A257278 -id:A257278 - cp's OEIS frontend

A257572 Prime root of n-th term in A257278.

2, 2, 2, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 3, 5, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 5, 2, 3, 7, 2, 3, 5, 2, 2, 3, 7, 2, 5, 3, 2, 2, 7, 3, 5, 2, 3, 2, 5, 2, 7, 3, 2, 2, 3, 5, 7, 2, 3, 2, 5, 2, 3, 7, 2, 5, 3, 2, 2, 3, 7, 2, 5, 2, 3, 11, 2, 7, 5, 3, 2, 2, 3

Offset: 1

Reinhard Zumkeller, May 01 2015

nonn

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

Cf. A257278, A020639, A025473, A257573.

Haskell
```
a257572 = a020639 . a257278
```

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

a(n) = A020639(A257278(n)).
A257278(n) = a(n) ^ A257573(n).
a(n) <= A257573(n) by definition of A257278.

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

A068936 Numbers having the sum of distinct prime factors not greater than the sum of exponents in prime factorization, A008472(k) <= A001222(k).

Original entry on oeis.org

1, 4, 8, 16, 27, 32, 48, 64, 72, 81, 96, 108, 128, 144, 162, 192, 216, 243, 256, 288, 320, 324, 384, 432, 486, 512, 576, 640, 648, 729, 768, 800, 864, 972, 1024, 1152, 1280, 1296, 1458, 1536, 1600, 1728, 1792, 1944, 2000, 2048, 2187, 2304, 2560, 2592, 2916

Offset: 1

Reinhard Zumkeller, Mar 08 2002

nonn

The product of any two terms is also a term. - Amiram Eldar, May 14 2025

			a(5) = 27 = 3^3, 3 = 3.
a(10) = 81 = 3^4, 3 < 4.
a(100) = 16000 = 2^7 * 5^3,  2+5 < 7+3.
a(1000) = 10321920 = 2^15 * 3^2 * 5 * 7, 2+3+5+7 < 15+2+1+1.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)

Cf. A001222, A008472, A068935, A068937, A068938.

Subsequences: A054411, A100716, A257278.

a068936 n = a068936_list !! (n-1)
a068936_list = [x | x <- [1..], a008472 x <= a001222 x]
-- Reinhard Zumkeller, Nov 10 2013

fQ[n_] := Block[{f = FactorInteger@n}, Plus @@ Last /@ f >= Plus @@ First /@ f]; Select[ Range@3000, fQ@ # &] (* Robert G. Wilson v, Jan 16 2006 *)
Select[Range@ 3000, First@ Differences@ Map[Total, Transpose@ FactorInteger@ #] >= 0 &] (* Michael De Vlieger, Dec 08 2016 *)

isok(k) = {my(f = factor(k)); vecsum(f[,1]) <= bigomega(f);} \\ Amiram Eldar, May 14 2025

More terms from Robert G. Wilson v, Jan 16 2006

A192135 Prime powers p^e with p < e.

Original entry on oeis.org

8, 16, 32, 64, 81, 128, 243, 256, 512, 729, 1024, 2048, 2187, 4096, 6561, 8192, 15625, 16384, 19683, 32768, 59049, 65536, 78125, 131072, 177147, 262144, 390625, 524288, 531441, 1048576, 1594323, 1953125, 2097152, 4194304, 4782969, 5764801, 8388608, 9765625, 14348907

Offset: 1

Reinhard Zumkeller, Jun 26 2011

nonn

Donovan Johnson, Table of n, a(n) for n = 1..10000

Complement to A074583 with respect to A000961.

Cf. A095874, A192187, A257278.

A192135 := proc(nmax) local s ,i,p,e ; s := {} ; for i from 1 do p := ithprime(i) ; if p^(p+1) > nmax then break; end if; for e from p+1 do if p^e > nmax then break; end if; s := s union {p^e} ; end do: end do: sort(s) ; end proc:
A192135(20000000) ; # R. J. Mathar, Jul 09 2011

seq[lim_] := Module[{s = {}, p = 2}, While[p^p <= lim, AppendTo[s, p^Range[p+1, Log[p, lim]]]; p = NextPrime[p]]; Sort[Flatten[s]]]; seq[10^7] (* Amiram Eldar, Apr 14 2025 *)

a(n) = A000961(A192187(n)).
A095874(a(n)) =  A192187(n).
Sum_{n>=1} 1/a(n) = Sum_{p prime} 1/(p^p*(p-1)) = 0.26859872089648243789... . - Amiram Eldar, Apr 14 2025

A122494 Numbers of the form a^b with 2<=a<=b.

Original entry on oeis.org

4, 8, 16, 27, 32, 64, 81, 128, 243, 256, 512, 729, 1024, 2048, 2187, 3125, 4096, 6561, 8192, 15625, 16384, 19683, 32768, 46656, 59049, 65536, 78125, 131072, 177147, 262144, 279936, 390625, 524288, 531441, 823543, 1048576, 1594323, 1679616

Offset: 1

Paul Stoeber (pstoeber(AT)uni-potsdam.de), Sep 16 2006

nonn

			279936 is there because it is 6^7.

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

Subsequences: A000312, A257278.

import Data.Set (singleton, deleteFindMin, insert)
a122494 n = a122494_list !! (n-1)
a122494_list = f (singleton (4, 2)) 27 [3..] where
   f s uu us@(u:us'@(u':_))
     | vv > uu = uu : f (insert (uu * u, u) s) (u' ^ u') us'
     | vv < uu = vv : f (insert (vv * v, v) s') uu us
     | otherwise = vv : f (insert (vv * v, v) s') (u' ^ u') us'
     where ((vv, v), s') = deleteFindMin s
-- Reinhard Zumkeller, May 01 2015

A257279 Zeroless prime powers p^m with p <= m.

Original entry on oeis.org

4, 8, 16, 27, 32, 64, 81, 128, 243, 256, 512, 729, 2187, 3125, 6561, 8192, 15625, 16384, 19683, 32768, 65536, 78125, 177147, 262144, 524288, 531441, 823543, 1594323, 1953125, 4782969, 9765625, 16777216, 33554432, 48828125, 134217728, 268435456, 282475249, 1162261467

Offset: 1

M. F. Hasler, Apr 28 2015

A few years ago, challenges had been launched to find a prime power p^n, n>1 as large as possible, cf. links. I have remarked that it is easy to find arbitrarily large examples by taking the square of very large primes, rather than high powers of smaller primes, and suggested a merit function to take into account and penalize such "trivial" solutions. This led to a new challenge including the condition n > p. This sequence lists such numbers with the last condition relaxed to n >= p, which is sufficient to make the search nontrivial but includes a few more terms, namely the zeroless powers p^p (A051674 intersect A052382).

Possibly is a(80) = 19^44 the largest term; there are no greater ones in the first 500000 terms of A257278. - Reinhard Zumkeller, May 01 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..80
S. S. Gupta, Can you find?, CYF n° 410 (Jan. 26, 2010) and CYF n° 412 (Nov. 13, 2012).
C. Rivera, Puzzle 607. A zeroless Prime power, primepuzzles.net, 2011

Cf. A052382, A000961, A025475; A122494.
Equals A257278 \ A011540 = intersection of A052382 and A257278.
Subsequence of A025475 \ A011540 and of A195943, see also A168046.

a257279 n = a257279_list !! (n-1)
a257279_list = filter ((== 1) . a168046) a257278_list
-- Reinhard Zumkeller, May 01 2015

is(n)=vecmin(digits(n)) && isprimepower(n,&n)>=n

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

cp's OEIS Frontend

A257572 Prime root of n-th term in A257278.

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

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A068936 Numbers having the sum of distinct prime factors not greater than the sum of exponents in prime factorization, A008472(k) <= A001222(k).

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A192135 Prime powers p^e with p < e.

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A122494 Numbers of the form a^b with 2<=a<=b.

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Haskell

A257279 Zeroless prime powers p^m with p <= m.

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PARI