107, 108, 109, 112, 113, 117, 120, 121, 125, 126
Offset: 1

Views

Author

Reikku Kulon, Sep 10 2008

Keywords

Comments

This is the set of all integers k such that there exists a full period linear congruential pseudorandom number generator x -> bx + c (mod k), where b is not a multiple of k, b - 1 is a multiple of f(k) and c is a positive integer relatively prime to k.
4 is the only prime power not a member of the set: f(4) = 2 * rad(4) = 4.
This sequence consists of the primes and 2*A013929. - Charlie Neder, Jan 28 2019

Examples

			2 is a member: f(2) = 1 and the sequence (0, 1, 0, ...) given by x -> x + 1 (mod 2) has period 2.
8 is a member: f(8) = 4 and the sequence (0, 1, 6, 7, 4, 5, 2, 3, 0, ...) given by x -> 5x + 1 (mod 8) has period 8.
18 is a member: f(18) = 6 and the sequence (0, 1, 14, 3, 4, 17, 6, 7, 2, 9, 10, 5, 12, 13, 8, 15, 16, 11, 0, ...) given by x -> 13x + 1 (mod 18) has period 18.

Links

Crossrefs

Cf. A000040, A007947, A071139, A082377, A086486, A089352, A133810, A133811.

Programs

  • Haskell
    a144100 n = a144100_list !! (n-1)
a144100_list = filter (\x -> a144907 x < x) [1..]
-- Reinhard Zumkeller, Mar 12 2014
  • PARI
    rad(n) = local(p); p=factor(n)[, 1]; prod(i=1, length(p), p[i]) ;
f(n) = if (isprime(n), 1, if ((n % 4)==0 , 2*rad(n), rad(n))); isok(n) = n > f(n); \\ Michel Marcus, Aug 09 2013

Formula

A144907(a(n)) < a(n). - Reinhard Zumkeller, Mar 12 2014

A133811 Numbers that are primally tight and have strictly ascending powers.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 18, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 54, 59, 61, 64, 67, 71, 73, 75, 79, 81, 83, 89, 97, 101, 103, 107, 108, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 162, 163, 167, 169, 173, 179, 181, 191
Offset: 1

Views

Author

Olivier Gérard, Sep 23 2007

Keywords

Comments

All numbers of the form p_1^k1*p_2^k2*...*p_n^k_n, where k1 < k2 < ... < k_n and the p_i are n successive primes.
Subset of A073491, A133810.
Different from A082377 starting n=16.
Different from A000961 (prime powers) starting n=13.

Links

Crossrefs

Cf. A025487, A087980, A073491, A133808-A133813.
Cf. A124010, A027748, A049084.

Programs

  • Haskell
    a133811 n = a133811_list !! (n-1)
a133811_list = 1 : filter f [2..] where
   f x = (and $ zipWith (<) eps $ tail eps) &&
         (all (== 1) $ zipWith (-) (tail ips) ips)
     where ips = map a049084 $ a027748_row x
           eps = a124010_row x
-- Reinhard Zumkeller, Nov 07 2012
  • PARI
    isok(n) = {my(f = factor(n)); my(nbf = #f~); my(lastp = 0); for (i=1, nbf, if (lastp && (f[i, 1] != nextprime(lastp+1)), return (0)); lastp = f[i, 1];); for (j=2, nbf, if (f[j,2] <= f[j-1,2], return (0));); return (1);} \\ Michel Marcus, Jun 04 2014

A145108 Multiples of 4 that are primally tight and have strictly ascending powers.

Original entry on oeis.org

4, 8, 16, 32, 64, 108, 128, 256, 324, 512, 648, 972, 1024, 1944, 2048, 2916, 3888, 4096, 5832, 8192, 8748, 11664, 16384, 17496, 23328, 26244, 32768, 34992, 52488, 65536, 67500, 69984, 78732, 104976, 131072, 139968, 157464, 209952, 236196, 262144
Offset: 1

Views

Author

Reikku Kulon, Oct 02 2008

Keywords

Comments

All numbers of the form 2^k0*p_1^k1*p_2^k2*...*p_n^k_n, where 2 <= k0 < k1 < k2 < ... < k_n and the p_i are n successive primes.
Subset of A073491, A133808, A133809, A133810 and A133811.

Links

Crossrefs

Cf. A025487, A087980, A073491, A133808, A133809, A133810, A133811, A133812, A133813.
Cf. A008586.

Programs

  • Haskell
    a145108 n = a145108_list !! (n-1)
a145108_list = filter ((== 0) . (`mod` 4)) a133809_list
-- Reinhard Zumkeller, Apr 14 2015
Showing 1-4 of 4 results.

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