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A373274
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Sigma-phi numbers: composite numbers k such that k-1 is a multiple of Sum_{prime^h|k} phi(prime^h), where h is the maximum integer such that prime^h|k and phi = A000010.
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0
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65, 66, 91, 154, 217, 369, 370, 451, 481, 561, 630, 703, 783, 793, 1065, 1106, 1353, 1407, 1463, 1729, 1854, 1891, 1921, 2002, 2059, 2146, 2201, 2415, 2501, 2701, 3451, 3550, 3870, 4033, 4081, 4097, 4225, 4625, 4642, 4681, 4699, 4921, 5215, 5457, 5626, 5830, 5833, 5929, 5985
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OFFSET
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1,1
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LINKS
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EXAMPLE
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For k = 3550 = 2 * 5^2 * 71, k-1 = 3549 is divisible by phi(2) + phi(5^2) + phi(71) = (2-1) + (5^2-5) + (71-1) = 91.
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MAPLE
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with(numtheory):
f := proc(n)
nops(numtheory[factorset](n))
end proc:
sum_phi := proc(n)
local prime_divisors, sum:
prime_divisors := ifactors(n)[2]:
if f(n) >= 2 then
sum := add(prime_divisors[i][1]^prime_divisors[i][2] - prime_divisors[i][1]^(prime_divisors[i][2]-1), i = 1 .. nops(prime_divisors)):
else
sum := 0:
end if:
sum:
end proc:
valid_n := proc(n)
local sum_val:
sum_val := sum_phi(n):
if sum_val <> 0 and (n-1) mod sum_val = 0 then
return true:
else
return false:
end if:
end proc:
result := select(valid_n, [$2 .. 5000]);
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MATHEMATICA
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f[p_, e_] := (p-1) * p^(e-1); Select[Range[6000], CompositeQ[#] && Divisible[# - 1, Plus @@ f @@@ FactorInteger[#]] &] (* Amiram Eldar, May 29 2024 *)
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PROG
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(PARI) isok(k) = if (k>1 && !isprime(k), my(f=factor(k)); (k-1) % sum(i=1, #f~, eulerphi(f[i, 1]^f[i, 2])) == 0; ) \\ Michel Marcus, May 29 2024
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CROSSREFS
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Cf. A000010 (Euler totient function phi).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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