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Sum of the largest exponents A025479 of the first n perfect powers > 1.
+10
3
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
EXAMPLE
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.
MATHEMATICA
Union@ Accumulate@ Table[If[Set[e, GCD @@ #[[All, -1]]] > 1, e, 0] &@ FactorInteger@ n, {n, 4, 2400}] (* Michael De Vlieger, Jan 01 2019 *)
PROG
(PARI) my(s=0); for(k=1, 3^7, if(j=ispower(k), print1(s+=j, ", ")))
Sum of perfect powers <= n.
+10
2
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
FORMULA
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
EXAMPLE
Sum of the 8 perfect powers <= 32: a(32) = 1+4+8+9+16+25+27+32 = 122.
MAPLE
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:
PROG
(PARI)
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) 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, ...
+10
0
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
EXAMPLE
For n=8: a(8) = |1 - 4 + 8 - 9 + 16 - 25 + 27 - 32|.
MATHEMATICA
t = Select[Range@2400, # == 1 || GCD @@ Last /@ FactorInteger@# > 1 &]; Abs@ Accumulate[t (-1)^Range@ Length[t]] (* Giovanni Resta, Sep 11 2019 *)
PROG
(PARI) 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
Triangular numbers that for some k >= 0 are also the sum of the first k perfect powers.
+10
0
EXAMPLE
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.
MATHEMATICA
Join[{0}, Select[Accumulate[Select[Range[574564], # == 1 || GCD @@ FactorInteger[#][[All, 2]] > 1 &]], IntegerQ[Sqrt[8 # + 1]] &]]
Product of the first n perfect powers ( A001597).
+10
0
1, 1, 4, 32, 288, 4608, 115200, 3110400, 99532800, 3583180800, 175575859200, 11236854988800, 910185254092800, 91018525409280000, 11013241574522880000, 1376655196815360000000, 176211865192366080000000, 25374508587700715520000000, 4288291951321420922880000000, 840505222458998500884480000000, 181549128051143676191047680000000
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