A255134 -id:A255134 - cp's OEIS frontend

A255136 Records in A255134.

12, 28, 36, 44, 52, 68, 76, 84, 92, 100, 116, 124, 132, 140, 148, 156, 172, 180, 188, 196, 204, 212, 228, 244, 252, 260, 268, 276, 284, 292, 300, 316, 324, 332, 340, 348, 356, 364, 372, 380, 396, 404, 412, 420, 428, 436, 444, 452, 460, 476, 484, 492, 500

Offset: 1

Reinhard Zumkeller, Feb 15 2015

nonn

a(n) = A255134(A255137(n)) and A255134(m) < a(n) for m < A255137(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A255137, A255134, A097764.

a255136 n = a255136_list !! (n-1)
(a255136_list, a255137_list) = unzip $ f [1..] a255134_list (-1) where
   f (x:xs) (y:ys) r = if y > r then (y, x) : f xs ys y else f xs ys r

A255137 Where records occur in A255134.

Original entry on oeis.org

1, 4, 5, 6, 7, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 25, 26, 27, 28, 29, 30, 33, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 48, 49, 50, 51, 52, 53, 54, 57, 58, 59, 60, 61, 62, 63, 64, 65, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 81, 82, 83, 84, 85, 86

Offset: 1

Reinhard Zumkeller, Feb 15 2015

nonn

A255134(a(n)) = A255136(n) and A255134(m) < A255136(n) for m < a(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A255136, A255134, A097764.

a255137 n = a255137_list !! (n-1)  -- a255137_list is defined in A255136.

A097764 Numbers of the form (kp)^p for prime p and k=1,2,3,....

Original entry on oeis.org

4, 16, 27, 36, 64, 100, 144, 196, 216, 256, 324, 400, 484, 576, 676, 729, 784, 900, 1024, 1156, 1296, 1444, 1600, 1728, 1764, 1936, 2116, 2304, 2500, 2704, 2916, 3125, 3136, 3364, 3375, 3600, 3844, 4096, 4356, 4624, 4900, 5184, 5476, 5776, 5832, 6084, 6400

Offset: 1

T. D. Noe, Aug 24 2004

The polynomial x^n - n is reducible over the integers for n in this sequence.
A result of Vahlen shows that the polynomial x^n - n is reducible over the integers for n in this sequence and no other n.
The representation (k*p)^p is generally not unique, e.g. a(120) = 46656 = (108*2)^2 = (12*3)^3. - Reinhard Zumkeller, Feb 14 2015
This is also numbers of the form (km)^m for any m > 1, not just primes. Let m be > 1; then m has a prime factor, so let m=pj, p a prime and j an integer > 0. Then (km)^m = (kpj)^pj = (k^j p^j j^j)^p = ((k^j p^(j-1) j^j) p) ^ p. - Franklin T. Adams-Watters, Sep 13 2015

T. D. Noe, Table of n, a(n) for n = 1..1000
A. Schinzel, Problems and results on polynomials.

Cf. A084746 (least k such that n^k-k is prime).
Cf. A097792 (numbers of the form 4k^4 or (kp)^p).
Cf. A000040, A051674, A255134 (first differences).

import Data.Set (singleton, deleteFindMin, insert)
a097764 n = a097764_list !! (n-1)
a097764_list = f 0 (singleton (4, 2, 2)) $
                 tail $ zip a051674_list a000040_list where
   f m s ppps'@((pp, p) : ppps)
     | pp < qq   = f m (insert (pp, p, 2) s) ppps
     | qq == m   = f m (insert ((k * q) ^ q, q, k + 1) s') ppps'
     | otherwise = qq : f qq (insert ((k * q) ^ q, q, k + 1) s') ppps'
     where ((qq, q, k), s') = deleteFindMin s
-- Reinhard Zumkeller, Feb 14 2015

nMax=10000; lst={}; n=1; While[p=Prime[n]; p^p<=nMax, k=1; While[(k*p)^p<=nMax, AppendTo[lst, (k*p)^p]; k++ ]; n++ ]; Union[lst]

is(n)=my(b,e=ispower(n,,&b),f); if(e==0, return(0)); f=factor(e)[,1]; for(i=1,#f, if(b%f[i]==0, return(1))); 0 \\ Charles R Greathouse IV, Aug 29 2016

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A255136 Records in A255134.

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A255137 Where records occur in A255134.

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A097764 Numbers of the form (kp)^p for prime p and k=1,2,3,....

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