A076467 -id:A076467 - cp's OEIS frontend

A378164 Smaller of consecutive terms b < c of A076467 such that the quality q=log(rad(c))/log(rad((c-b)bc)) of the abc-triple c-b,b,c with gcd(c-b,b,c)=1 sets a new record.

1, 81, 1296, 2187, 1419857

Offset: 1

Hugo Pfoertner, Nov 18 2024

If it exists, a(6)>5*10^27.

			           Pairs b,c of consecutive
           terms of A076467
  A378165
    c-b        b        c     Quality = log(rad(c))/log(rad((c-b)*b*c))
      7,       1,       8,    0.262649535...
     44,      81,     125,    0.277532712...
     35,    1296,    1331,    0.309605475...
     10,    2187,    2197,    0.429917243...
  23040, 1419857, 1442897,    0.431260235...

Wikipedia, abc conjecture

A378165 gives the corresponding values of c-b.

Cf. A007947 (rad), A076467, A377933, A377934, A378166, A378167.

\\ Uses M. F. Hasler's A076467_vec from A076467
a378164_5(upto) = {my(W=A076467_vec(upto), qw=0); for(k=2, #W, my(d=W[k]-W[k-1]); if(gcd([d,W[k],W[k-1]])==1, my(C=factor(W[k])[,1], B=factor(W[k-1])[,1], A=factor(d)[,1], P=vecprod(setunion(setunion(Set(B),Set(C)),Set(A))), q=log(vecprod(C))/log(P)); if(q>qw, print([d,W[k-1],W[k],q]); qw=q)))};
a378164_5(10^16)

A378165 Differences between adjacent terms of A076467 that correspond to the locations of abc-quality records of A378164.

Original entry on oeis.org

7, 44, 35, 10, 23440

Offset: 1

Hugo Pfoertner, Nov 19 2024

See A378164 for more information and examples.

Cf. A076467, A378164.

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

Original entry on oeis.org

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

A377933 First differences of consecutive perfect powers m^k with k>=3 (A076467).

Original entry on oeis.org

7, 8, 11, 5, 32, 17, 44, 3, 88, 27, 13, 87, 169, 113, 104, 271, 24, 272, 35, 397, 320, 139, 10, 204, 343, 381, 250, 721, 817, 919, 729, 298, 917, 224, 192, 1069, 739, 648, 1519, 1657, 817, 984, 759, 423, 769, 2107, 1053, 1216, 2437, 2611, 1561, 1230, 2977, 3169, 2479, 888

Offset: 1

Hugo Pfoertner, Nov 24 2024

Hugo Pfoertner, Table of n, a(n) for n = 1..10000
Plot of log_10(a(n)) vs log_10(n), using Plot 2.

Cf. A001597, A053289, A076467.

N:= 10^5: # for terms <= N of A076467
S:= sort(convert({1, seq(seq(m^k, m = 2 .. floor(N^(1/k))), k=3..ilog2(N))},list)):
S[2..-1]-S[1..-2]; # Robert Israel, Nov 24 2024

lista(nn) = my(S=List(1)); for(x=2, sqrtnint(nn, 3), for(k=3, logint(nn, x), listput(S, x^k))); my(v=Set(S)); vector(#v-1, k, v[k+1]-v[k]); \\ Michel Marcus, Nov 24 2024

a(n) = A076467(n+1) - A076467(n).

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

A378166 Terms c = A076467(k) such that the distinct prime factors of b = A076467(k-1) and of c-b are subsets of the prime factors of c, i.e., rad(c)/rad((c-b)bc) = 1.

Original entry on oeis.org

16, 64, 2744, 474552, 157529610000, 407165596771032, 1491025241529616, 173903694695292024, 661905356066769705912, 14918256451377811247508792, 19801061641727872277815512, 2718924063971620383558231552

Offset: 1

Hugo Pfoertner, Nov 20 2024

a(13) > 5*10^27.

			                      Pairs b,c of consecutive terms of A076467
          A378167
              c-b                      b               c = a(n)
                8,                     8,                    16,
               32,                    32,                    64,
              343,                  2401,                  2744,
            17576,                456976,                474552,
         65610000,          157464000000,          157529610000,
      11329982936,       407154266788096,       407165596771032,
      26102469128,      1490999139060488,      1491025241529616,
     315404039943,    173903379291252081,    173903694695292024,
  152838610998696, 661905203228158707216, 661905356066769705912.

A378167 gives the corresponding values of c-b.

Cf. A007947 (rad), A076467, A378164, A378165.

\\ Uses M. F. Hasler's A076467_vec from A076467
rad(x) = vecprod(factor(x)[,1]);
a378166_7(upto) = {my(W=A076467_vec(upto)); for(k=2, #W, my(d=W[k]-W[k-1], q=rad(W[k])/rad(W[k]*W[k-1]*d)); if(q==1, print([d, W[k-1], W[k]])))};
\\ Alternative program not using rad, more efficient
a378166_7(upto) = {my(W=A076467_vec(upto)); for(k=2, #W, my(C=Set(factor(W[k])[,1]), d=W[k]-W[k-1]); if(#setminus(Set(factor(d)[,1]), C)>0, , if(#setminus(Set(factor(W[k-1])[,1]), C)==0, print([d, W[k-1], W[k]]))))};
a378166_7(10^18)

A378167 Differences between adjacent terms of A076467 that correspond to the locations described by A378166.

Original entry on oeis.org

8, 32, 343, 17576, 65610000, 11329982936, 26102469128, 315404039943, 152838610998696, 7327416190396311, 146668341275463896, 1097750613982270976

Offset: 1

Hugo Pfoertner, Nov 20 2024

See A378166 for more information and examples.

Cf. A076467, A378164, A378165, A378166.

A378287 Numbers not of the form m^k for some k>=3. Complement of A076467.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 1

Chai Wah Wu, Nov 21 2024

Differs from A362147 at a(419).

Cf. A076467, A362147.

from sympy import integer_nthroot, mobius
def A378287(n):
    def f(x): return int(n+integer_nthroot(x,4)[0]-sum(mobius(k)*(integer_nthroot(x,k)[0]+integer_nthroot(x,k<<1)[0]-2) for k in range(3,x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m

A153158 a(n) = A007916(n)^2.

Original entry on oeis.org

4, 9, 25, 36, 49, 100, 121, 144, 169, 196, 225, 289, 324, 361, 400, 441, 484, 529, 576, 676, 784, 841, 900, 961, 1089, 1156, 1225, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2500, 2601, 2704, 2809, 2916, 3025, 3136, 3249, 3364, 3481

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 19 2008

nonn

A378168(n) is the number of terms <= 10^n. - Chai Wah Wu, Nov 21 2024

			2^2 = 4, 3^2 = 9, 4^2 = 16 = 2^4 is not in the sequence, 5^2 = 25, 6^2 = 36, ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A007916, A153147, A153157, A153159, A153160, A062503, A076467, A378168, A378287.

a153158 n = a153158_list !! (n-1)
a153158_list = filter ((== 2) . foldl1 gcd . a124010_row) [2..]
-- Reinhard Zumkeller, Apr 13 2012

q:= n-> is(igcd(seq(i[2], i=ifactors(n)[2]))=2):
select(q, [i^2$i=2..60])[];  # Alois P. Heinz, Nov 26 2024

Select[Range[2,100],GCD@@Last/@FactorInteger@#==1&]^2

from sympy import mobius, integer_nthroot
def A153158(n):
    def f(x): return int(n+1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m**2 # Chai Wah Wu, Aug 13 2024

GCD(exponents in prime factorization of a(n)) = 2, cf. A124010. - Reinhard Zumkeller, Apr 13 2012
Sum_{n>=1} 1/a(n) = zeta(2) - 1 - Sum_{k>=2} mu(k)*(1 - zeta(2*k)) = 0.5444587396... - Amiram Eldar, Jul 02 2022
Intersection of A000290 and A378287. Squares that are not of the form m^k for some k>=3. - Chai Wah Wu, Nov 21 2024

Edited by Ray Chandler, Dec 22 2008

A340700 Lower of a pair of adjacent perfect powers, both with exponents > 2.

Original entry on oeis.org

27, 64, 125, 243, 1000, 1296, 2187, 50625, 59049, 194481, 279841, 456533, 614125, 3111696, 6434856, 22665187, 25411681, 38950081, 62742241, 96059601, 131079601, 418161601, 506250000, 741200625, 796594176, 1249198336, 2136719872, 2217342464, 5554571841, 5802782976

Offset: 1

Hugo Pfoertner, Jan 16 2021

nonn

It is conjectured that the intersection of A340700 and A340701 is empty, i.e., that no 3 immediately consecutive perfect powers with all exponents > 2 (A076467) exist. No counterexample < 3.4*10^30 was found.

			Initial terms of sequences A340700 .. A340706:
a(n) = x^p,
A340701(n) = A340703(n)^A340705(n) = y^q,
A340706(n) = A340701(n) - a(n) = y^q - x^p.
.
  n  a(n)    x ^  p  A340701    y ^  q  A340706 adjacent squares
  1    27 =  3 ^  3,      32 =  2 ^  5,      5  5^2=25, 6^2=36
  2    64 =  2 ^  6,      81 =  3 ^  4,     17  8^2=64, 9^2=81
  3   125 =  5 ^  3,     128 =  2 ^  7,      3  11^2=121, 12^2=144
  4   243 =  3 ^  5,     256 =  2 ^  8,     13  15^2=225, 16^2=256
  5  1000 = 10 ^  3,    1024 =  2 ^ 10,     24  31^2=961, 32^2=1024
  6  1296 =  6 ^  4,    1331 = 11 ^  3,     35  36^2=1296, 37^2=1369
  7  2187 =  3 ^  7,    2197 = 13 ^  3,     10  46^2=2116, 47^2=2209
  8 50625 = 15 ^  4,   50653 = 37 ^  3,     28  225^2=50625, 226^2=51076
  9 59049 =  3 ^ 10,   59319 = 39 ^  3,    270  243^2=59049, 244^2=59536

Hugo Pfoertner, Table of n, a(n) for n = 1..1670
StackExchange MathOverflow, Are there ever three perfect powers between consecutive squares? Answers by Gjergji Zaimi and Felipe Voloch (2011).
Michel Waldschmidt, Perfect Powers: Pillai's works and their developments, arXiv:0908.4031 [math.NT], 27 Aug 2009.

The corresponding upper members of the pairs are A340701.
Cf. A001597, A076467, A097056, A340642, A340702, A340703, A340704, A340705, A340706.
Cf. A117934 (excluding pairs where one of the members is a square).

a(n) = A340702(n)^A340704(n) = A340701(n) - A340706(n).

cp's OEIS Frontend

A378164 Smaller of consecutive terms b < c of A076467 such that the quality q=log(rad(c))/log(rad((c-b)*b*c)) of the abc-triple c-b,b,c with gcd(c-b,b,c)=1 sets a new record.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A378165 Differences between adjacent terms of A076467 that correspond to the locations of abc-quality records of A378164.

Views

Author

Keywords

Comments

Crossrefs

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A377933 First differences of consecutive perfect powers m^k with k>=3 (A076467).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

Formula

Extensions

A378166 Terms c = A076467(k) such that the distinct prime factors of b = A076467(k-1) and of c-b are subsets of the prime factors of c, i.e., rad(c)/rad((c-b)*b*c) = 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A378167 Differences between adjacent terms of A076467 that correspond to the locations described by A378166.

Views

Author

Keywords

Comments

Crossrefs

A378287 Numbers not of the form m^k for some k>=3. Complement of A076467.

Views

Author

Keywords

Comments

Crossrefs

Programs

Python

A153158 a(n) = A007916(n)^2.

Views

Author

Keywords

Comments

Examples

Links

A378164 Smaller of consecutive terms b < c of A076467 such that the quality q=log(rad(c))/log(rad((c-b)bc)) of the abc-triple c-b,b,c with gcd(c-b,b,c)=1 sets a new record.

A378166 Terms c = A076467(k) such that the distinct prime factors of b = A076467(k-1) and of c-b are subsets of the prime factors of c, i.e., rad(c)/rad((c-b)bc) = 1.