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User: Philippe Guglielmetti

Philippe Guglielmetti has authored 2 sequences.

A318530 Numbers that can be written in more than one way as p^2 + q^3 + r^4 with p, q and r primes.

145, 210, 637, 754, 2317, 2530, 2917, 5218, 5437, 5890, 6447, 6997, 7469, 7653, 7738, 8650, 9333, 11818, 12417, 12796, 14770, 15178, 15197, 15295, 15513, 16349, 16501, 17367, 18389, 19709

Offset: 1

Philippe Guglielmetti, Aug 28 2018

nonn

			a(1) = 145 = 2^2 + 5^3 + 2^4 = 11^2 + 2^3 + 2^4 .
The first term which can be written in three different ways is
17367 = 23^2 + 13^3 + 11^4 = 113^2 + 13^3 + 7^4 = 131^2 + 5^3 + 3^4 .

Robert Israel, Table of n, a(n) for n = 1..10000

Subset of A134657.

N:= 10^5: # to get terms <= N
P:= select(isprime, [2,seq(i,i=3..floor(sqrt(N)))]):
Psq:= map(`^`,P,2):
P3:= select(`<=`,map(`^`,P,3),N):
P4:= select(`<=`,map(`^`,P,4),N):
V:= Vector(N):
for a in Psq do for b in P3 do for c in P4 do
   s:= a+b+c;
   if s <= N then V[s]:= V[s]+1 fi
od od od:
select(t -> V[t]>=2, [$1..N]); # Robert Israel, Jan 30 2019

A303935 Size of orbit of n under repeated application of sum of factorial of digits of n.

Original entry on oeis.org

2, 1, 1, 16, 8, 10, 15, 32, 36, 35, 2, 2, 17, 33, 13, 10, 15, 32, 36, 35, 17, 17, 9, 37, 7, 12, 6, 8, 33, 31, 33, 33, 37, 18, 34, 31, 48, 39, 24, 8, 13, 13, 7, 34, 30, 54, 42, 39, 29, 52, 10, 10, 12, 31, 54, 10, 24, 21, 41, 24, 15, 15, 6, 48, 42, 24, 12, 42

Offset: 0

Philippe Guglielmetti, May 03 2018

Numbers n for which a(n)=1 are called factorions (A014080).
Apart from factorions, only 3 cycles exist:
169 -> 363601 -> 1454 -> 169, so a(169) = a(363601) = a(1454) = 3.
871 -> 45361 -> 871, so a(871) = a(45361) = 2.
872 -> 45362 -> 872, so a(872) = a(45362) = 2.
All other n produce a chain reaching either a factorion or a cycle.

			For n = 4, 4!=24, 2!+4!=26, 2!+6!=722, 7!+2!+2!=5044, 5!+0!+4!+4!=169, 1!+6!+9!=363601, 3!+6!+3!+6!+0!+1!=1454, then 1!+4!+5!+4!=169 which already belongs to the chain, so a(4) = length of [4, 24, 26, 722, 5044, 169, 363601, 1454] = 8.

Philippe Guglielmetti, Table of n, a(n) for n = 0..1000
S. S. Gupta, Sum of the factorials of the digits of integers, The Mathematical Gazette, 88-512 (2004), 258-261.
Project Euler, Problem 74: Digit factorial chains

Cf. A061602, A014080 (contains n for which a(n) = 1).

Array[Length@ NestWhileList[Total@ Factorial@ IntegerDigits@ # &, #, UnsameQ, All, 100, -1] &, 68, 0] (* Michael De Vlieger, May 10 2018 *)

f(n) = if (!n, n=1); my(d=digits(n)); sum(k=1, #d~, d[k]!);
a(n) = {my(v = [n], vs = Set(v)); for (k=1, oo, new = f(n); if (vecsearch(vs, new), return (#vs)); v = concat(v, new); vs = Set(v); n = new;);} \\ Michel Marcus, May 18 2018

for n in count(0):
    l=[]
    i=n
    while i not in l:
        l.append(i)
        i=sum(map(factorial,map(int,str(i))))
    print(n,len(l))

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User: Philippe Guglielmetti

A318530 Numbers that can be written in more than one way as p^2 + q^3 + r^4 with p, q and r primes.

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A303935 Size of orbit of n under repeated application of sum of factorial of digits of n.

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