cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Olivier Gérard

Keywords

Comments

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

Examples

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

References

  • 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.

Links

Crossrefs

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.

Programs

  • Haskell
    a008475 1 = 0
a008475 n = sum $ a141809_row n
-- Reinhard Zumkeller, Jan 29 2013, Oct 10 2011
  • Maple
    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
  • Mathematica
    f[n_] := Plus @@ Power @@@ FactorInteger@ n; f[1] = 0; Array[f, 73]
  • PARI
    for(n=1,100,print1(sum(i=1,omega(n), component(component(factor(n),1),i)^component(component(factor(n),2),i)),","))
  • PARI
    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 */
  • PARI
    A008475(n) = { my(f=factor(n)); vecsum(vector(#f~,i,f[i,1]^f[i,2])); }; \\ Antti Karttunen, Nov 17 2017
  • Python
    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

Formula

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

A081403 a(n) = A008475(n^2).

Original entry on oeis.org

0, 4, 9, 16, 25, 13, 49, 64, 81, 29, 121, 25, 169, 53, 34, 256, 289, 85, 361, 41, 58, 125, 529, 73, 625, 173, 729, 65, 841, 38, 961, 1024, 130, 293, 74, 97, 1369, 365, 178, 89, 1681, 62, 1849, 137, 106, 533, 2209, 265, 2401, 629, 298, 185, 2809, 733, 146, 113
Offset: 1

Views

Author

Labos Elemer, Mar 31 2003

Keywords

Examples

			a(1) = 0 since 1 has no prime factor; n = p^2: a(p^2) = p^2; n = 6: a(6) = 4+9 = 13; a(u*w) = a(u)+a(w) if gcd(u,w) = 1; a(21) = a(7)+a(3) = 49+9 = 58; additive with respect of unitary prime divisor decompositions.

Links

Crossrefs

Cf. A008475, A000142, A081403, A081404.

Programs

  • Maple
    a:= n-> add(i[1]^i[2], i=ifactors(n^2)[2]):
seq(a(n), n=1..60);  # Alois P. Heinz, Sep 03 2019
  • Mathematica
    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}]
  • PARI
    f(n) = { my(f=factor(n)); vecsum(vector(#f~, i, f[i, 1]^f[i, 2])); }; \\ A008475
a(n) = f(n^2); \\ Michel Marcus, Sep 03 2019
  • Python
    from sympy import factorint
def A081403(n): return sum(p**(e<<1) for p,e in factorint(n).items()) # Chai Wah Wu, Jul 01 2024

Formula

Additive with a(p^e) = p^(2e).

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

Views

Author

Labos Elemer, Mar 31 2003

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

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

Formula

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)
Showing 1-3 of 3 results.

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