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.
n=213 gives {213,713,3123,3473,15123,713},
n=323 gives {323,1917,713,3123,3473,15123,713},
n=639 gives {639,713,3123,3473,15123,713}.
a(13) = 313 because A192137(13) = 39 = 3 * 13.
f[n_] := FromDigits[Flatten[IntegerDigits /@ FactorInteger[n][[;; , 1]]]]; Select[f /@ Range[2, 1900], PalindromeQ] (* Amiram Eldar, Aug 06 2024 *)
Concatenation of prime divisors of number 747 = 3^2 * 83 is 383 (palindromic number).
f[n_] := FromDigits[Flatten[IntegerDigits /@ FactorInteger[n][[;; , 1]]]]; Select[Range[2, 20000], And @@ (PalindromeQ /@ {#, f[#]}) &] (* Amiram Eldar, Aug 06 2024 *)
a(21) = 383 because A192140(21) = 747 = 3 * 83.
f[n_] := FromDigits[Flatten[IntegerDigits /@ FactorInteger[n][[;; , 1]]]]; f /@ Select[Range[2, 20000], And @@ (PalindromeQ /@ {#, f[#]}) &] (* Amiram Eldar, Aug 06 2024 *)
1117 is in the sequence since pi(1117) = 11*17, 2163 is in the sequence since pi(2163) = 2*163, 2537 is in the sequence since pi(2537) = 2*5*37, and 5137 is in the sequence since pi(5137) = 5*137.
a251360[n_Integer] := Select[Range[n], # == FromDigits[Flatten@IntegerDigits[First@ Transpose@ FactorInteger[PrimePi[#]]]] &]; a251360[10^5] (* Michael De Vlieger, Dec 03 2014 *)
from sympy import prime, factorint A251360_list, p = [], 3 for n in range(2,10**6): q, fn = prime(n+1), factorint(n) m = int(''.join(str(d)*fn[d] for d in sorted(fn))) if p <= m < q: A251360_list.append(m) p = q # Chai Wah Wu, Dec 10 2014, corrected Apr 04 2018
4 is in the sequence since 4=2^2 and pi(4)=2, 100 is in the sequence since 100=2^2*5^2 and pi(100)=25, 31509 is in the sequence since 31509=3^4*389 and pi(31509)=3389, and 7560625 is in the sequence since 7560625=5^4*12097 and pi(7560625)=512097.
a251361[n_Integer] := Select[Range[n], PrimePi[#] == FromDigits[ Flatten@ IntegerDigits[First@ Transpose@ FactorInteger[#]]] &]; a251361[10^6] (* Michael De Vlieger, Dec 03 2014 *)
is(n)=eval(fold((x,y)->Str(x,y),factor(n)[,1]))==primepi(n) \\ Charles R Greathouse IV, Dec 06 2014
70 is in the sequence because the prime divisors of 70 are {2,5,7} and 257 is prime.
read("transforms") ;
isA209799 := proc(n)
local pdivs ;
if isprime(n) or n < 4 then
return false;
end if;
pdivs := sort(convert(numtheory[factorset](n),list)) ;
isprime(digcatL(pdivs)) ;
end proc:
for n from 4 to 200 do
if isA209799(n) then printf("%d,",n) ;
end if;
end do: # R. J. Mathar, Mar 19 2012
Select[Range[200],CompositeQ[#]&&PrimeQ[FromDigits[Flatten[ IntegerDigits/@ FactorInteger[#] [[;;,1]]]]]&] (* Harvey P. Dale, Apr 10 2023 *)
25 is in the sequence since phi(25)=2^2*5, 235741 is in the sequence since phi(235741)=2^4*3^2*5*7*41, 23517131 is in the sequence since phi(23517131)=2^7*3*5^2*17*131.
a251362[n_Integer] := Rest@ Select[Range[n], # == FromDigits[Flatten@IntegerDigits[First@Transpose@FactorInteger[EulerPhi[#]]]] &]; a251362[10^6] (* Michael De Vlieger, Dec 03 2014 *)
a(60) = a(2^2 * 3 * 5) = a(prime(1)^2 * prime(2) * prime(3)) = 123.
a[n_] := FromDigits[Flatten@IntegerDigits@(PrimePi[#[[1]]] & /@ FactorInteger[n])]; Table[a[n], {n, 1, 70}]
n=4: primorial[4]=2310; a(4)=4359293547691=A084318[2310]; the list of iterations:
{2310, 235711, 7151223, 34495309, 41841349, 1116722777, 1958774883, 313113444469, 744730492067, 4359293547691};
at each step the ordered prime factors of previous term are concatenated.
ffi[x_] := Flatten[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] q[x_] := Apply[Times, Table[Prime[w], {w, 1, x}]] lf[x_] := Length[FactorInteger[x]] nd[x_, y_] := 10*x+y tn[x_] := Fold[nd, 0, x] coc[x_] := Fold[nd, 0, Flatten[IntegerDigits[ba[x]], 1]] Table[FixedPoint[coc, q[w]], {w, 1, 7}]
Comments