A001597 -id:A001597 - cp's OEIS frontend

A251753 n!/pp, where pp is the largest perfect power (A001597) which divides n!.

1, 1, 2, 6, 3, 15, 5, 35, 70, 70, 7, 77, 231, 3003, 858, 1430, 1430, 24310, 12155, 230945, 46189, 230945, 176358, 4056234, 676039, 676039, 104006, 312018, 44574, 1292646, 1077205, 33393355, 66786710, 2203961430, 64822395, 90751353, 90751353, 3357800061, 353452638, 1531628098, 3829070245, 156991880045

Offset: 0

Robert G. Wilson v, Dec 07 2014

nonn

Robert G. Wilson v, Table of n, a(n) for n = 0..100

Cf. A000142, A001597, A090630.

perfectPowerQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; f[n_] := Block[{d = Divisors[n!], k = 1},  While[ ! perfectPowerQ[ d[[-k]]], k++]; n!/d[[-k]]]; Array[f, 41, 0] (* or *)
f[n_] := Block[{fi = FactorInteger[n!]}, n!/Times @@ (#1[[1]] ^ (2 Quotient[#1[[2]],2])&) /@ fi]; f[4] = 3; f[5] = 15; f[21] = 230945; Array[f, 40]

If p is prime, then a(p) = p*a(p-1).

a(n) = n! / A090630(n). - Joerg Arndt, Dec 08 2014

A070228 Number of perfect powers (A001597) not exceeding 2^n.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 11, 16, 23, 31, 42, 58, 82, 114, 156, 217, 299, 417, 583, 814, 1136, 1589, 2224, 3116, 4369, 6136, 8623, 12128, 17064, 24023, 33839, 47689, 67227, 94805, 133738, 188710, 266351, 376019, 530941, 749820, 1059097, 1496144, 2113802, 2986770, 4220666

Offset: 0

Donald S. McDonald, May 14 2002

			How many powers are there not exceeding 2^4?: 1, 4, 8, 9, 16. Hence a(4) = 5.
a(22)=2224: there are 2224 powers not exceeding 2^22.

Amiram Eldar, Table of n, a(n) for n = 0..6643 (terms 0..1000 from Robert G. Wilson v)

Cf. A070428, A188951.

f[n_] := 1 - Sum[ MoebiusMu[x]*Floor[2^(n/x) - 1], {x, 2, n}]; Array[f, 44, 0] (* Robert G. Wilson v, Jan 20 2015 *)

a(n) = 1 - sum(k=2, n, moebius(k)*(sqrtnint(2^n,k)-1));

from sympy import mobius, integer_nthroot
def A070228(n): return int(1+sum(mobius(x)*(1-integer_nthroot(1<Chai Wah Wu, Aug 13 2024

a(n) = 1 - Sum_{k=2..n} Moebius(k)*floor(2^(n/k)-1). - Robert G. Wilson v, Jan 20 2015

a(n) = A188951(n) + 1 for n > 1. - Amiram Eldar, May 19 2022

a(39)-a(44) from Alex Ratushnyak, Jan 02 2014

A118715 Numbers n such that the digital reversal of n is a perfect power (A001597).

Original entry on oeis.org

0, 1, 4, 8, 9, 10, 18, 23, 40, 46, 52, 61, 63, 72, 80, 90, 94, 100, 121, 144, 148, 163, 169, 180, 215, 230, 342, 343, 400, 423, 441, 460, 484, 487, 520, 521, 522, 526, 610, 612, 630, 652, 675, 676, 691, 720, 800, 821, 900, 925, 927, 940, 961, 982, 1000, 1042, 1062

Offset: 1

Giovanni Teofilatto, May 21 2006

			63 is a member of the sequence since its digital reversal, 36=6^2, is the ninth perfect power.

Cf. A001597.

fQ[n_] := Block[{rid = FromDigits@ Reverse@ IntegerDigits@n}, rid == 0 || rid == 1 || GCD @@ Last /@ FactorInteger@ rid > 1];
Select[ Range[0, 1088], fQ@# &] (* Robert G. Wilson v, May 22 2006 *)

Edited and extended by Robert G. Wilson v, May 22 2006

A128948 Primes p for which the period length of 1/p is a perfect power, A001597.

Original entry on oeis.org

3, 17, 73, 101, 137, 163, 257, 353, 449, 577, 641, 751, 757, 883, 1297, 1409, 1801, 3137, 3529, 5477, 7057, 7351, 8929, 9397, 10753, 11831, 12101, 13457, 13553, 14401, 15361, 15377, 15973, 18523, 19841, 20809, 21401, 21601, 23549, 24001, 24337

Offset: 1

Robert G. Wilson v, May 05 2007

Number of primes p < 10^n whose period length of 1/p is a perfect power: 1,3,14,24,78,173,461,1190,3235,8933,....

The primes modulo any integer do not seem to be equally distributed.

			The prime 73 has a period of 8 = 2^3 which is a member of A001597, hence is a member of this sequence.

Ray Chandler & Robert G. Wilson v, Table of n, a(n) for n = 1..30000
Index entries for sequences related to decimal expansion of 1/n

Cf. A001597, A072859, A072982.

lst = {3}; p = 1; While[p < 10^8, p = NextPrime@p; If[GCD @@ Last /@ FactorInteger@ MultiplicativeOrder[10, p] > 1, AppendTo[lst, p]; Print@p]]; lst (* Ray Chandler, May 11 2007 *)

Correction (3 is a member of the sequence) from Ray Chandler, May 11 2007

B-file corrected by Ray Chandler, Oct 23 2011

A175050 Positive integers n where both n and the number of divisors of n are perfect powers. (Both n and d(n) are elements of A001597.)

Original entry on oeis.org

1, 8, 27, 36, 100, 125, 128, 196, 216, 225, 256, 343, 441, 484, 676, 900, 1000, 1089, 1156, 1225, 1296, 1331, 1444, 1521, 1764, 2116, 2187, 2197, 2304, 2601, 2744, 3025, 3249, 3364, 3375, 3844, 4225, 4356, 4761, 4900, 4913, 5476, 5929, 6084, 6400, 6561

Offset: 1

Leroy Quet, Dec 08 2009

nonn

			128 has 8 divisors. Since 128 is a perfect power (128 = 2^7), and since 8 is also a perfect power (8 = 2^3), then 128 is in this sequence.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001597, A175051.

Block[{nn = 10^4, s}, s = Union@ Flatten@ Table[n^e, {e, Prime@ Range@ PrimePi@ Log2@ nn}, {n, nn^(1/e)}]; Select[s, MemberQ[s, DivisorSigma[0, #]] &]] (* Michael De Vlieger, Nov 24 2017, after T. D. Noe at A001597 *)

Extended by Ray Chandler, Dec 10 2009

A189117 Conjectured number of pairs of consecutive perfect powers (A001597) differing by n.

Original entry on oeis.org

1, 1, 2, 3, 1, 0, 2, 1, 3, 1, 2, 1, 3, 0, 2, 1, 5, 2, 3, 1, 1, 0, 1, 2, 1, 2, 1, 3, 0, 1, 0, 1, 1, 0, 2, 1, 1, 1, 3, 1, 1, 0, 1, 0, 1, 0, 3, 1, 2, 0, 1, 0, 2, 0, 2, 1, 1, 0, 1, 2, 1, 0, 1, 0, 3, 0, 2, 2, 1, 0, 2, 0, 2, 1, 1, 1, 1, 0, 3, 1, 1, 0, 1, 0, 1, 0, 1, 0, 3, 0, 0, 1, 1, 1, 2, 0, 2, 0, 1, 5

Offset: 1

T. D. Noe, Apr 16 2011

nonn

Only a(1) is proved. Perfect powers examined up to 10^21. This is similar to A076427, but more restrictive.
Hence, through 10^21, there is only one value in the sequence: Semiprimes which are both one more than a perfect power and one less than another perfect power. This is to perfect powers A001597 approximately as A108278 is to squares. A more exact analogy would be to the set of integers such as 30^2 = 900 since 900-1 = 899 = 29 * 31, and 900+1 = 901 = 17 * 53. A189045 INTERSECTION A189047. a(1) = 26 because 26 = 2 * 13 is semiprime, 26-1 = 25 = 5^2, and 26+1 = 27 = 3^3. - Jonathan Vos Post, Apr 16 2011
Pillai's conjecture is that a(n) is finite for all n. - Charles R Greathouse IV, Apr 30 2012

			1 = 3^2 - 2^3;
2 = 3^3 - 2^5;
3 = 2^2 - 1^2 = 2^7 - 5^3;
4 = 2^3 - 2^2 = 6^2 - 2^5 = 5^3 - 11^2.

Cf. A023056 (least k such that k and k+n are consecutive perfect powers).

Cf. A023057 (conjectured n such that a(n)=0).

nn = 10^12; pp = Join[{1}, Union[Flatten[Table[n^i, {i, 2, Log[2, nn]}, {n, 2, nn^(1/i)}]]]]; d = Select[Differences[pp], # <= 100 &]; Table[Count[d, n], {n, 100}]

A304574 Number of perfect powers (A001597) less than n and relatively prime to n.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 3, 2, 4, 1, 4, 2, 3, 2, 5, 1, 5, 2, 4, 2, 5, 1, 5, 3, 5, 4, 7, 1, 7, 4, 6, 4, 7, 2, 9, 4, 6, 3, 9, 2, 9, 4, 5, 4, 9, 2, 9, 4, 7, 5, 10, 3, 9, 4, 7, 5, 10, 2, 10, 5, 6, 5, 10, 3, 11, 5, 8, 3, 11, 3, 11, 5, 7, 5, 10, 3, 11, 4, 8, 6, 12, 2

Offset: 1

Gus Wiseman, May 14 2018

nonn

			The a(33) = 6 perfect powers less than and relatively prime to 33 are {1, 4, 8, 16, 25, 32}.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 2..2^16, showing prime n in red, proper prime powers n in gold, squarefree composite n in green, and numbers n that are neither squarefree nor prime powers in blue and purple, where purple represents powerful n that are not prime powers. Large green points highlight composite primorials n.

Cf. A000010, A000961, A001597, A005117, A007916, A073311, A139555, A304326, A304362, A304573, A304575, A304576.

Table[Length[Select[Range[n-1],And[#==1||GCD@@FactorInteger[#][[All,2]]>1,GCD[n,#]==1]&]],{n,100}] (* Corrected by Peter Luschny, May 17 2018 *)

ispow(n) = (n==1) || ispower(n);
a(n) = #select(x->(ispow(x) && (gcd(n, x) == 1)), [1..n-1]); \\ Michel Marcus, May 17 2018

def a(n):
    return len([k for k in (1..n-1) if k.is_perfect_power() and gcd(n,k) == 1])
[a(n) for n in (1..84)] # Peter Luschny, May 16 2018

a(1) = 0 corrected by Zak Seidov, May 15 2018

A362424 Number of partitions of n into 2 distinct perfect powers (A001597).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 2, 1, 1, 2, 1, 0, 0, 2, 2, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 2, 1, 0, 0, 0, 2, 1, 1, 0, 1, 0, 1, 0, 2, 0, 0, 2, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 2, 0, 0, 0, 2, 1, 1

Offset: 0

Ilya Gutkovskiy, Apr 19 2023

nonn

Karl-Heinz Hofmann, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Perfect Power.

Cf. A001597, A072103, A362425.

perfectPowerQ[n_] := n == 1 || GCD @@ FactorInteger[n][[;; , 2]] > 1; a[n_] := Count[IntegerPartitions[n, {2}], ?(AllTrue[#, perfectPowerQ] && UnsameQ @@ # &)]; Array[a, 100, 0] (* _Amiram Eldar, May 05 2023 *)

import numpy
from math import isqrt
A072103 = []
for m in range(2,isqrt(10001)+1):
    k = 2
    while m**k < 10001:
        A072103.append(m**k)
        k += 1
A001597 = sorted(set(A072103)) # eliminates multiples and sorting
A362424 = numpy.zeros(10001+1, dtype="i4")
A001597 = [1] + A001597 # we need a "1" in front of A001597
a = 0
while A001597[a] < 10001 // 2:
    b = a + 1
    while b < len(A001597) and A001597[a] + A001597[b] < 10001:
        A362424[A001597[a] + A001597[b]] += 1
        b += 1
    a += 1
print(list(A362424[0:92])) # Karl-Heinz Hofmann, Sep 16 2023

A379962 Numbers k such that A276086(k)+1 is a perfect power (A001597), where A276086 is the primorial base exp-function.

Original entry on oeis.org

2, 8, 30, 34, 36, 214, 248, 254, 421, 2311, 2318, 2350, 2520, 2564, 2776, 4654, 5076, 30038, 30092, 30120, 30480, 32374, 510515, 510542, 510547, 510728, 510746, 512886, 515134, 540540, 540818, 542862, 542888, 1021442, 9699702, 9699722, 9699772, 9699788, 9702010, 9702256, 9729938, 9734358, 10210414, 10217558, 10240472, 10240724

Offset: 1

Antti Karttunen, Jan 24 2025

nonn

			A276086(30) = 7, +1 = 8 = 2^3, therefore 30 is included.
A276086(2311) = 26, +1 = 27 = 3^3, therefore 2311 is included.

Antti Karttunen, Table of n, a(n) for n = 1..93
Index entries for sequences related to primorial base.

Setwise difference A379960 \ A379961.

Cf. A001597, A276086, A379963 (subsequence).

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
is_A379962(n) = ispower(1+A276086(n));

A127913 Least k >= 0 such that A001597(n)+k is an even semiprime.

Original entry on oeis.org

3, 0, 2, 1, 6, 1, 7, 2, 2, 9, 10, 1, 6, 1, 9, 6, 2, 9, 6, 2, 1, 11, 6, 9, 2, 3, 1, 22, 5, 18, 2, 9, 10, 1, 18, 5, 10, 1, 14, 13, 6, 18, 5, 18, 1, 10, 15, 13, 10, 1, 18, 25, 26, 2, 9, 6, 1, 14, 6, 7, 9, 9, 2, 1, 18, 1, 18, 2, 9, 2, 21, 9, 6, 5, 22, 11, 1, 2, 1, 18, 5, 10, 1, 2, 13, 42, 1, 18, 5, 1, 2

Offset: 1

Giovanni Teofilatto, Apr 06 2007

			A001597(5) = 16. Among 16+0 = 16, 16+1 = 17, 16+2 = 18 = 2*3*3, 16+3 = 19, 16+4 = 20 = 2*2*5, 16+5 = 21 = 3*7 there is no even semiprime, but 16+6 = 22 = 2*11 is an even semiprime. Hence a(5) = 6.
A001597(14) = 121. 121+0 = 121 = 11*11 is not even, but 121+1 = 122 = 2*61 is an even semiprime. Hence a(14) = 1.

Cf. A001597 (perfect powers).

PP:=[1] cat [ n: n in [2..5184] | IsPower(n) ]; [ k: p in PP | exists(k) {x: x in [0..100000] | IsEven(p+x) and IsPrime((p+x) div 2) } ]; /* Klaus Brockhaus, Apr 09 2007 */

Edited, corrected and extended by Klaus Brockhaus, Apr 09 2007

cp's OEIS Frontend

A251753 n!/pp, where pp is the largest perfect power (A001597) which divides n!.

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A070228 Number of perfect powers (A001597) not exceeding 2^n.

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A118715 Numbers n such that the digital reversal of n is a perfect power (A001597).

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A128948 Primes p for which the period length of 1/p is a perfect power, A001597.

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A175050 Positive integers n where both n and the number of divisors of n are perfect powers. (Both n and d(n) are elements of A001597.)

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A189117 Conjectured number of pairs of consecutive perfect powers (A001597) differing by n.

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A304574 Number of perfect powers (A001597) less than n and relatively prime to n.

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A362424 Number of partitions of n into 2 distinct perfect powers (A001597).

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