A006145 -id:A006145 - cp's OEIS frontend

A039753 Numbers common to A006145 and A039752.

5, 77, 714, 5405, 26642, 52554, 95709, 154842, 173162, 204258, 208581, 248109, 278277, 332994, 417162, 445305, 529194, 554682, 693610, 851709, 869054, 1232746, 1252509, 1275546, 1275730, 1549454, 1600962, 1607045, 1671333, 1672710, 1777026

Offset: 1

David W. Wilson

nonn

Mostly the subset of either A006145 or A039752 where both a(n) and a(n)+1 are squarefree (members of A005117), but also members of A178214. - Hans Havermann, Dec 16 2010, Dec 19 2010

			77 (7*11, with 78 = 2*3*13) is a member of both A006145 and A039752 (7+11 = 2+3+13) and so 77 is a member of this sequence.
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) and so 7129199 is also a member of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006145, A039752, A005117, A178214.

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

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

A228126 Sum of prime divisors of n (with repetition) is one less than the sum of prime divisors (with repetition) of n+1.

Original entry on oeis.org

2, 3, 4, 9, 20, 24, 98, 170, 1104, 1274, 2079, 2255, 3438, 4233, 4345, 4716, 5368, 7105, 7625, 10620, 13350, 13775, 14905, 20220, 21385, 23408, 25592, 26123, 28518, 30457, 34945, 35167, 38180, 45548, 49230, 51911, 52206, 53456, 56563, 61456, 65429, 66585

Offset: 1

Abhiram R Devesh, Aug 11 2013

This is an extension to Ruth-Aaron pairs. Sum of prime factors (inclusive of multiplicity) of pair of Consecutive positive integers are also consecutive.
The number of pairs less than 10^k (k=1,2,3,4,5,6,..) with this property are 4,7,8,19,55,149,...
Up to 10^13 there are only 5 sets of consecutive terms, namely, (2, 3), (3,4), (27574665988, 27574665989), (862179264458, 1862179264459) and (9600314395008, 9600314395009). - Giovanni Resta, Dec 24 2013
The sum of reciprocals of this sequence is approximately equal to 1.3077. - Abhiram R Devesh, Jun 14 2014

			For n=20: prime factors = 2,2,5; sum of prime factors = 9.
For n+1=21: prime factors = 3,7; sum of prime factors = 10.

Harvey P. Dale, Table of n, a(n) for n = 1..300
Giovanni Resta, eRAPs: the 446139 terms < 10^12
Carlos Rivera, Extension to Ruth Aaron pairs

Cf. A001414, A006145 Ruth-Aaron numbers (1): sum of prime divisors of n = sum of prime divisors of n+1.

spd[n_]:=Total[Flatten[Table[#[[1]],#[[2]]]&/@FactorInteger[n]]]; Rest[ Position[ Partition[Array[spd,70000],2,1],?(#[[2]]-#[[1]]==1&),{1}, Heads->False]//Flatten] (* _Harvey P. Dale, Sep 07 2016 *)

sopfm(n)=my(f=factor(n));sum(i=1,#f[,1],f[i,1]*f[i,2])
for(n=1,10^5,if(sopfm(n)==sopfm(n+1)-1,print1(n,","))) /* Ralf Stephan, Aug 12 2013 */

## sumdivisors(n) is a function that would return the sum of prime
## divisors of n. (See A001414)
i=2
while i < 100000:
  sdi=sumdivisors(i)
  sdip=sumdivisors(i+1)
  if sdi==sdip-1:
    print(i)
  i=i+1

More terms from Ralf Stephan, Aug 12 2013

A129316 Positive integers k such that sopfr(k) divides sopfr(k+1), where sopfr(k) is the sum of the prime factors of k, counting multiplicity.

Original entry on oeis.org

5, 8, 15, 77, 125, 160, 252, 496, 714, 948, 980, 1045, 1053, 1260, 1330, 1378, 1404, 1430, 1508, 1520, 1610, 1750, 1862, 1890, 2170, 2491, 2680, 2821, 3094, 3100, 3248, 3400, 3591, 3610, 3652, 3808, 4185, 4191, 4384, 4452, 4500, 4598, 4906, 5120, 5145

Offset: 1

Walter Kehowski, Apr 09 2007

A generalization of Ruth-Aaron pairs (A006145).

			a(6)=160 since sopfr(160)=sopfr(2^5*5)=10+5=15 and sopfr(161)=sopfr(7*23)=30.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A001414, A006145, A039752, A129317, A129318, A129319.

sopf[n_]:=Total[Flatten[Table[#[[1]],{#[[2]]}]&/@FactorInteger[n]]]; Rest[ Flatten[Position[Partition[Table[sopf[n],{n,5200}],2,1],?(Divisible[#[[2]],#[[1]]]&),{1},Heads->False]]] (* _Harvey P. Dale, Jul 18 2013 *)

sopfr(k+1) mod sopfr(k) = 0.

A129317 The second of the pair of consecutive integers k and k+1 such that sopfr(k) divides sopfr(k+1), where sopfr(k) is the sum of the prime factors of k, counting multiplicity.

Original entry on oeis.org

6, 9, 16, 78, 126, 161, 253, 497, 715, 949, 981, 1046, 1054, 1261, 1331, 1379, 1405, 1431, 1509, 1521, 1611, 1751, 1863, 1891, 2171, 2492, 2681, 2822, 3095, 3101, 3249, 3401, 3592, 3611, 3653, 3809, 4186, 4192, 4385, 4453, 4501, 4599, 4907, 5121, 5146

Offset: 1

Walter Kehowski, Apr 09 2007

A129316 is the first element of the pair.

A generalization of Ruth-Aaron pairs (A006145).

			a(6)=161 since sopfr(160)=sopfr(2^5*5)=10+5=15 and sopfr(161)=sopfr(7*23)=30.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001414, A006145, A039752, A129316, A129318, A129319.

sopfr(k+1) mod sopfr(k) = 0.

a(n) = A129316(n+1). - Amiram Eldar, Oct 26 2019

A129318 a(n) = sopfr(A129316(n)), where sopfr(x) is the sum of the prime factors of x, counting multiplicity.

Original entry on oeis.org

5, 6, 8, 18, 15, 15, 17, 39, 29, 86, 23, 35, 25, 22, 33, 68, 26, 31, 46, 32, 37, 24, 35, 23, 45, 100, 78, 51, 39, 45, 44, 33, 35, 45, 98, 34, 45, 141, 147, 67, 25, 43, 236, 25, 29, 75, 150, 29, 76, 125, 160, 26, 121, 122, 73, 37, 52, 56, 40, 39, 54, 58, 239, 69, 36, 39

Offset: 1

Walter Kehowski, Apr 09 2007, corrected Apr 09 2007

			a(6)=15 since sopfr(A129316(6))=sopfr(160)=sopfr(2^5*5)=15.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001414, A006145, A039752, A129316, A129317, A129319, A129320.

A129319 a(n) = sopfr(A129317(n)), where sopfr(x) is the sum of the prime factors of x, counting multiplicity.

Original entry on oeis.org

5, 6, 8, 18, 15, 30, 34, 78, 29, 86, 115, 525, 50, 110, 33, 204, 286, 62, 506, 32, 185, 120, 35, 92, 180, 100, 390, 102, 624, 450, 44, 198, 455, 180, 294, 306, 45, 141, 882, 134, 650, 86, 708, 575, 116, 75, 150, 174, 152, 375, 160, 286, 242, 122, 584, 518, 1456, 448

Offset: 1

Walter Kehowski, Apr 09 2007

			a(6)=30 since sopfr(A129317(6))=sopfr(161)=sopfr(7*23)=30.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001414, A006145, A039752, A129316, A129317, A129318, A129320.

A054378 Sums of prime factors of A039752, including multiplicities.

Original entry on oeis.org

5, 6, 8, 18, 15, 29, 86, 33, 32, 35, 100, 44, 45, 141, 75, 150, 160, 122, 40, 54, 39, 205, 532, 107, 79, 93, 421, 401, 193, 66, 144, 117, 149, 211, 93, 64, 57, 118, 480, 82, 299, 1242, 485, 176, 774, 152, 78, 89, 216, 72, 53, 218, 429, 587, 245, 295, 1924

Offset: 1

Eric W. Weisstein

nonn

John L. Drost, Ruth/Aaron Pairs, J. Recreational Math. 28 (No. 2), 120-122.

Hans Havermann, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Ruth-Aaron Pair.

Cf. A006146, A039752, A039753, A006145.

A045759 Maris-McGwire numbers: numbers k such that f(k) = f(k+1), where f(k) = sum of digits of k + sum of digits of prime factors of k (including multiplicities).

Original entry on oeis.org

7, 14, 43, 50, 61, 63, 67, 80, 84, 118, 122, 134, 137, 163, 196, 212, 213, 224, 241, 273, 274, 277, 279, 283, 351, 352, 373, 375, 390, 398, 421, 457, 462, 474, 475, 489, 495, 510, 516, 523, 526, 537, 547, 555, 558, 577, 584, 590, 592, 616, 638, 644, 660, 673, 687, 691

Offset: 1

Mike Keith

Named "Maris-McGwire-Sosa Numbers" by Keith (1998) after the baseball players Roger Maris, Mark McGwire and Sammy Sosa. Both McGwire and Sosa hit their 62nd home runs for the season, breaking Maris's record of 61 (A006145 is a similarly named sequence). - Amiram Eldar, Jun 27 2021

			(61, 62) is such a pair, hence the name.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Mike Keith, Maris-McGwire-Sosa Numbers, 1998.
Ivars Peterson, Home Run Numbers, MathTrek, 1998.
Wikipedia, Maris-McGwire-Sosa pair.

Cf. A006145 (Ruth-Aaron numbers), A039945.

ds[n_] := Plus @@ IntegerDigits[n]; f[n_] := ds[n] + Total[(fi = FactorInteger[n])[[;; , 2]] *( ds /@fi[[;; , 1]])]; s={}; f1 = 1; Do[f2=f[n]; If[f1 == f2, AppendTo[s, n-1]]; f1 = f2, {n, 2, 700}]; s (* Amiram Eldar, Nov 24 2019 *)

from sympy import factorint
def sd(n): return sum(map(int, str(n)))
def  f(n): return sd(n) + sum(sd(p)*e for p, e in factorint(n).items())
def ok(n): return f(n) == f(n+1)
print(list(filter(ok, range(692)))) # Michael S. Branicky, Jul 14 2021

Corrected and extended by David W. Wilson

Offset corrected by Amiram Eldar, Nov 24 2019

cp's OEIS Frontend

A039753 Numbers common to A006145 and A039752.

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

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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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A228126 Sum of prime divisors of n (with repetition) is one less than the sum of prime divisors (with repetition) of n+1.

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A129316 Positive integers k such that sopfr(k) divides sopfr(k+1), where sopfr(k) is the sum of the prime factors of k, counting multiplicity.

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A129317 The second of the pair of consecutive integers k and k+1 such that sopfr(k) divides sopfr(k+1), where sopfr(k) is the sum of the prime factors of k, counting multiplicity.

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A129318 a(n) = sopfr(A129316(n)), where sopfr(x) is the sum of the prime factors of x, counting multiplicity.

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A129319 a(n) = sopfr(A129317(n)), where sopfr(x) is the sum of the prime factors of x, counting multiplicity.

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