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A039752 Ruth-Aaron numbers (2): sum of prime divisors of n = sum of prime divisors of n+1 (both taken with multiplicity). 15
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 01 2010, Dec 13 2010

The number of terms <= x is at most O(x (loglog x)^4 / (log x)^2) (Pomerance 1999/2002). - Tomohiro Yamada, Apr 22 2017

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.

Science, vol. 275, p. 759, 1997.

LINKS

P. Weisenhorn and Michael De Vlieger, Table of n, a(n) for n = 1..1121 (first 215 terms from P. Weisenhorn)

Brady Haran and Carl Pomerance, Aaron Numbers - Numberphile (2017)

Hans Havermann, Ruth-Aaron pairs, indexed and factored

Hans Havermann, A Large Ruth-Aaron Pair

C. Nelson, D. E. Penney and C. Pomerance, 714 and 715, J. Recreational Math. 7 (No. 2) 1974, 87-89.

Ivars Peterson, Playing with Ruth-Aaron pairs

Carl Pomerance, Ruth-Aaron Numbers Revisited, Paul Erdős and his Mathematics, (Budapest, 1999), Bolyai Soc. Math. Stud. 11, János Bolyai Math. Soc., Budapest, 2002, pp. 567-579.

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

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] &] (* Harvey P. Dale, Apr 27 2009 *)

PROG

(PARI) is_A039752(n)=A001414(n)==A001414(n+1) \\ M. F. Hasler, Mar 01 2014

CROSSREFS

Cf. A006145, A006146, A039753, A054378, A101805, A175513.

Sequence in context: A220034 A063731 A129316 * A141536 A065905 A286056

Adjacent sequences:  A039749 A039750 A039751 * A039753 A039754 A039755

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved

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Last modified November 21 13:53 EST 2017. Contains 295001 sequences.