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A006145
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Ruth-Aaron numbers (1): sum of prime divisors of n = sum of prime divisors of n+1.
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32
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5, 24, 49, 77, 104, 153, 369, 492, 714, 1682, 2107, 2299, 2600, 2783, 5405, 6556, 6811, 8855, 9800, 12726, 13775, 18655, 21183, 24024, 24432, 24880, 25839, 26642, 35456, 40081, 43680, 48203, 48762, 52554, 61760, 63665, 64232, 75140, 79118, 95709, 106893, 109939
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OFFSET
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1,1
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COMMENTS
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Nelson, Penney, & Pomerance call these "Aaron numbers" 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. - David W. Wilson
Number of terms < 10^n: 1, 4, 9, 19, 40, 139, 494, 1748, 6650, ..., . - Robert G. Wilson v, Jan 23 2012
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REFERENCES
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John L. Drost, Ruth/Aaron Pairs, J. Recreational Math. 28 (No. 2), 120-122.
P. Hoffman, The Man Who Loved Only Numbers, pp. 179-181, Hyperion, NY 1998.
J. Roberts, Lure of Integers, pp. 250, MAA 1992.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 159-160, Penguin 1986.
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LINKS
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Brady Haran and Carl Pomerance, Aaron Numbers, Numberphile video (2017).
C. Nelson, D. E. Penney and C. Pomerance, 714 and 715, J. Recreational Math. 7:2 (1994), pp. 87-89.
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MAPLE
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with(numtheory): for n from 1 to 10000 do t0 := 0; t1 := factorset(n);
for j from 1 to nops(t1) do t0 := t0+t1[ j ]; od: s[ n ] := t0; od:
for n from 1 to 9999 do if s[ n ] = s[ n+1 ] then lprint(n, s[ n ]); fi; od:
# Alternative:
SumPF := proc(n) option remember; add(NumberTheory:-PrimeFactors(n)) end:
seq(ifelse(SumPF(n) = SumPF(n+1), n, NULL), n = 1..3000); # Peter Luschny, Jun 11 2024
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MATHEMATICA
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fQ[n_] := Plus @@ (First@# & /@ FactorInteger[n]) == Plus @@ (First@# & /@ FactorInteger[n + 1]); Select[ Range@ 100000, fQ] (* Robert G. Wilson v, Jan 22 2012 *)
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PROG
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(PARI) sopf(n)=my(f=factor(n)); sum(i=1, #f[, 1], f[i, 1])
(Python)
from sympy import factorint
def aupton(terms):
alst, k, sopfk, sopfkp1 = [], 2, 2, 3
while len(alst) < terms:
if sopfkp1 == sopfk: alst.append(k)
k, sopfk, sopfkp1 = k+1, sopfkp1, sum(p for p in factorint(k+2))
return alst
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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