A081403 -id:A081403 - cp's OEIS frontend

A008475 If n = Product (p_j^k_j) then a(n) = Sum (p_j^k_j) (a(1) = 0 by convention).

0, 2, 3, 4, 5, 5, 7, 8, 9, 7, 11, 7, 13, 9, 8, 16, 17, 11, 19, 9, 10, 13, 23, 11, 25, 15, 27, 11, 29, 10, 31, 32, 14, 19, 12, 13, 37, 21, 16, 13, 41, 12, 43, 15, 14, 25, 47, 19, 49, 27, 20, 17, 53, 29, 16, 15, 22, 31, 59, 12, 61, 33, 16, 64, 18, 16, 67, 21, 26, 14, 71, 17, 73

Offset: 1

Olivier Gérard

For n>1, a(n) is the minimal number m such that the symmetric group S_m has an element of order n. - Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 26 2001

If gcd(u,w) = 1, then a(u*w) = a(u) + a(w); behaves like logarithm; compare A001414 or A056239. - Labos Elemer, Mar 31 2003

			a(180) = a(2^2 * 3^2 * 5) = 2^2 + 3^2 + 5 = 18.

F. J. Budden, The Fascination of Groups, Cambridge, 1972; pp. 322, 573.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter IV, p. 147.
T. Z. Xuan, On some sums of large additive number theoretic functions (in Chinese), Journal of Beijing normal university, No. 2 (1984), pp. 11-18.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
John Bamberg, Grant Cairns and Devin Kilminster, The crystallographic restriction, permutations and Goldbach's conjecture, Amer. Math. Monthly, Vol. 110, No. 3 (March 2003), pp. 202-209.
Roger B. Eggleton and William P. Galvin, Upper Bounds on the Sum of Principal Divisors of an Integer, Mathematics Magazine, Vol. 77, No. 3 (Jun., 2004), pp. 190-200.

Cf. A001414, A000961, A005117, A051613, A072691, A081402, A081403, A081404, A027748, A124010, A001221, A028233, A034684, A053585, A159077, A023888, A078771, A092509, A286875.

See A222416 for the variant with a(1)=1.

a008475 1 = 0
a008475 n = sum $ a141809_row n
-- Reinhard Zumkeller, Jan 29 2013, Oct 10 2011

A008475 := proc(n) local e,j; e := ifactors(n)[2]:
add(e[j][1]^e[j][2], j=1..nops(e)) end:
seq(A008475(n), n=1..60); # Peter Luschny, Jan 17 2010

f[n_] := Plus @@ Power @@@ FactorInteger@ n; f[1] = 0; Array[f, 73]

for(n=1,100,print1(sum(i=1,omega(n), component(component(factor(n),1),i)^component(component(factor(n),2),i)),","))

a(n)=local(t);if(n<1,0,t=factor(n);sum(k=1,matsize(t)[1],t[k,1]^t[k,2])) /* Michael Somos, Oct 20 2004 */

A008475(n) = { my(f=factor(n)); vecsum(vector(#f~,i,f[i,1]^f[i,2])); }; \\ Antti Karttunen, Nov 17 2017

from sympy import factorint
def a(n):
    f=factorint(n)
    return 0 if n==1 else sum([i**f[i] for i in f]) # Indranil Ghosh, May 20 2017

Additive with a(p^e) = p^e.
a(A000961(n)) = A000961(n); a(A005117(n)) = A001414(A005117(n)).
a(n) = Sum_{k=1..A001221(n)} A027748(n,k) ^ A124010(n,k) for n>1. - Reinhard Zumkeller, Oct 10 2011
a(n) = Sum_{k=1..A001221(n)} A141809(n,k) for n > 1. - Reinhard Zumkeller, Jan 29 2013
Sum_{k=1..n} a(k) ~ (Pi^2/12)* n^2/log(n) + O(n^2/log(n)^2) (Xuan, 1984). - Amiram Eldar, Mar 04 2021

A005063 Sum of squares of primes dividing n.

Original entry on oeis.org

0, 4, 9, 4, 25, 13, 49, 4, 9, 29, 121, 13, 169, 53, 34, 4, 289, 13, 361, 29, 58, 125, 529, 13, 25, 173, 9, 53, 841, 38, 961, 4, 130, 293, 74, 13, 1369, 365, 178, 29, 1681, 62, 1849, 125, 34, 533, 2209, 13, 49, 29, 298, 173, 2809, 13, 146, 53, 370, 845, 3481, 38, 3721

Offset: 1

N. J. A. Sloane

nonn

The set of these terms apart from 0 is A048261. - Bernard Schott, Feb 10 2022

Inverse Möbius transform of n^2 * c(n), where c(n) is the prime characteristic (A010051). - Wesley Ivan Hurt, Jun 22 2024

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000290, A005066, A005071, A005075, A005079, A005083, A059841, A079978.
Cf. A067666, A081403, A048261. - Franklin T. Adams-Watters, May 03 2009
Sum of the k-th powers of the primes dividing n for k=0..10 : A001221 (k=0), A008472 (k=1), this sequence (k=2), A005064 (k=3), A005065 (k=4), A351193 (k=5), A351194 (k=6), A351195 (k=7), this sequence (k=8), A351197 (k=9), A351198 (k=10).
Cf. A010051.

A005063 := proc(n)
        add(d^2, d= numtheory[factorset](n)) ;
end proc;
seq(A005063(n),n=1..40) ; # R. J. Mathar, Nov 08 2011

a[n_] := Total[FactorInteger[n][[All, 1]]^2]; a[1]=0; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 20 2017 *)
Array[DivisorSum[#, #^2 &, PrimeQ] &, 61] (* Michael De Vlieger, Jul 11 2017 *)

a(n)=local(fm,t);fm=factor(n);t=0;for(k=1,matsize(fm)[1],t+=fm[k,1]^2);t \\ Franklin T. Adams-Watters, May 03 2009

a(n) = vecsum(apply(x->x^2, factor(n)[, 1])); \\ Michel Marcus, Sep 19 2020

from sympy import primefactors
def a(n): return sum(p**2 for p in primefactors(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jul 11 2017

(define (A005063 n) (if (= 1 n) 0 (+ (A000290 (A020639 n)) (A005063 (A028234 n))))) ;; Antti Karttunen, Jul 10 2017

Additive with a(p^e) = p^2.
G.f.: Sum_{k>=1} prime(k)^2*x^prime(k)/(1 - x^prime(k)). - Ilya Gutkovskiy, Dec 24 2016
From Antti Karttunen, Jul 11 2017: (Start)
a(n) = A005066(n) + 4*A059841(n).
a(n) = A005079(n) + A005083(n) + 4*A059841(n).
a(n) = A005071(n) + A005075(n) + 9*A079978(n).
(End)
Dirichlet g.f.: primezeta(s-2)*zeta(s). - Benedict W. J. Irwin, Jul 11 2018
a(n) = Sum_{p|n, p prime} p^2. - Wesley Ivan Hurt, Feb 04 2022
a(n) = Sum_{d|n} d^2 * c(d), where c = A010051. - Wesley Ivan Hurt, Jun 22 2024

More terms from Franklin T. Adams-Watters, May 03 2009

A067666 Sum of squares of prime factors of n (counted with multiplicity).

Original entry on oeis.org

0, 4, 9, 8, 25, 13, 49, 12, 18, 29, 121, 17, 169, 53, 34, 16, 289, 22, 361, 33, 58, 125, 529, 21, 50, 173, 27, 57, 841, 38, 961, 20, 130, 293, 74, 26, 1369, 365, 178, 37, 1681, 62, 1849, 129, 43, 533, 2209, 25, 98, 54, 298, 177, 2809, 31, 146, 61, 370, 845, 3481

Offset: 1

Henry Bottomley, Feb 04 2002

nonn

16 and 27 are fixed points, ... and see Rivera link. - Michel Marcus, Sep 19 2020

			a(2) = 2^2 = 4;
a(45) = a(3*3*5) = 3^2 + 3^2 + 5^2 = 43.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
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. A166319 (where a(n)<=n), A001222, A001414, A005063, A078137, A081403.

A067666 := proc(n)
    add(op(2,pe)*op(1,pe)^2, pe=ifactors(n)[2]) ;
end proc:
seq(A067666(n),n=1..100) ;# R. J. Mathar, Jul 31 2024

Join[{0},Table[Total[Flatten[Table[#[[1]],{#[[2]]}]&/@ FactorInteger[ n]]^2],{n,2,60}]] (* Harvey P. Dale, Dec 24 2012 *)
Join[{0}, Table[Total[#[[1]]^2*#[[2]] & /@ FactorInteger[n]], {n, 2, 60}]] (* Zak Seidov, Apr 18 2013 *)

a(n)=local(fm,t);fm=factor(n);t=0;for(k=1,matsize(fm)[1],t+=fm[k,1]^2*fm[k,2]);t \\ Franklin T. Adams-Watters, May 03 2009

a(n) = my(f=factor(n)); sum(k=1, #f~, f[k,1]^2*f[k,2]); \\ Michel Marcus, Sep 19 2020

a(x*y) = a(x) + a(y); a(p^k) = k*p^2 for p prime.

Totally additive with a(p) = p^2.

Values through a(59) verified by Franklin T. Adams-Watters, May 03 2009

A081404 a(n) = A008475(prime(n)-1).

Original entry on oeis.org

0, 2, 4, 5, 7, 7, 16, 11, 13, 11, 10, 13, 13, 12, 25, 17, 31, 12, 16, 14, 17, 18, 43, 19, 35, 29, 22, 55, 31, 23, 18, 20, 25, 28, 41, 30, 20, 83, 85, 47, 91, 18, 26, 67, 53, 22, 17, 42, 115, 26, 37, 26, 24, 127, 256, 133, 71, 34, 30, 20, 52, 77, 28, 38, 24, 83, 21, 26, 175, 36

Offset: 1

Labos Elemer, Mar 31 2003

nonn

			a(1) = 0 since 1 has no prime factor.
a(100) = A008475(541-1) = A008475(4*27*5) = 4+27+5 = 36.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A008475, A000142, A081402, A081403.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] supo[x_] := Apply[Plus, ba[x]^ep[x]] Table[supo[w], {w, 1, 25}]

A081402 a(n) = A008475(n!).

Original entry on oeis.org

0, 2, 5, 11, 16, 30, 37, 149, 221, 369, 380, 1310, 1323, 2389, 2975, 33695, 33712, 72312, 72331, 269439, 282855, 545109, 545132, 4254514, 4269514, 8463974, 9999248, 35167130, 35167159, 71972737, 71972768, 2152347552, 2161914700

Offset: 1

Labos Elemer, Mar 31 2003

nonn

			a(1) = 0 since 1! = 1 has no prime factor.
a(8) = 2^7 + 3^2 + 5 + 7 = 149 since 8! = 2^7*3^2*5*7.

Jean-Marie De Koninck and William Verreault, Arithmetic functions at factorial arguments, Publications de l'Institut Mathematique, Vol. 115, No. 129 (2024), pp. 45-76.

Cf. A008475, A000120, A000142, A081403, A081404.

ffi[x_] := Flatten[FactorInteger[x]]; lf[x_] := Length[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}]; supo[x_] := Apply[Plus, ba[x]^ep[x]]; Table[supo[w], {w, 1, 25}]

a(n) = my(f=factor(n!)); sum(k=1, #f~, f[k,1]^f[k,2]); \\ Michel Marcus, Jul 09 2018

From Amiram Eldar, Dec 10 2024: (Start)
a(n) = 2^(n-s_2(n)) + O(sqrt(3)^n), where s_2(n) = A000120(n).
Sum_{k=1..n} a(k) = 2^(n+O(log(n))).
Both formulas from De Koninck and Verreault (2024, pp. 51-52, eq. (3.10) and (3.16)). (End)

cp's OEIS Frontend

A008475 If n = Product (p_j^k_j) then a(n) = Sum (p_j^k_j) (a(1) = 0 by convention).

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A005063 Sum of squares of primes dividing n.

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A067666 Sum of squares of prime factors of n (counted with multiplicity).

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A081404 a(n) = A008475(prime(n)-1).

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A081402 a(n) = A008475(n!).

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