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(67500) = a(2^2*3^3*5^4) = a(4*27*625) = 427625.
Table[FromDigits@Flatten@IntegerDigits[#[[1]]^#[[2]] & /@ FactorInteger[n]], {n, 64}] (* Ivan Neretin, May 31 2016 *)
a(6)=223 because 6th composite number is 12: A002808(6)=12, then 12=2*2*3 and merging prime factors we get a(6)=223.
Reap[Do[If[!PrimeQ[n],Sow[FromDigits@Flatten[Table[IntegerDigits[ #[[1]]],{#[[2]]}]&/@FactorInteger@(n)]]],{n,2,115}]][[2,1]]
Some examples for the calculation of a(n):
(For digits 10,11...36 the letters A,B...Z are used.)
n -> prime factors -> a(n)(base) -> a(n)(base 10)
6 -> 2 * 3 -> 23 (4) -> 11
20 -> 2 * 2 * 5 -> 225 (6) -> 89
33 -> 3 * 11 -> 3B (12) -> 47
56 -> 2 * 2 * 2 * 7 -> 2227 (8) -> 1175
62 -> 2 * 31 -> 2U (32) -> 95
72 -> 2 * 2 * 2 * 3 * 3 ->22233 (4) -> 687
100 -> 2 * 2 * 5 * 5 -> 2255 (6) -> 539
910 -> 2 * 5 * 7 * 13 -> 257D (14) -> 6579
a:= n-> (l-> (m-> add(l[-i]*m^(i-1), i=1..nops(l)))(1+
max(l)))(map(i-> i[1]$i[2], sort(ifactors(n)[2]))):
seq(a(n), n=2..77); # Alois P. Heinz, Jan 09 2021
a(n) = my(f=factor(n), list=List()); for (k=1, #f~, for (j=1, f[k, 2], listput(list, f[k,1]))); fromdigits(Vec(list), vecmax(f[,1])+1); \\ Michel Marcus, Jan 06 2021
def A(startn,lastn=0):
a,n,lastn=[],startn,max(lastn,startn)
while n<=lastn:
i,j,v,m,f=2,0,0,n,[]
while i
from sympy import factorint def fromdigits(d, b): n = 0 for di in d: n *= b; n += di return n def a(n): f = sorted(factorint(n, multiple=True)) return fromdigits(f, f[-1]+1) print([a(n) for n in range(2, 76)]) # Michael S. Branicky, Jan 06 2021
c(1) = 1 not prime -> 1 is not a term. c(2) = 2 prime -> 2 is a term. c(3) = 3 prime -> 3 is a term. c(4) = 22 not prime -> 4 is not a term. c(5) = 5 prime -> 5 is a term. c(6) = 23 prime -> 6 is a term.
q:= n-> isprime(parse(cat(sort(map(i-> i[1]$i[2], ifactors(n)[2]))[]))): select(q, [$2..222])[]; # Alois P. Heinz, Mar 27 2024
m[{p_, e_}] := Table[p, {e}]; c[w_] := FromDigits[Join @@ IntegerDigits@ w]; Select[ Range@ 150, PrimeQ@ c@ Flatten[m /@ FactorInteger[#]] &] (* Giovanni Resta, Apr 23 2021 *)
from sympy import *
def b(n):
f=factorint(n)
l=sorted(f)
return 1 if n==1 else int("".join(str(i)*f[i] for i in l))
# print([b(n) for n in range(1, 101)])
for n in range(1,200):
if isprime(b(n)):
print (n)
a(7) = 14 is a term because A103168(14) = 41 mod 14 = 13 and A340592(14) = 27 mod 14 = 13.
revdigs:= proc(n) local L,i; L:= convert(n,base,10); add(L[-i]*10^(i-1),i=1..nops(L)) end proc: f:= proc(n) local L,p,i,r; L:= sort(map(t -> t[1]$t[2],ifactors(n)[2])); r:= L[1]; for i from 2 to nops(L) do r:= r*10^(1+max(0,ilog10(L[i])))+L[i] od; r end proc: f(1):= 1: select(n -> (f(n) - revdigs(n)) mod n = 0, [$1..20000]);
from sympy import factorint def A103168(n): return int(str(n)[::-1])%n def A340592(n): if n == 1: return 0 return int("".join(str(f) for f in factorint(n, multiple=True)))%n def ok(n): return A103168(n) == A340592(n) print([k for k in range(1, 20000) if ok(k)]) # Michael S. Branicky, Jan 18 2022
802 = 2*401 and 2401 = 49^2. 1131 = 3*13*29 and 31329 = 177^2.
f(n)={ n<4 & return(n); for(i=1, #n=factor(n)~, n[1, i]=concat(vector(n[2, i], j, Str(n[1, i])))); eval(concat(n[1, ]));} \\ A037276
isok(n) = (n>1) && issquare(f(n)); \\ Michel Marcus, Jan 28 2021
Since the prime factorization of 95 is (5, 19), and both 9 and 5 occur in (5, 19), the number 95 is on the list. Since the prime factorization of 1255 is (5, 251), and 1, 2, and both 5s occur in (5, 251), the number 1255 is on the list. 22 is not on the list because its prime factorization is (2, 11) and that does not have enough 2s. Nor is 25 on the list because for this sequence we express its factorization as (5, 5) rather than (5^2).
S:=[]; for n in [1..10000] do if not IsPrime(n) then u:=Intseq(n); f:=Factorization(n); v:=&cat[ [ f[j, 1]: i in [1..f[j, 2]] ]: j in [1..#f] ]; w:=&cat[ Intseq(p): p in v ]; if forall{ a: a in [0..9] | Multiplicity(SequenceToMultiset(u), a) le Multiplicity(SequenceToMultiset(w), a) } then Append(~S, n); end if; end if; end for; S; // Klaus Brockhaus, Jul 09 2011
Select[Range[2, 5000], Not[PrimeQ[#]] && Sort[DigitCount[FromDigits[Flatten[IntegerDigits/@Flatten[Table[#1, {#2}]&@@@FactorInteger[#]]]]] - DigitCount[#]][[1]] >= 0 &] (* Alonso del Arte, Jun 28 2011, based on HomePrimeStep function by Eric W. Weisstein *)
co[n_,k_]:=Nest[Flatten[IntegerDigits[{#, n}]]&, n, k-1]; A037276=Table[FromDigits[Flatten[IntegerDigits[co@@@FactorInteger[n]]]], {n, 3 10^5}]; Table[Sum[A037276[[k]], {k, 1, 2^n}], {n, 0, 18}] (* Vincenzo Librandi, Oct 01 2017 *)
A037276(k) = {for(i=1, #k=factor(k)~, k[1, i]=concat(vector(k[2, i], j, Str(k[1, i])))); eval(concat(k[1, ])); } lista(nn) = {my(s=1); print1(s); for(n=0, nn, s+=sum(k=2^n+1, 2*2^n, A037276(k)); print1(", ", s)); } \\ Jinyuan Wang, Feb 19 2020
2 = prime(1)^1 => a(2) = 11, 3 = prime(2)^1 => a(3) = 21, 4 = prime(1)^2 => a(4) = 12, 5 = prime(3)^1 => a(5) = 31, 6 = prime(1)^1*prime(2)^1 => a(1) = 1121, 7 = prime(3)^1 => a(7) = 41, 8 = prime(1)^3 => a(8) = 13, and so on.
a:= n-> `if`(n=1, 0, parse(cat(seq([numtheory[pi]
(i[1]), i[2]][], i=sort(ifactors(n)[2]))))):
seq(a(n), n=1..60); # Alois P. Heinz, Mar 16 2018
Array[FromDigits@ Flatten@ Map[{PrimePi@ #1, #2} & @@ # &, FactorInteger@ #] &, 51] (* Michael De Vlieger, Mar 16 2018 *)
A299400(n)=if(n=factor(n),eval(concat(apply(f->Str(primepi(f[1]),f[2]), Col(n)~))))
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