This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(n)=prime(n)^58 \\ Charles R Greathouse IV, Jul 31 2011
a(n)=prime(n)^60
a(1)=793 because 2^6+3^6=793.
a = 6; Table[Prime[n]^a + Prime[n + 1]^a, {n, 1, 100}]
[1+(&+[NthPrime(n)^(k): k in [1..5]]): n in [1..100]]; // Berselli - Librandi, Apr 20 2011
Total[#^Range[0,5]]&/@Prime[Range[30]] (* Harvey P. Dale, Apr 20 2011 *)
From _R. J. Mathar_, Nov 05 2016: (Start) a(4)=2: 4^1 = 2^2. a(8)=2: 8^1 = 2^3. a(9)=2: 9^1 = 3^2. a(12)=2: 12^1 = 2^2*3^1. a(16)=4: 16^1 = 4^2 = 2^2*4^1 = 2^4. a(18)=2: 18^1 = 2*3^2. a(20)=2: 20^1 = 2^2*5^1. a(24)=3: 24^1 = 2^2*6^1 = 2^3*3^1. a(32)=5: 32^1 = 2^1*4^2 = 2^2*8^1 = 2^3*4^1 = 2^5. a(36)=4: 36^1 = 6^2 = 3^2*4^1 = 2^2*9^1. a(48)=5: 48^1 = 3^1*4^2 = 2^2*12^1 = 2^3*6^1 = 2^4*3^1. a(60)=2 : 60^1 = 2^2*15^1. a(64)=7: 64^1 = 8^2 = 4^3 = 2^2*16^1 = 2^3*8^1 = 2^4*4^1 = 2^6. a(72)=6 : 72^1 = 3^2*8^1 = 2^1*6^2 = 2^2*18^1 = 2^3*9^1 = 2^3*3^2. (End)
# Count solutions for products if n = dvs_i^exps(i) where i=1..pividx are fixed
Apiv := proc(n,dvs,exps,pividx)
local dvscnt, expscopy,i,a,expsrt,e ;
dvscnt := nops(dvs) ;
a := 0 ;
if pividx > dvscnt then
# have exhausted the exponent list: leave of the recursion
# check that dvs_i^exps(i) is a representation
if n = mul( op(i,dvs)^op(i,exps),i=1..dvscnt) then
# construct list of non-0 exponents
expsrt := [];
for i from 1 to dvscnt do
if op(i,exps) > 0 then
expsrt := [op(expsrt),op(i,exps)] ;
end if;
end do;
# check that list is duplicate-free
if nops(expsrt) = nops( convert(expsrt,set)) then
return 1;
else
return 0;
end if;
else
return 0 ;
end if;
end if;
# need a local copy of the list to modify it
expscopy := [] ;
for i from 1 to nops(exps) do
expscopy := [op(expscopy),op(i,exps)] ;
end do:
# loop over all exponents assigned to the next base in the list.
for e from 0 do
candf := op(pividx,dvs)^e ;
if modp(n,candf) <> 0 then
break;
end if;
# assign e to the local copy of exponents
expscopy := subsop(pividx=e,expscopy) ;
a := a+procname(n,dvs,expscopy,pividx+1) ;
end do:
return a;
end proc:
A255231 := proc(n)
local dvs,dvscnt,exps ;
if n = 1 then
return 1;
end if;
# candidates for the bases are all divisors except 1
dvs := convert(numtheory[divisors](n) minus {1},list) ;
dvscnt := nops(dvs) ;
# list of exponents starts at all-0 and is
# increased recursively
exps := [seq(0,e=1..dvscnt)] ;
# take any subset of dvs for the bases, i.e. exponents 0 upwards
Apiv(n,dvs,exps,1) ;
end proc:
seq(A255231(n),n=1..120) ; # R. J. Mathar, Nov 05 2016
1; 2,3; 4,9,25; 6,8,10,14; 16,81,625,2401,14641; ...
d = Table[Length[Divisors[n]], {n, 2000}]; t = {}; n = 0; ok = True; While[ok, n++; If[PrimeQ[n], AppendTo[t, Prime[Range[n]]^(n - 1)], c = Flatten[Position[d, n, 1, n]]; If[Length[c] >= n, AppendTo[t, c], ok = False]]]; Flatten[t] (* T. D. Noe, Jun 23 2013 *)
[NthPrime((n))^6 - NthPrime((n))^2: n in [1..30] ]; // Vincenzo Librandi, Jun 17 2011
a = {}; Do[p = Prime[n]; AppendTo[a, p^6 - p^2], {n, 1, 50}]; a
#^6-#^2&/@Prime[Range[20]] (* Harvey P. Dale, Jul 04 2023 *)
forprime(p=2,1e3,print1(p^6-p^2", ")) \\ Charles R Greathouse IV, Jun 16 2011
A232106 := Concatenation([267, 504], List(Filtered([5..10^5], IsPrime), p -> 3 * p^2 + 39 * p + 344 + 24 * Gcd(p-1, 3) + 11 * Gcd(p-1, 4) + 2 * Gcd(p-1, 5))); # Muniru A Asiru, Nov 16 2017
a:= n-> `if`(n<3, [267, 504][n], (c-> 386 +(45 +3*c)*c+
24*igcd(c, 3) +11*igcd(c, 4) +2*igcd(c, 5))(ithprime(n)-1)):
seq(a(n), n=1..40); # Alois P. Heinz, Nov 17 2017
Table[FiniteGroupCount[Prime[n]^6], {n, 40}] (* Michael De Vlieger, Apr 12 2016 *)
a(n) = if(n==1, 267, if (n==2, 504, my(p=prime(n)); 3*p^2 + 39*p + 344 + 24*gcd(p - 1, 3) + 11*gcd(p - 1, 4) + 2*gcd(p - 1, 5))); \\ Altug Alkan, Apr 12 2016
def A232106(n) : p = nth_prime(n); return 267 if p==2 else 504 if p==3 else 3*p^2 + 39*p + 344 + 24*gcd(p - 1, 3) + 11*gcd(p - 1, 4) + 2*gcd(p - 1, 5)
. n | p | a(n) | A069493(a(n)) = A030516(n) = p^6 . ----+----+-------+--------------------------------- . 1 | 2 | 2 | 64 . 2 | 3 | 6 | 729 . 3 | 5 | 13 | 15625 . 4 | 7 | 22 | 117649 . 5 | 11 | 45 | 1771561 . 6 | 13 | 58 | 4826809 . 7 | 17 | 87 | 24137569 . 8 | 19 | 102 | 47045881 . 9 | 23 | 135 | 148035889 . 10 | 29 | 181 | 594823321 . 11 | 31 | 199 | 887503681 . 12 | 37 | 252 | 2565726409 . 13 | 41 | 287 | 4750104241 . 14 | 43 | 306 | 6321363049 . 15 | 47 | 342 | 10779215329 . 16 | 53 | 401 | 22164361129 . 17 | 59 | 461 | 42180533641 . 18 | 61 | 479 | 51520374361 . 19 | 67 | 536 | 90458382169 . 20 | 71 | 583 | 128100283921 . 21 | 73 | 602 | 151334226289 . 22 | 79 | 665 | 243087455521 . 23 | 83 | 712 | 326940373369 . 24 | 89 | 776 | 496981290961 . 25 | 97 | 860 | 832972004929 .
import Data.List (elemIndex); import Data.Maybe (fromJust) a258603 = (+ 1) . fromJust . (`elemIndex` a258571_list) . a000040
\\ Gen(limit,k) defined in A036967. a(n)=#Gen(prime(n)^6,6) \\ Andrew Howroyd, Sep 10 2024
from math import gcd from sympy import prime, integer_nthroot, factorint def A258603(n): c, m = 0, prime(n)**6 for y1 in range(1,integer_nthroot(m,11)[0]+1): if all(d<=1 for d in factorint(y1).values()): for y2 in range(1,integer_nthroot(z2:=m//y1**11,10)[0]+1): if gcd(y2,y1)==1 and all(d<=1 for d in factorint(y2).values()): for y3 in range(1,integer_nthroot(z3:=z2//y2**10,9)[0]+1): if all(gcd(y3,x)==1 for x in (y1,y2)) and all(d<=1 for d in factorint(y3).values()): for y4 in range(1,integer_nthroot(z4:=z3//y3**9,8)[0]+1): if all(gcd(y4,x)==1 for x in (y1,y2,y3)) and all(d<=1 for d in factorint(y4).values()): for y5 in range(1,integer_nthroot(z5:=z4//y4**8,7)[0]+1): if all(gcd(y5,x)==1 for x in (y1,y2,y3,y4)) and all(d<=1 for d in factorint(y5).values()): c += integer_nthroot(z5//y5**7,6)[0] return c # Chai Wah Wu, Sep 10 2024
a(1) = 2^100, a(2) = 3^100, a(3) = 5^100, a(4) = 7^100.
a(n)=prime(n)^100 \\ Charles R Greathouse IV, Jun 19 2016
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