|
| |
|
|
A039752
|
|
Ruth-Aaron numbers (2): sum of prime divisors of n = sum of prime divisors of n+1 (both taken with multiplicity).
|
|
13
|
|
|
|
5, 8, 15, 77, 125, 714, 948, 1330, 1520, 1862, 2491, 3248, 4185, 4191, 5405, 5560, 5959, 6867, 8280, 8463, 10647, 12351, 14587, 16932, 17080, 18490, 20450, 24895, 26642, 26649, 28448, 28809, 33019, 37828, 37881, 41261, 42624, 43215
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
So called because 714 is Babe Ruth's lifetime home run record, Hank Aaron's 715th home run broke this record and 714 and 715 have the same sum of prime divisors, taken with multiplicity.
An infinite number of terms would follow from A175513 and the assumption of Schinzel's Hypothesis H. - Hans Havermann, Dec 15 2010.
A 3109-digit term determined by Jens Kruse Andersen is currently the largest-known. - Hans Havermann, Dec 21 2010.
The sum of this sequence's reciprocals is 0.42069... - Hans Havermann, Dec 21 2010.
Both 417162 and 417163 are in the sequence. Hence these two numbers along with 417164 constitute a Ruth-Aaron "triplet". The smallest member of the next triplet is 6913943284. - Hans Havermann, Dec 1 2010, Dec 13 2010.
|
|
|
REFERENCES
|
John L. Drost, Ruth/Aaron Pairs, J. Recreational Math. 28 (No. 2), 120-122.
S. G. Krantz, Mathematical Apocrypha, MAA, 2002, see p. 26.
C. Nelson, D. E. Penney and C. Pomerance, "714 and 715", J. Recreational Math. 7 (No. 2) 1974, 87-89.
Science, vol. 275, p. 759, 1997.
|
|
|
LINKS
|
P. Weisenhorn, Table of n, a(n) for n=1,..,215
Hans Havermann, Ruth-Aaron pairs, indexed and factored
Hans Havermann, A Large Ruth-Aaron Pair
Ivars Petersen, Playing with Ruth-Aaron Pairs (in MathTrek)
Carl Pomerance, Ruth-Aaron Numbers Revisited
Carlos Rivera, Ruth-Aaron Pairs Revisited
Terrel Trotter, Jr., Ruth-Aaron Numbers
Terrel Trotter, Jr., 714 and 715
Eric Weisstein, Ruth-Aaron Pair (in Wolfram MathWorld)
|
|
|
EXAMPLE
|
7129199 (7*11^2*19*443, with 7129200 = 2^4*3*5^2*13*457) is in this sequence because 7+11+11+19+443 = 2+2+2+2+3+5+5+13+457.
|
|
|
MAPLE
|
Contribution from Paul Weisenhorn, Jul 02 2009: (Start)
anzahl:=0: n:=4: nr:=0: g:=nops(ifactors(n)[2]):
s[nr]:=sum(ifactors(n)[2, u][1]*ifactors(n)[2, u][2], u=1..g):
for j from n+1 to 1000000 do nr:=(nr+1) mod 2: g:=nops(ifactors(j)[2]):
s[nr]:=sum(ifactors(j)[2, u][1]*ifactors(j)[2, u][2], u=1..g):
if (s[0]=s[1]) then anzahl):=anzahl+1: print(anzahl, j-1, j, s[0]): end if:
end do:
(End)
|
|
|
MATHEMATICA
|
ppf[n_] := Plus @@ ((#[[1]] #[[2]]) & /@ FactorInteger[n]); Select[Range[50000], ppf[#] == ppf[#+1] &] (* from Harvey P. Dale, Apr 27 2009 *)
|
|
|
CROSSREFS
|
Cf. A006145, A006146, A039753, A054378, A101805, A175513.
Sequence in context: A220034 A063731 A129316 * A141536 A065905 A126695
Adjacent sequences: A039749 A039750 A039751 * A039753 A039754 A039755
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
David W. Wilson
|
|
|
STATUS
|
approved
|
| |
|
|
