A067666 -id:A067666 - cp's OEIS frontend

A338093 Composite numbers which are multiples of the sum of the squares of their prime factors (taken with multiplicity).

16, 27, 256, 540, 756, 1200, 1890, 2940, 3060, 3125, 4050, 4200, 4320, 5460, 6000, 6048, 7920, 8232, 10080, 10164, 10368, 10530, 11232, 11286, 12960, 13104, 13524, 13800, 14000, 14157, 14175, 15708, 15960, 17280, 18200, 18480, 19278, 19683, 19992, 20295, 23814

Offset: 1

Giorgos Kalogeropoulos, Oct 09 2020

nonn

If a(n)=p1*p2*..*pk where p1,p2,..pk primes, then a(n)=m(p1^2+p2^2+..+pk^2) with m a positive integer.
For the special case of m=1, a(n) is equal to the sum of the squares of its prime factors.
There are only 5 known numbers to have this property:
16, 27 and three more numbers with 123, 163 and 179 digits found by Giorgos Kalogeropoulos (see Rivera links).
It is not known if any smaller numbers than those three exist for the case of m=1.
From Robert Israel, Oct 16 2020: (Start)
Suppose n is in the sequence with n = k*A067666(n).  Then n^m is in the sequence if m divides k^m (in particular for m=k).
For any prime p, p^(p^j) is in the sequence if j >= 1 (except j>=2 if p=2). (End)

			16 = 2*2*2*2 = 1*(2^2 + 2^2 + 2^2 + 2^2).
7920 = 2*2*2*2*3*3*5*11 = 44*(2^2 + 2^2 + 2^2 + 2^2 + 3^2 + 3^2 + 5^2 + 11^2).

Robert Israel, Table of n, a(n) for n = 1..2000
Carlos Rivera, Puzzle 625. Sum of squares of prime divisors, The Prime Puzzles and Problems Connection.
Carlos Rivera, Puzzle 1019. Follow-up to Puzzle 625, The Prime Puzzles and Problems Connection.

Cf. A067666.

filter:= proc(n) local t;
  if isprime(n) then return false fi;
  n mod add(t[1]^2*t[2],t=ifactors(n)[2]) = 0
end proc:
select(filter, [$4..30000]); # Robert Israel, Oct 16 2020

Select[Range@20000,Mod[#,Total[Flatten[Table@@@FactorInteger@#]^2]]==0&]

isok(m) = if (!isprime(m) && (m>1), my(f=factor(m)); (m % sum(k=1, #f~, f[k,1]^2*f[k,2])) == 0); \\ Michel Marcus, Oct 11 2020

A118585 Sum of squares of digits of prime factors of n, with multiplicity.

Original entry on oeis.org

0, 4, 9, 8, 25, 13, 49, 12, 18, 29, 2, 17, 10, 53, 34, 16, 50, 22, 82, 33, 58, 6, 13, 21, 50, 14, 27, 57, 85, 38, 10, 20, 11, 54, 74, 26, 58, 86, 19, 37, 17, 62, 25, 10, 43, 17, 65, 25, 98, 54, 59, 18, 34, 31, 27, 61, 91, 89, 106, 42

Offset: 1

Jonathan Vos Post, May 07 2006

Differs from A067666 if any prime factor exceeds 1 digit. Fixed points include 16, 27. See also: A067666 Sum of squares of prime factors of n (counted with multiplicity). See also: A003132 Sum of squares of digits of n. See also: A118503 Sum of digits of prime factors of n, with multiplicity.

			a(22) = 6 because 22 = 2 * 11 and the sum of squares of digits of prime factors is 2^2 + 1^2 + 1^2.
a(121) = 4 because 121 = 11^2 = 11 * 11, so 1^2 + 1^2 + 1^2 + 1^2 = 4.

Cf. A001221, A001222, A003132, A005063, A007953, A007954, A067666, A095402, A118503.

Join[{0},Table[Total[Flatten[IntegerDigits/@(Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[ n]])]^2],{n,2,60}]] (* Harvey P. Dale, Nov 17 2022 *)

a(n) = SUM[i=1..k] (e_i)*A003132(p_i) where prime decomposition of n = (p_1)^(e_1) * (p_2)^(e_2) * ... * (p_k)^(e_k).

a(0) removed by Andrey Zabolotskiy, Jun 08 2024

A337395 a(n) is the largest exponent k such that the sums, with multiplicity, of the i-th powers of the prime factors of A100118(n) are all prime for i=1 to k.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 9, 1, 1, 4, 1, 1, 1, 1, 3, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 2, 3, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Torlach Rush, Aug 25 2020

nonn

			a(4) = 2 because (2^1) + (3^1) = 5 and (2^2) + (3^2) = 13.
a(6) = 2 because (2^1) + (5^1) = 7 and (2^2) + (5^2) = 29.
a(8) = 6 because (2^1) + (2^1) + (3^1) = 7 and (2^2) + (2^2) + (3^2) = 17 and (2^3) + (2^3) + (3^3) = 43 and (2^4) + (2^4) + (3^4) = 113 and (2^5) + (2^5) + (3^5) = 307 and (2^6) + (2^6) + (3^6) = 857.

Cf. A001414, A100118.

Cf. A067666, A224787.

a(n) = {my(f=factor(n), x = 1, y = 1); while(y, if(isprime(sum(i=1, #f~, f[i, 1]^x*f[i, 2])), x++, y = 0)); return(x - 1)}
for (n = 2, 220, if(a(n) > 0, print1(a(n), ", ")))

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A338093 Composite numbers which are multiples of the sum of the squares of their prime factors (taken with multiplicity).

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A118585 Sum of squares of digits of prime factors of n, with multiplicity.

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A337395 a(n) is the largest exponent k such that the sums, with multiplicity, of the i-th powers of the prime factors of A100118(n) are all prime for i=1 to k.

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PARI