A222416 -id:A222416 - cp's OEIS frontend

A222415 a(n) = max{A222416(q): q in P(n)}, where P(n) is the set of integers defined in Defn. 2.6 of Nussbaum and Verduyn Lunel (2003).

1, 2, 3, 4, 5, 7, 7, 11, 11, 13, 13, 16, 16, 18, 18, 26, 26, 26, 26, 31, 31, 32, 32, 35, 35, 37, 43, 43, 43, 47, 47, 58, 58, 59, 59, 60, 60, 61, 64, 64, 66, 66, 66, 70, 71, 71, 72, 75, 75, 75, 77, 79, 79, 84, 84, 84, 84, 84, 85, 87, 88, 89, 90, 122, 122, 127, 127, 131, 131, 131, 131, 131, 131, 131, 131, 133, 134, 144, 144, 146

Offset: 1

N. J. A. Sloane, Feb 28 2013

nonn

See Nussbaum and Verduyn Lunel (2003) for precise definition.

Nussbaum, Roger D.; Verduyn Lunel, Sjoerd M., Asymptotic estimates for the periods of periodic points of non-expansive maps, Ergodic Theory Dynam. Systems 23 (2003), no. 4, 1199--1226. See Table 1 and the function n*d_n. MR1997973 (2004m:37033)

Cf. A222416, A008475.

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

Original entry on oeis.org

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

A304037 If n = Product (p_j^k_j) then a(n) = Sum (pi(p_j)^k_j), where pi() = A000720.

Original entry on oeis.org

0, 1, 2, 1, 3, 3, 4, 1, 4, 4, 5, 3, 6, 5, 5, 1, 7, 5, 8, 4, 6, 6, 9, 3, 9, 7, 8, 5, 10, 6, 11, 1, 7, 8, 7, 5, 12, 9, 8, 4, 13, 7, 14, 6, 7, 10, 15, 3, 16, 10, 9, 7, 16, 9, 8, 5, 10, 11, 17, 6, 18, 12, 8, 1, 9, 8, 19, 8, 11, 8, 20, 5, 21, 13, 11, 9, 9, 9, 22, 4, 16, 14, 23, 7, 10, 15, 12, 6

Offset: 1

Ilya Gutkovskiy, May 05 2018

nonn

			a(72) = 5 because 72 = 2^3*3^2 = prime(1)^3*prime(2)^2 and 1^3 + 2^2 = 5.

Ilya Gutkovskiy, Extended graphical example
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A000720, A002110, A003963, A008475, A056239, A066328, A156061, A222416.

a[n_] := Plus @@ (PrimePi[#[[1]]]^#[[2]]& /@ FactorInteger[n]); a[1] = 0; Table[a[n], {n, 1, 88}]

If gcd(u,v) = 1 then a(u*v) = a(u) + a(v).
a(p^k) = A000720(p)^k where p is a prime.
a(A002110(m)^k) = 1^k + 2^k + ... + m^k.
As an example:
a(A000040(k)) = k.
a(A006450(k)) = A000040(k).
a(A038580(k)) = A006450(k).
a(A001248(k)) = a(A011757(k)) = A000290(k).
a(A030078(k)) = a(A055875(k)) = A000578(k).
a(A002110(k)) = a(A011756(k)) = A000217(k).
a(A061742(k)) = A000330(k).
a(A115964(k)) = A000537(k).
a(A080696(k)) = A007504(k).
a(A076954(k)) = A001923(k).

A304251 If n = Product (p_j^k_j) then a(n) = Sum (prime(p_j)^k_j).

Original entry on oeis.org

0, 3, 5, 9, 11, 8, 17, 27, 25, 14, 31, 14, 41, 20, 16, 81, 59, 28, 67, 20, 22, 34, 83, 32, 121, 44, 125, 26, 109, 19, 127, 243, 36, 62, 28, 34, 157, 70, 46, 38, 179, 25, 191, 40, 36, 86, 211, 86, 289, 124, 64, 50, 241, 128, 42, 44, 72, 112, 277, 25, 283, 130, 42, 729, 52, 39, 331, 68, 88, 31

Offset: 1

Ilya Gutkovskiy, May 09 2018

nonn

			a(12) = 14 because 12 = 2^2*3 and prime(2)^2 + prime(3) = 3^2 + 5 = 14.

Robert Israel, Table of n, a(n) for n = 1..10000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=100000
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A002110, A006450, A008475, A038580, A064988, A083186, A222416, A304037, A304253 (fixed points).

f:= proc(n) local t;
   add(ithprime(t[1])^t[2],t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Apr 25 2024

a[n_] := Plus @@ (Prime[#[[1]]]^#[[2]] & /@ FactorInteger[n]); a[1] = 0; Table[a[n], {n, 70}]

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

a(prime(i)^k) = prime(prime(i))^k.
a(A000040(k)) = A006450(k).
a(A006450(k)) = A038580(k).
a(A002110(k)) = A083186(k).

A286875 If n = Product (p_j^k_j) then a(n) = Sum (k_j >= 2, p_j^k_j).

Original entry on oeis.org

0, 0, 0, 4, 0, 0, 0, 8, 9, 0, 0, 4, 0, 0, 0, 16, 0, 9, 0, 4, 0, 0, 0, 8, 25, 0, 27, 4, 0, 0, 0, 32, 0, 0, 0, 13, 0, 0, 0, 8, 0, 0, 0, 4, 9, 0, 0, 16, 49, 25, 0, 4, 0, 27, 0, 8, 0, 0, 0, 4, 0, 0, 9, 64, 0, 0, 0, 4, 0, 0, 0, 17, 0, 0, 25, 4, 0, 0, 0, 16, 81, 0, 0, 4, 0, 0, 0, 8, 0, 9, 0, 4, 0, 0, 0, 32, 0, 49, 9, 29, 0, 0, 0, 8, 0, 0, 0, 31

Offset: 1

Ilya Gutkovskiy, Aug 02 2017

Sum of unitary, proper prime power divisors of n.

			a(360) = a(2^3*3^2*5) = 2^3 + 3^2 = 17.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors.

Cf. A005117, A008475, A222416, A023888, A023889, A034448, A063956, A077610, A092261, A246547, A284117.

Table[DivisorSum[n, # &, CoprimeQ[#, n/#] && PrimePowerQ[#] && !PrimeQ[#] &], {n, 108}]
f[p_, e_] := If[e == 1, 0, p^e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 24 2024 *)

A286875(n) = { my(f=factor(n)); for (i=1, #f~, if(f[i, 2] < 2, f[i, 1] = 0)); vecsum(vector(#f~,i,f[i,1]^f[i,2])); }; \\ Antti Karttunen, Oct 07 2017

from sympy import primefactors, isprime, gcd, divisors
def a(n): return sum(d for d in divisors(n) if gcd(d, n//d)==1 and len(primefactors(d))==1 and not isprime(d))
print([a(n) for n in range(1, 109)]) # Indranil Ghosh, Aug 02 2017

a(n) = Sum_{d|n, d = p^k, p prime, k >= 2, gcd(d, n/d) = 1} d.
a(A246547(k)) = A246547(k).
a(A005117(k)) = 0.
Additive with a(p^e) = p^e if e >= 2, and 0 otherwise. - Amiram Eldar, Jul 24 2024

A322177 If n = Product (p_j^k_j) then a(n) = Sum (prime(p_j)^prime(k_j)).

Original entry on oeis.org

0, 9, 25, 27, 121, 34, 289, 243, 125, 130, 961, 52, 1681, 298, 146, 2187, 3481, 134, 4489, 148, 314, 970, 6889, 268, 1331, 1690, 3125, 316, 11881, 155, 16129, 177147, 986, 3490, 410, 152, 24649, 4498, 1706, 364, 32041, 323, 36481, 988, 246, 6898, 44521, 2212, 4913, 1340

Offset: 1

Ilya Gutkovskiy, Nov 30 2018

nonn

			a(12) = a(2^2 * 3^1) = prime(2)^prime(2) + prime(3)^prime(1) = 3^3 + 5^2 = 52.

Cf. A000040, A008475, A222416, A304251, A321874.

a[n_] := Plus @@ (Prime[#[[1]]]^Prime[#[[2]]] & /@ FactorInteger[n]); a[1] = 0; Table[a[n], {n, 50}]

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = prime(f[k,1])^prime(f[k,2]);); vecsum(f[,1]); \\ Michel Marcus, Nov 30 2018

cp's OEIS Frontend

A222415 a(n) = max{A222416(q): q in P(n)}, where P(n) is the set of integers defined in Defn. 2.6 of Nussbaum and Verduyn Lunel (2003).

Views

Author

Keywords

Comments

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

PARI

Python

Formula

A304037 If n = Product (p_j^k_j) then a(n) = Sum (pi(p_j)^k_j), where pi() = A000720.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A304251 If n = Product (p_j^k_j) then a(n) = Sum (prime(p_j)^k_j).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A286875 If n = Product (p_j^k_j) then a(n) = Sum (k_j >= 2, p_j^k_j).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A322177 If n = Product (p_j^k_j) then a(n) = Sum (prime(p_j)^prime(k_j)).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI