A257572 -id:A257572 - cp's OEIS frontend

A257278 Prime powers p^m with p <= m.

4, 8, 16, 27, 32, 64, 81, 128, 243, 256, 512, 729, 1024, 2048, 2187, 3125, 4096, 6561, 8192, 15625, 16384, 19683, 32768, 59049, 65536, 78125, 131072, 177147, 262144, 390625, 524288, 531441, 823543, 1048576, 1594323, 1953125, 2097152, 4194304, 4782969, 5764801, 8388608, 9765625

Offset: 1

M. F. Hasler, Apr 28 2015

nonn

Might be called "high" powers of primes. Motivated by challenges for which low powers of large primes provide somewhat trivial solutions, cf. A257279. The definition also avoids the question of the whether prime itself is to be considered as a prime power or not, cf. A000961 vs. A025475. In view of the condition p <= n, up to 10^10, only powers of the primes 2, 3, 5 and 7 (namely, less than 10) can occur.

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

Cf. A000961, A025475, A257279.
Cf. A000040, A051674 (subsequence).
Subsequence of A122494 and A192135 (p < m, subsequence).
Cf. A257572, A257573.

import Data.Set (singleton, deleteFindMin, insert)
a257278 n = a257278_list !! (n-1)
a257278_list = f (singleton (4, 2)) 27 (tail a000040_list) where
   f s pp ps@(p:ps'@(p':_))
     | qq > pp   = pp : f (insert (pp * p, p) s) (p' ^ p') ps'
     | otherwise = qq : f (insert (qq * q, q) s') pp ps
     where ((qq, q), s') = deleteFindMin s
-- Reinhard Zumkeller, May 01 2015

seq[lim_] := Module[{s = {}, p = 2}, While[p^p <= lim, AppendTo[s, p^Range[p, Log[p, lim]]]; p = NextPrime[p]]; Sort[Flatten[s]]]; seq[10^7] (* Amiram Eldar, Apr 14 2025 *)

L=List();lim=10;forprime(p=1,lim,for(n=p,lim*log(lim)\log(p),listput(L,p^n)));listsort(L);L

a(n) = A257572(n) ^ A257573(n). - Reinhard Zumkeller, May 01 2015

Sum_{n>=1} 1/a(n) = Sum_{p prime} 1/(p^(p-1)*(p-1)) = 0.55595697220270661763... - Amiram Eldar, Oct 24 2020

A257569 Triangular array read by rows: T(h,k) = number of steps from (h,k) to (0,0), where one step is (x,y) -> (x-1, y) if x is odd or (x,y) -> (y, x/2) if x is even.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 01 2015

The number of pairs (h,k) satisfying T(h,k) = n is F(n), where F = A000045, the Fibonacci numbers.  The number of such pairs having odd h is F(n-2), and the number having even h is F(n-1).
Let c(n,k) be the number of pairs (h,k) satisfying T(h,k) = n; in particular, c(n,0) is the number of integers (pairs of the form (h,0)) satisfying T(h,0) = n.  Let p(n) = A000931(n).  Then c(n,0) = p(n+3) for n >= 0.  More generally, for fixed k >=0, the sequence satisfies the recurrence r(n) = r(n-2) + r(n-3) except for initial terms.
The greatest h for which some (h,k) is n steps from (0,0) is H = A029744(n-1) for n >= 2, and the only such pair is (H,0).
T(n,k) is also the number of steps from h + k*sqrt(2) to 0, where one step is x + y*sqrt(2) -> x-1 + y*sqrt(2) if x is odd, and x + sqrt(y) -> y + (x/2)*sqrt(2) if x is even.

			First ten rows:
0
1   2
3   3   4
4   4   5   5
5   5   5   6   6
6   6   6   6   7   7
6   7   6   7   7   8   7
7   7   7   7   8   8   8   8
7   8   7   8   7   9   8   9   8
8   8   8   8   8   8   9   9   9   9
Row 3 counts the pairs (2,0), (1,1), (0,2), for which the paths to (0,0) are as shown here:
(2,0) -> (0,1) -> (1,0) -> (0,0)  (3 steps);
(1,1) -> (0,1) -> (1,0) -> (0,0)  (3 steps);
(0,2) -> (2,0) -> (0,1) -> (1,0) -> (0,0) (4 steps).

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A257570, A257571, A257572, A000045, A000931, A029744.

f[{x_, y_}] := If[EvenQ[x], {y, x/2}, {x - 1, y}];
g[{x_, y_}] := Drop[FixedPointList[f, {x, y}], -1];
h[{x_, y_}] := -1 + Length[g[{x, y}]];
t = Table[h[{n - k, k}], {n, 0, 16}, {k, 0, n}];
TableForm[t] (* A257569 array *)
Flatten[t]   (* A257569 sequence *)

A257573 Exponents in A257278 = powers of primes p^m, p <= m.

Original entry on oeis.org

2, 3, 4, 3, 5, 6, 4, 7, 5, 8, 9, 6, 10, 11, 7, 5, 12, 8, 13, 6, 14, 9, 15, 10, 16, 7, 17, 11, 18, 8, 19, 12, 7, 20, 13, 9, 21, 22, 14, 8, 23, 10, 15, 24, 25, 9, 16, 11, 26, 17, 27, 12, 28, 10, 18, 29, 30, 19, 13, 11, 31, 20, 32, 14, 33, 21, 12, 34, 15, 22

Offset: 1

Reinhard Zumkeller, May 01 2015

nonn

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

Cf. A257278, A001222, A025474, A257572.

Haskell
```
a257573 = a001222 . a257278
```

seq[lim_] := Module[{s = {}, p = 2, r}, While[p^p <= lim, r = Range[p, Log[p, lim]]; AppendTo[s, Transpose[{r, p^r}]]; p = NextPrime[p]]; SortBy[Flatten[s, 1], Last][[;; , 1]]]; seq[10^13] (* Amiram Eldar, Apr 14 2025 *)

apply(bigomega,A257278) \\ (A257278 assumed to be defined as vector). - M. F. Hasler, May 02 2015

a(n) = A001222(A257278(n)).
A257278(n) = A257572(n) ^ a(n).
A257572(n) <= a(n) by definition of A257278.

Edited by M. F. Hasler, May 02 2015

cp's OEIS Frontend

A257278 Prime powers p^m with p <= m.

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A257569 Triangular array read by rows: T(h,k) = number of steps from (h,k) to (0,0), where one step is (x,y) -> (x-1, y) if x is odd or (x,y) -> (y, x/2) if x is even.

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A257573 Exponents in A257278 = powers of primes p^m, p <= m.

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Programs

Haskell

Mathematica

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