A076408 -id:A076408 - cp's OEIS frontend

A322969 Sum of the largest exponents A025479 of the first n perfect powers > 1.

2, 5, 7, 11, 13, 16, 21, 23, 25, 31, 35, 37, 39, 42, 49, 51, 53, 55, 58, 60, 65, 73, 75, 77, 80, 82, 84, 86, 88, 97, 99, 101, 105, 107, 113, 115, 117, 119, 121, 124, 134, 136, 138, 140, 144, 147, 149, 151, 153, 155, 157, 160, 162, 164, 166, 168, 179, 181, 188

Offset: 1

Hugo Pfoertner, Jan 01 2019

nonn

			a(1) = 2 because the first perfect power 4 = 2^2,
a(2) = 5: added exponent 3 from 8 = 2^3,
a(3) = 7: added exponent 2 from 9 = 3^2,
a(4) = 11: added largest exponent 4 from 16=2^4.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A001597, A025479, A076408.

Union@ Accumulate@ Table[If[Set[e, GCD @@ #[[All, -1]]] > 1, e, 0] &@ FactorInteger@ n, {n, 4, 2400}] (* Michael De Vlieger, Jan 01 2019 *)

my(s=0); for(k=1, 3^7, if(j=ispower(k), print1(s+=j, ", ")))

A076407 Sum of perfect powers <= n.

Original entry on oeis.org

1, 1, 1, 5, 5, 5, 5, 13, 22, 22, 22, 22, 22, 22, 22, 38, 38, 38, 38, 38, 38, 38, 38, 38, 63, 63, 90, 90, 90, 90, 90, 122, 122, 122, 122, 158, 158, 158, 158, 158, 158, 158, 158, 158, 158, 158, 158, 158, 207, 207, 207, 207, 207, 207, 207, 207, 207, 207, 207, 207, 207

Offset: 1

Reinhard Zumkeller, Oct 09 2002

nonn

			Sum of the 8 perfect powers <= 32: a(32) = 1+4+8+9+16+25+27+32 = 122.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Perfect Powers.

Cf. A001597, A076408, A069623.

N:= 100: # for a(1)..a(N)
V:= Vector(N,1):
pps:= {seq(seq(x^k,k=2..floor(log[x](N))),x=2..floor(sqrt(N)))}:
for y in pps do
   V[y..N]:= V[y..N] +~ y
od:
convert(V,list); # Robert Israel, Oct 19 2023

F(k,n) = (subst(bernpol(k+1), x, n+1) - subst(bernpol(k+1), x, 1)) / (k+1);
a(n) = 1 - sum(k=2, logint(n,2), moebius(k) * (F(k, sqrtnint(n,k)) - 1)); \\ Daniel Suteu, Aug 19 2023

a(n) = 1 - Sum_{k=2..floor(log_2(n))} mu(k) * (F(k, floor(n^(1/k))) - 1), where F(k, n) = Sum_{j=1..n} j^k = (Bernoulli(k+1, n+1) - Bernoulli(k+1, 1))/(k+1). - Daniel Suteu, Aug 19 2023

A326119 a(n) is the absolute value of the alternating sum of the first n increasing perfect powers (A001597): 1, 1-4, 1-4+8, 1-4+8-9, ...

Original entry on oeis.org

1, 3, 5, 4, 12, 13, 14, 18, 18, 31, 33, 48, 52, 69, 56, 72, 72, 97, 99, 117, 108, 135, 121, 168, 156, 187, 174, 226, 215, 269, 243, 286, 290, 335, 341, 388, 396, 445, 455, 506, 494, 530, 559, 597, 628, 668, 663, 706, 738, 783, 817, 864, 864, 900, 949, 987, 1038

Offset: 1

Richard Locke Peterson, Sep 10 2019

nonn

			For n=8: a(8) = |1 - 4 + 8 - 9 + 16 - 25 + 27 - 32|.

Cf. A001597, A076408.

t = Select[Range@2400, # == 1 || GCD @@ Last /@ FactorInteger@# > 1 &]; Abs@ Accumulate[t (-1)^Range@ Length[t]] (* Giovanni Resta, Sep 11 2019 *)

seq(n)={my(v=vector(n), i=0, k=0, s=0); while(i<#v, k++; if(ispower(k)||k==1, s=k-s; i++; v[i]=abs(s))); v} \\ Andrew Howroyd, Sep 10 2019

a(n) = abs(Sum_{k=1..n} (-1)^k*A001597(k)). - Andrew Howroyd, Sep 10 2019

A365019 Triangular numbers that for some k >= 0 are also the sum of the first k perfect powers.

Original entry on oeis.org

0, 1, 159284476

Offset: 1

Ilya Gutkovskiy, Aug 16 2023

			159284476 is a term because 159284476 = 1 + 2 + 3 + 4 + ... + 17848 = 1 + 4 + 8 + 9 + ... + 574564 = 1^2 + 2^2 + 2^3 + 3^2 + ... + 758^2.

Cf. A000217, A001597, A066527, A076408, A298270.

Join[{0}, Select[Accumulate[Select[Range[574564], # == 1 || GCD @@ FactorInteger[#][[All, 2]] > 1 &]], IntegerQ[Sqrt[8 # + 1]] &]]

A380318 Product of the first n perfect powers (A001597).

Original entry on oeis.org

1, 1, 4, 32, 288, 4608, 115200, 3110400, 99532800, 3583180800, 175575859200, 11236854988800, 910185254092800, 91018525409280000, 11013241574522880000, 1376655196815360000000, 176211865192366080000000, 25374508587700715520000000, 4288291951321420922880000000, 840505222458998500884480000000, 181549128051143676191047680000000

Offset: 0

Ilya Gutkovskiy, Jan 20 2025

nonn

Eric Weisstein's World of Mathematics, Perfect Power.

Cf. A000442, A001044, A001597, A002110, A024923, A036691, A076408, A111059.

Join[{1},FoldList[Times,Join[{1},Select[Range[250],GCD@@FactorInteger[#][[All,2]]>1&]]]] (* Harvey P. Dale, May 03 2025 *)

cp's OEIS Frontend

A322969 Sum of the largest exponents A025479 of the first n perfect powers > 1.

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A076407 Sum of perfect powers <= n.

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A326119 a(n) is the absolute value of the alternating sum of the first n increasing perfect powers (A001597): 1, 1-4, 1-4+8, 1-4+8-9, ...

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A365019 Triangular numbers that for some k >= 0 are also the sum of the first k perfect powers.

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Mathematica

A380318 Product of the first n perfect powers (A001597).

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Mathematica