A070428 -id:A070428 - cp's OEIS frontend

A378168 a(n) is the number of squares <= 10^n that are not higher powers, i.e., terms of A076467.

2, 6, 24, 87, 292, 959, 3089, 9875, 31410, 99633, 315589, 998889, 3160340, 9996605, 31616816, 99989509, 316209268, 999967330, 3162219896, 9999897769, 31622595517, 99999679010, 316227196708, 999998989804, 3162275866962, 9999996815862, 31622770946248, 99999989953079

Offset: 1

Hugo Pfoertner, Nov 20 2024

nonn

			a(1) = 2: squares <= 10 are 2^2 and 3^2;
a(2) = 6: 2 squares <= 10 and 5^2, 6^2, 7^2, 10^2, but not 4^2=2^4, 8^2=2^6, and 9^2=3^4;
a(3) = 24: 6 squares <= 100 and all squares between 11^2 and 31^2, except for 16^2=2^8, 25^2=5^4, and 27^2=3^6.

Chai Wah Wu, Table of n, a(n) for n = 1..2000

Cf. A000290, A001597, A070428, A076467, A089579.

Table[Sum[MoebiusMu[k]*Floor[10^(n/(2k))-1],{k,Floor[Log2[10^n]-1]}],{n,28}] (* James C. McMahon, Nov 21 2024 *)

from math import gcd
from sympy import integer_nthroot, mobius
def A378168(n): return sum(mobius(k)*(integer_nthroot(10**(n//(a:=gcd(n,b:=k<<1))), b//a)[0]-1) for k in range(1, (10**n).bit_length()-1)) # Chai Wah Wu, Nov 20 2024

a(n) = Sum_{k=1..floor(log_2(10^n)-1)} mu(k)*floor(10^(n/(2k))-1). - Chai Wah Wu, Nov 20 2024

a(20) onwards from Chai Wah Wu, Nov 20 2024

A377934 a(n) is the number of perfect powers m^k with k>=3 (A076467) <= 10^n.

Original entry on oeis.org

1, 2, 7, 17, 38, 75, 152, 306, 616, 1260, 2598, 5401, 11307, 23798, 50316, 106776, 227236, 484737, 1036002, 2217529, 4752349, 10194727, 21887147, 47020054, 101065880, 217325603, 467484989, 1005881993, 2164843035, 4660016778, 10032642455, 21602193212, 46518438071

Offset: 0

Hugo Pfoertner, Nov 24 2024

nonn

			a(0) = 1: 1^k with any k>2 (<= 10^0);
a(1) = 2: 1 and 2^3 (<=10^1);
a(2) = 7: 2 powers <= 10 and 16, 27, 32, 64, 81 (<=10^2).

Chai Wah Wu, Table of n, a(n) for n = 0..2999

Cf. A070428, A076467, A378168.

from math import gcd
from sympy import integer_nthroot, mobius
def A377934(n): return int(integer_nthroot(10**(n//(a:=gcd(n,4))),4//a)[0]-sum(mobius(k)*(integer_nthroot(10**(n//(b:=gcd(n,k))),k//b)[0]+integer_nthroot(10**(n//(c:=gcd(n,d:=k<<1))),d//c)[0]-2) for k in range(3,(10**n).bit_length()))) # Chai Wah Wu, Nov 24 2024

a(n) = 10^n - Sum_{k=1..floor(log2(10^n))} mu(k)*(floor(10^(n/k))+floor(10^(n/(2k)))-2). - Chai Wah Wu, Nov 24 2024

a(28) onwards from Chai Wah Wu, Nov 24 2024

A380337 Number of perfect powers (in A001597) that do not exceed primorial A002110(n).

Original entry on oeis.org

1, 1, 2, 7, 19, 63, 208, 802, 3344, 15576, 82368, 453834, 2743903, 17510668, 114616907, 785002449, 5711892439, 43861741799, 342522899289, 2803468693325, 23621594605383, 201819398349092, 1793794228847381, 16342173067958793, 154171432351500060, 1518411003599957803

Offset: 0

Michael De Vlieger, Jan 21 2025

nonn

In other words, A001597(a(n)) is the largest perfect power less than or equal to A002110(n).

			Let P = A002110 and let s = A001597.
a(0) = 1 since P(0) = 1, and the set s(1) = {1} contains k that do not exceed 1.
a(1) = 1 since P(1) = 2, and the set s(1) = {1} contains k <= 2.
a(2) = 2 since P(2) = 6, and the set s(1..2) = {1, 4} contains k <= 6.
a(3) = 7 since P(3) = 30, and the set s(1..7) = {1, 4, 8, 9, 16, 25, 27} contains k <= 30.
a(4) = 19 since P(4) = 210, and the set s(1..19) = {1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196} contains k <= 210, etc.

Michael De Vlieger, Table of n, a(n) for n = 0..632

Cf. A001597, A002110, A070228, A070428, A380254.

Map[1 - Sum[MoebiusMu[k]*Floor[#^(1/k) - 1], {k, 2, Floor[Log2[#]]}] &, FoldList[Times, 1, Prime[Range[30]]] ]

from sympy import primorial, mobius, integer_nthroot
def A380337(n):
    if n == 0: return 1
    p = primorial(n)
    return int(1-sum(mobius(k)*(integer_nthroot(p,k)[0]-1) for k in range(2,p.bit_length()))) # Chai Wah Wu, Jan 23 2025

A075127 Safe perfect powers: perfect powers n such that (n-1)/2 is also a perfect power.

Original entry on oeis.org

9, 243, 289, 9801, 332929, 11309769, 384199201, 13051463049, 443365544449, 15061377048201, 511643454094369, 17380816062160329, 590436102659356801, 20057446674355970889, 681362750825443653409

Offset: 1

Zak Seidov, Oct 11 2002

nonn

If both powers are squares, the smaller square is a triangular number, and all square triangular numbers (A001110) correspond to a member in this sequence. This proves that this sequence is infinite. Are there only finitely many other members, i.e., is A075127 \ A055792 finite? - Charles R Greathouse IV, Dec 12 2010

Cf. A001110, A001597, A055792, A070428, A075114.

pp = Select[ Range[10^8], Apply[ GCD, Last[ Transpose[ FactorInteger[ # ]]]] > 1 & ]; Select[pp, Apply[GCD, Last[ Transpose[ FactorInteger[( # - 1)/2]]]] > 1 & ]

for(n=1, 1e10, if(ispower(n) && ispower((n-1)/2), print1(n, ", "))) \\ Altug Alkan, Oct 28 2015

Conjectures from Colin Barker, Oct 28 2015: (Start)
a(n) = 35*a(n-1)-35*a(n-2)+a(n-3) for n>5.
G.f.: x*(234*x^4-8182*x^3+7901*x^2+72*x-9) / ((x-1)*(x^2-34*x+1)).
(End)

One more term from Robert G. Wilson v, Oct 16 2002

a(7)-a(15) from Donovan Johnson, Mar 10 2010

A111026 Perfect powers (A001597) of the form 3p + q + 3, p & q are primes.

Original entry on oeis.org

16, 25, 27, 32, 49, 121, 125, 128, 169, 225, 243, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1000, 1225, 1331, 1369, 1681, 1849, 2025, 2048, 2187, 2197, 2209, 2401, 2809, 3025, 3125, 3375, 3481, 3721, 3969, 4225, 4489, 4913, 5041, 5329, 5625, 5929, 6241

Offset: 1

Walter Kehowski, Oct 05 2005

nonn

The sequence has repetitions since different p's and q's will give the same perfect power. Remove the andmap in the program if you want the repetitions.
Includes all perfect powers, pp, (A001597) congruent +/- 1 (modulo 6). Also if pp-9 or pp-12 is a prime or if (pp -2)/3 or (pp-3)/3 is a prime.
The number of perfect powers of the form 3p + q + 3 <= 10^n: 0,5,21,56,157,433,...,. - Robert G. Wilson v, Jun 21 2006
In the first one million integers there are 1111 perfect powers (A070428) of which only 433 of them are of the form 3p + q + 3.

			a(5)=49 since 3*3+37+3=49 = 5*3+31+3 = 3*11+13+3 = 3*13+7+7 = 7^2.
6859 = 19^3 is in the sequence because there are 116 different ways to combine primes of the form 3p + q + 3, beginning with p=5 & q=6841 and ending with p=2281 & q=13.

Cf. A000040, A001597.

with(numtheory); egcd := proc(n) local L; L:=map(proc(z) z[2] end, ifactors(n)[2]); igcd(op(L)) end: PW:=[]: for z to 1 do for j from 1 to 100 do for k from 1 to 100 do p:=ithprime(j); q:=ithprime(k); x:=3*p+q+3; if egcd(x)>1 and andmap(proc(w) not(w[3]=x) end, PW) then PW:=[op(PW), [p,q,x]] fi od od od; PW; map(proc(z) z[3] end, PW);

fQ[n_] := GCD @@ Last /@ FactorInteger@n > 1; lst = {}; Do[p = Prime@j; q = Prime@k; x = 3p + q + 3; If[fQ@x, AppendTo[lst, x]], {j, 340}, {k, PrimePi[6856 - 3Prime@j]}]; Union@lst (* Robert G. Wilson v *)

a(n)=3p+q+3 where p and q are primes and a(n) is a perfect power.

Edited, corrected and extended by Robert G. Wilson v, Jun 21 2006

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A378168 a(n) is the number of squares <= 10^n that are not higher powers, i.e., terms of A076467.

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A377934 a(n) is the number of perfect powers m^k with k>=3 (A076467) <= 10^n.

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A380337 Number of perfect powers (in A001597) that do not exceed primorial A002110(n).

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A075127 Safe perfect powers: perfect powers n such that (n-1)/2 is also a perfect power.

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A111026 Perfect powers (A001597) of the form 3p + q + 3, p & q are primes.

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