A005063 -id:A005063 - cp's OEIS frontend

A228181 Numbers k such that sum of square of prime divisors of k equals sum of prime divisors of k+1.

12, 27, 385, 1120, 4840, 9936, 14500, 29440, 95795, 105875, 178904, 223155, 341248, 343343, 754985, 830908, 1059630, 1841049, 2408832, 2949375, 3564704, 4934358, 5368792, 5500312, 6695000, 6805372, 8332831, 8846656, 10126336, 12956040, 13157235, 17254600

Offset: 1

Michel Lagneau, Aug 15 2013

nonn

Numbers k such that A005063(k) = A008472(k+1).

			The prime divisors of 9936 are {2, 3, 23} and the prime divisors of 9937 are {19, 523} => 2^2 + 3^2 + 23^2 = 19 + 523 = 542, hence 9936 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..333 (terms below 10^10)

Cf. A000961, A005063, A008472, A164722.

[k:k in [2..2500000]| &+PrimeDivisors(k+1) eq &+[PrimeDivisors(k)[i]^2: i in [1..#PrimeDivisors(k)]]]; // Marius A. Burtea, Feb 18 2020

fQ[n_] := Plus @@ (First@# & /@ FactorInteger[n]^2) == Plus @@ (First@# & /@ FactorInteger[n + 1]); Select[ Range@ 100000, fQ]

A322080 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{p|n, p prime} p^k.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 0, 4, 3, 1, 0, 8, 9, 2, 1, 0, 16, 27, 4, 5, 2, 0, 32, 81, 8, 25, 5, 1, 0, 64, 243, 16, 125, 13, 7, 1, 0, 128, 729, 32, 625, 35, 49, 2, 1, 0, 256, 2187, 64, 3125, 97, 343, 4, 3, 2, 0, 512, 6561, 128, 15625, 275, 2401, 8, 9, 7, 1, 0, 1024, 19683, 256, 78125, 793, 16807, 16, 27, 29, 11, 2

Offset: 1

Ilya Gutkovskiy, Nov 26 2018

			Square array begins:
  0,  0,   0,    0,    0,     0,  ...
  1,  2,   4,    8,   16,    32,  ...
  1,  3,   9,   27,   81,   243,  ...
  1,  2,   4,    8,   16,    32,  ...
  1,  5,  25,  125,  625,  3125,  ...
  2,  5,  13,   35,   97,   275,  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
Index entries for sequences related to sums of divisors

Columns k=0..4 give A001221, A008472, A005063, A005064, A005065.

Cf. A109974, A200768 (diagonal), A285425, A286880, A321258.

Table[Function[k, Sum[Boole[PrimeQ[d]] d^k, {d, Divisors[n]}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[Prime[j]^k x^Prime[j]/(1 - x^Prime[j]), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

T(n,k)={vecsum([p^k | p<-factor(n)[,1]])}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018

G.f. of column k: Sum_{j>=1} prime(j)^k*x^prime(j)/(1 - x^prime(j)).

A332385 Sum of squares of indices of distinct prime factors of n.

Original entry on oeis.org

0, 1, 4, 1, 9, 5, 16, 1, 4, 10, 25, 5, 36, 17, 13, 1, 49, 5, 64, 10, 20, 26, 81, 5, 9, 37, 4, 17, 100, 14, 121, 1, 29, 50, 25, 5, 144, 65, 40, 10, 169, 21, 196, 26, 13, 82, 225, 5, 16, 10, 53, 37, 256, 5, 34, 17, 68, 101, 289, 14, 324, 122, 20, 1, 45, 30, 361, 50, 85, 26, 400, 5, 441, 145, 13

Offset: 1

Ilya Gutkovskiy, Feb 10 2020

nonn

			a(21) = a(3 * 7) = a(prime(2) * prime(4)) = 2^2 + 4^2 = 20.

Cf. A000720, A005063, A056239, A066328, A067666, A090885, A289506.

a:= n-> add(numtheory[pi](i[1])^2, i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 10 2020

nmax = 75; CoefficientList[Series[Sum[k^2 x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := Plus @@ (PrimePi[#[[1]]]^2 & /@ FactorInteger[n]); Table[a[n], {n, 1, 75}]

G.f.: Sum_{k>=1} k^2 * x^prime(k) / (1 - x^prime(k)).

If n = Product (p_j^k_j) then a(n) = Sum (pi(p_j)^2), where pi = A000720.

A347156 Sum of squares of distinct prime divisors of n that are < sqrt(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 0, 4, 0, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 38, 0, 4, 9, 4, 25, 13, 0, 4, 9, 29, 0, 13, 0, 4, 34, 4, 0, 13, 0, 29, 9, 4, 0, 13, 25, 53, 9, 4, 0, 38, 0, 4, 58, 4, 25, 13, 0, 4, 9, 78, 0, 13, 0, 4, 34, 4, 49, 13, 0, 29

Offset: 1

Ilya Gutkovskiy, Aug 20 2021

nonn

Cf. A001157, A005063, A005066, A098002, A333806, A333808, A339353, A347157, A347158.

Table[DivisorSum[n, #^2 &, # < Sqrt[n] && PrimeQ[#] &], {n, 1, 80}]
nmax = 80; CoefficientList[Series[Sum[Prime[k]^2 x^(Prime[k] (Prime[k] + 1))/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

G.f.: Sum_{k>=1} prime(k)^2 * x^(prime(k)*(prime(k) + 1)) / (1 - x^prime(k)).

A138296 Table T(k,n) read along antidiagonals: sum of the k-th powers of the distinct prime factors of A024619(n).

Original entry on oeis.org

5, 13, 7, 35, 29, 5, 97, 133, 13, 9, 275, 641, 35, 53, 8, 793, 3157, 97, 351, 34, 5, 2315, 15689, 275, 2417, 152, 13, 7, 6817, 78253, 793, 16839, 706, 35, 29, 10, 20195, 390881, 2315, 117713, 3368, 97, 133, 58, 13, 60073, 1953637, 6817, 823671, 16354, 275, 641

Offset: 1

R. J. Mathar, May 07 2008

Row k=1 is A109353. Rows k=2,3 and 4 are subsequences of A005063-A005065.

			Upper left corner of the table starting at row k=1, column n=1:
1|......5.......7.......5.......9.......8.......5.......7.
2|.....13......29......13......53......34......13......29.
3|.....35.....133......35.....351.....152......35.....133.
4|.....97.....641......97....2417.....706......97.....641.
5|....275....3157.....275...16839....3368.....275....3157.
6|....793...15689.....793..117713...16354.....793...15689.
7|...2315...78253....2315..823671...80312....2315...78253.
8|...6817..390881....6817.5765057..397186....6817..390881.

J.-M de Koninck, F. Luca, Integers divisible by sums of powers of their prime factors, J. Num. Theory vol 128 (2008) 557-563.

A024619 := proc(n)
    local a;
    if n = 1 then
        RETURN(6);
    else
        for a from A024619(n-1)+1 do
            if A001221(a) > 1 then
               RETURN(a) ;
            fi ;
        od:
    fi ;
end:
A138296 := proc(n,j)
    local f,beta ;
    beta := 0 ;
    for f in ifactors( A024619(n) )[2] do
        beta := beta+op(1,f)^j ;
    od:
    RETURN(beta) ;
end:
for d from 1 to 10 do for n from 1 to d do printf("%d,",A138296(n,d-n+1)) ; od: od: # R. J. Mathar, May 07 2008

T(k,n) = sum_{d in A000040, d| A024619(n)} d^k.

A199583 a(n) is the smallest number such that the sum of the n-th powers of its distinct prime divisors is divisible by n.

Original entry on oeis.org

2, 2, 3, 2, 5, 70, 7, 2, 3, 33, 11, 1155, 13, 78, 26, 2, 17, 2156564410, 19, 6006, 26, 114, 23, 2156564410, 5, 33, 3, 1365, 29, 110, 31, 2, 62, 15, 201, 2156564410, 37, 30, 14, 961380175077106319535, 41, 1385670, 43, 2805, 26, 266, 47, 961380175077106319535

Offset: 1

Michel Lagneau, Nov 08 2011

nonn

a(n) > 1 and a(n) = n if n prime. All terms are squarefree.

			a(6) = 70 = 2*5*7; 2^6 + 5^6 + 7^6 = 133338 = 22223*6.
a(18)= 2*5*7*11*13*17*19*23*29 = 2156564410 because:
p^18 == 10, 9 (mod 18) for p = 2,3 respectively, and p^18 == 1 (mod 18) for p prime > 3. The minimum sum divisible by 18 is s = 2^18 + Sum_{k=3..10} prime(k)^18 whose residues sum to 10 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 18. Hence a(18) = 2156564410.

Cf. A005063, A005064, A005065, A199167.

with(numtheory): T:=array(1..50):for n from 1 to 50 do:q:=0:for k from 2 to 7000 while(q=0)do:x:=factorset(k):s:=sum(x[j]^n ,j=1..nops(x)) :if irem(s,n)=0 then printf ( "%d %d \n",n,k):q:=1:else fi:od:if q=0 then for i from 1 to n do: T[i]:=irem(ithprime(i)^n,n):od:W:=convert(T,set):n1:=nops(W):n2:=W[n1]:n3:=W[n1-1]:
s:=0:p:=1:for a from 1 to n  while(s<>n) do: if T[a]= 1 or T[a]=n2 or (T[a] = n3 and n2+n3

A252424 Numbers k such that sum of odd divisors of k equals sum of squares of primes dividing k.

Original entry on oeis.org

18, 36, 72, 144, 234, 288, 468, 576, 936, 1152, 1872, 2304, 3744, 4608, 7488, 9216, 14976, 18432, 29952, 36864, 59904, 73728, 119808, 147456, 239616, 294912, 479232, 589824, 958464, 1179648, 1916928, 2359296, 3833856, 4718592, 7667712, 9437184, 15335424, 18874368

Offset: 1

Michel Lagneau, Dec 17 2014

nonn

Numbers k such that A000593(k) = A005063(k).
a(n) == 0 (mod 18), and the numbers 18*2^m, m = 0,1,... are in the sequence because the odd divisors are {1, 3, 9}, the prime factors are {2, 3} => 2^2 + 3^2 = 1 + 3 + 9 = 13.
The numbers of the form 18*13*2^m are in the sequence because the odd divisors are {1, 3, 9, 13, 39, 117}, the prime factors are {2, 3, 13} => 2^2 + 3^2 + 13^2 = 1 + 3 + 9 + 13 + 39 + 117 = 182.

			18 is in the sequence because the prime factors of 18 are {2, 3}, the odd divisors of 18 are {1, 3, 9} => 2^2 + 3^2 = 1 + 3 + 9 = 13.
Or 18 => A000593(18) = A005063(18) = 13.

Robert G. Wilson v, Table of n, a(n) for n = 1..56

Cf. A000593, A005063.

with(numtheory):nn:=10^5:
for n from 2 to nn do:
   x:=factorset(n):n0:=nops(x):
   s0:=sum('x[i]^2','i'=1..n0):
   y:=divisors(n):n1:=nops(y):
   s :=0 :
        for j from 1 to n1 do :
       if irem (y[j],2)=1 then s:=s+y[j]:
      else
      fi:
    od:
     if s=s0
    then
   printf(`%d, `,n):
   else
   fi:
od:

a252424[n_Integer] := Module[{f, g},
  f[x_] := Plus @@ Select[Divisors[x], OddQ[#] &];
  g[x_] := Plus @@ (First@Transpose@FactorInteger[x]^2);
Rest@Select[Range[n], f[#] == g[#] &]]; a252424[10^6] (* Michael De Vlieger, Dec 17 2014 *)
Select[Range[19*10^6],Total[Select[Divisors[#],OddQ]]==Total[ FactorInteger[ #][[All,1]]^2]&] (* Harvey P. Dale, May 11 2020 *)
f[p_, e_] := If[p == 2, 1, (p^(e + 1) - 1)/(p - 1)]; q[n_] := Times @@ f @@@ (fct = FactorInteger[n]) == Total[fct[[;; , 1]]^2]; Select[Range[2, 10^6], q] (* Amiram Eldar, Jul 09 2022 *)

isok(n) = my(f = factor(n)); sum(i=1, #f~, f[i,1]^2) == sumdiv(n, d, d*(d%2)); \\ Michel Marcus, Dec 17 2014

A291108 Expansion of Sum_{k>=2} k^2x^(2k)/(1 - x^k).

Original entry on oeis.org

0, 0, 0, 4, 0, 13, 0, 20, 9, 29, 0, 65, 0, 53, 34, 84, 0, 130, 0, 145, 58, 125, 0, 273, 25, 173, 90, 265, 0, 399, 0, 340, 130, 293, 74, 614, 0, 365, 178, 609, 0, 735, 0, 625, 340, 533, 0, 1105, 49, 754, 298, 865, 0, 1183, 146, 1113, 370, 845, 0, 1859, 0, 965, 580, 1364, 194, 1743, 0, 1465, 538, 1599, 0, 2550, 0, 1373, 884

Offset: 1

Ilya Gutkovskiy, Aug 17 2017

nonn

Sum of squares of divisors of n except 1 and n^2 (sum of squares of nontrivial divisors of n).

			a(6) = 13 because 6 has 4 divisors {1, 2, 3, 6} among which 2 are nontrivial {2, 3} and 2^2 + 3^2 = 13.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sums of divisors

Cf. A000290, A001157, A001248, A001358, A005063, A008578, A027751, A048050, A053807, A067558, A070824.

nmax = 75; Rest[CoefficientList[Series[Sum[k^2 x^(2 k)/(1 - x^k), {k, 2, nmax}], {x, 0, nmax}], x]]
Join[{0}, Table[DivisorSigma[2, n] - n^2 - 1, {n, 2, 75}]]

A291108(n) = sumdiv(n,d,if((1==d)||(n==d),0,d^2)); \\ Antti Karttunen, Jan 22 2025

G.f.: Sum_{k>=2} k^2*x^(2*k)/(1 - x^k).
a(n) = A001157(n) -  A000290(n) - 1 for n > 1.
a(n) = A067558(n) - 1 for n > 1.
a(n) = A005063(n) if n is a semiprime (A001358).
a(n) = 0 if n is a prime or 1 (A008578).
a(n) = n if n is a square of prime (A001248).
a(p^k) = (p^(2*k) - p^2)/(p^2 - 1) for p is a prime and k > 0.

A380444 Sum of the nonprimes dividing n and the squares of the primes dividing n.

Original entry on oeis.org

1, 5, 10, 9, 26, 20, 50, 17, 19, 40, 122, 36, 170, 68, 50, 33, 290, 47, 362, 64, 80, 148, 530, 68, 51, 200, 46, 100, 842, 100, 962, 65, 164, 328, 110, 99, 1370, 404, 218, 112, 1682, 146, 1850, 196, 104, 580, 2210, 132, 99, 115, 350, 256, 2810, 128, 202, 164, 428, 904, 3482, 196, 3722, 1028, 152, 129, 260, 262, 4490, 400, 608, 208, 5042, 203, 5330, 1448, 150, 484

Offset: 1

Wesley Ivan Hurt, Jun 21 2025

Inverse Möbius transform of A103164(n).

			a(12) = 1 + 2^2 + 3^2 + 4 + 6 + 12 = 36.

Cf. A000005 (tau), A000203 (sigma), A005063, A008472 (sopf), A010051, A023890, A103164.

Table[DivisorSigma[1, n] + Sum[p (p - 1), {p, Select[Divisors[n], PrimeQ]}], {n, 100}]

a(n) = sigma(n) - sopf(n) + sopf_2(n), where sopf_2(n) = Sum_{p|n, p prime} p^2.
a(n) = Sum_{d|n} d^tau(d^c(d)), where c = A010051.
a(n) = A023890(n) + A005063(n).
a(p^k) = (p^(k+1)+p^3-2*p^2+p-1)/(p-1) for p prime, k >= 1. - Wesley Ivan Hurt, Jul 02 2025

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

cp's OEIS Frontend

A228181 Numbers k such that sum of square of prime divisors of k equals sum of prime divisors of k+1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

A322080 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{p|n, p prime} p^k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A332385 Sum of squares of indices of distinct prime factors of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A347156 Sum of squares of distinct prime divisors of n that are < sqrt(n).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A138296 Table T(k,n) read along antidiagonals: sum of the k-th powers of the distinct prime factors of A024619(n).

Views

Author

Keywords

Comments

Examples

Links

Programs

Maple

Formula

A199583 a(n) is the smallest number such that the sum of the n-th powers of its distinct prime divisors is divisible by n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

A252424 Numbers k such that sum of odd divisors of k equals sum of squares of primes dividing k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A291108 Expansion of Sum_{k>=2} k^2*x^(2*k)/(1 - x^k).

Views

Author

Keywords

Comments

Examples

Links

A291108 Expansion of Sum_{k>=2} k^2x^(2k)/(1 - x^k).