A039753 -id:A039753 - cp's OEIS frontend

A178214 Numbers in A039753 with neither of their Ruth-Aaron pairs squarefree.

7129199, 9867275, 18918704, 43009524, 43882488, 45828324, 126511280, 132082191, 150786063, 252625743, 285816464, 303792200, 313887275, 330130475, 336945392, 337795524, 361652075, 380035664, 480297824, 579423924, 647037215, 650012724, 756098624, 986677500, 1000308015, 1001438136, 1139325668, 1140314075, 1205629524, 1315277147

Offset: 1

Hans Havermann, Dec 19 2010

nonn

			7129199 (7*11^2*19*443, with 7129200 = 2^4*3*5^2*13*457) is a member of both A006145 (7+11+19+443 = 2+3+5+13+457) and A039752 (7+11+11+19+443 = 2+2+2+2+3+5+5+13+457) but neither 7129199 nor 7129200 is squarefree, so 7129199 is a member of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..385

Cf. A039753, A006145, A039752, A013929.

A006145 Ruth-Aaron numbers (1): sum of prime divisors of n = sum of prime divisors of n+1.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

nonn

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

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.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..6651 from Robert G. Wilson v)
Joe K. Crump, Ruth-Aaron Pairs-an algorithm
Shyam Sunder Gupta, Smith Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 4, 127-157.
Brady Haran and Carl Pomerance, Aaron Numbers, Numberphile video (2017).
G. Kreweras and Y. Poupard, Sur les partitions en paires d'un ensemble fini totalement ordonné, Publications de l'Institut de Statistique de l'Université de Paris, 23 (1978), 57-74. (Annotated scanned copy)
Dana Mackenzie, Homage to an Itinerant Master, Science, Vol. 275 (1997), p. 759; alternative link.
Carol Nelson, David E. Penney, and Carl Pomerance, 714 and 715, J. Recreational Math. 7:2 (1994), pp. 87-89.
Ivars Peterson, Playing with Ruth-Aaron pairs
Terry Trotter, Jr., Ruth-Aaron Numbers.
Eric Weisstein's World of Mathematics, Ruth-Aaron Pair.

Cf. A006146, A039752, A039753, A054378.

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

fQ[n_] := Plus @@ (First@# & /@ FactorInteger[n]) == Plus @@ (First@# & /@ FactorInteger[n + 1]); Select[ Range@ 100000, fQ] (* Robert G. Wilson v, Jan 22 2012 *)

sopf(n)=my(f=factor(n));sum(i=1,#f[,1],f[i,1])
is(n)=sopf(n)==sopf(n+1) \\ Charles R Greathouse IV, Jan 27 2012

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
print(aupton(42)) # Michael S. Branicky, May 24 2021

A039752 Ruth-Aaron numbers (2): sum of prime divisors of n = sum of prime divisors of n+1 (both taken with multiplicity).

Original entry on oeis.org

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, 44831, 44891, 47544, 49240

Offset: 1

David W. Wilson

nonn

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 "triple". The smallest member of the next triple 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

			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.

John L. Drost, Ruth/Aaron Pairs, J. Recreational Math. 28 (No. 2), 120-122.
S. G. Krantz, Mathematical Apocrypha, MAA, 2002, see p. 26.
Dana Mackenzie, Homage to an itinerant master, Science 275, p. 759, 1997.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..215 from P. Weisenhorn, terms 216..1121 from Michael De Vlieger)
Shyam Sunder Gupta, Smith Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 4, 127-157.
Brady Haran and Carl Pomerance, Aaron Numbers, Numberphile video (2017)
Hans Havermann, Ruth-Aaron pairs, indexed and factored
Hans Havermann, A Large Ruth-Aaron Pair
Carol Nelson, David E. Penney and Carl 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)

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

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:
# Paul Weisenhorn, Jul 02 2009

ppf[n_] := Plus @@ ((#[[1]] #[[2]]) & /@ FactorInteger[n]); Select[Range[50000], ppf[#] == ppf[#+1] &] (* Harvey P. Dale, Apr 27 2009 *)

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

from sympy import factorint
def aupton(terms):
  alst, k, sopfrk, sopfrkp1 = [], 2, 2, 3
  while len(alst) < terms:
    if sopfrkp1 == sopfrk: alst.append(k)
    k += 1
    fkp1 = factorint(k+1)
    sopfrk, sopfrkp1 = sopfrkp1, sum(p*fkp1[p] for p in fkp1)
  return alst
print(aupton(42)) # Michael S. Branicky, May 08 2021

A006146 Sums of prime divisors of Ruth-Aaron numbers (A006145).

Original entry on oeis.org

5, 5, 7, 18, 15, 20, 44, 46, 29, 31, 50, 30, 20, 34, 75, 162, 146, 46, 14, 113, 53, 66, 333, 36, 514, 318, 43, 193, 279, 418, 30, 121, 55, 485, 200, 136, 77, 37, 211, 587, 147, 269, 477, 108, 136, 235, 185, 290, 333, 309, 493, 177, 199, 223, 641, 531, 182, 368

Offset: 1

N. J. A. Sloane

nonn

John L. Drost, Ruth/Aaron Pairs, J. Recreational Math. 28 (No. 2), 120-122.
Dana Mackenzie, Homage to an itinerant master, Science, vol. 275, p. 759, 1997.
Carol Nelson, David E. Penney, and Carl Pomerance, 714 and 715. Journal of Recreational Mathematics 7(2):87-89, 1974.

Amiram Eldar, Table of n, a(n) for n = 1..10000
G. Kreweras and Y. Poupard, Sur les partitions en paires d'un ensemble fini totalement ordonné, Publications de l'Institut de Statistique de l'Université de Paris, 23 (1978), 57-74. (Annotated scanned copy)
Ivars Peterson, Playing with Ruth-Aaron pairs
Ivars Peterson's MathTrek, Playing with Ruth-Aaron Pairs [In the internet archive]
Eric Weisstein's World of Mathematics, Ruth-Aaron Pair

Cf. A006145, A008472, A039752, A039753, A039703, A054378.

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:

Cases[Partition[(Plus@@(First@#&/@FactorInteger@#)&/@Range@100000),2,1],{a_,a_}:>a] (* Hans Rudolf Widmer, May 31 2024 *)

from sympy import primefactors
def aupton(terms):
  alst, k, sopfk, sopfkp1 = [], 0, 0, 1
  while len(alst) < terms:
    k, sopfk, sopfkp1 = k+1, sopfkp1, sum(p for p in primefactors(k+1))
    if sopfkp1 == sopfk: alst.append(sopfk)
  return alst
print(aupton(58)) # Michael S. Branicky, May 05 2021

a(n) = A008472(A006145(n)) = A008472(A006145(n) + 1). - Amiram Eldar, Nov 24 2019

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A178214 Numbers in A039753 with neither of their Ruth-Aaron pairs squarefree.

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A006145 Ruth-Aaron numbers (1): sum of prime divisors of n = sum of prime divisors of n+1.

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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).

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A006146 Sums of prime divisors of Ruth-Aaron numbers (A006145).

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A054378 Sums of prime factors of A039752, including multiplicities.

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