A039752 -id:A039752 - 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.

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.

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

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.

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

A090340 Difference between the sums of the prime factors, including multiplicity, of n and those of n + 1.

Original entry on oeis.org

-2, -1, -1, -1, 0, -2, 1, 0, -1, -4, 4, -6, 4, 1, 0, -9, 9, -11, 10, -1, -3, -10, 14, -1, -5, 6, -2, -18, 19, -21, 21, -4, -5, 7, 2, -27, 16, 5, 5, -30, 29, -31, 28, 4, -14, -22, 36, -3, 2, -8, 3, -36, 42, -5, 3, -9, -9, -28, 47, -49, 28, 20, 1, -6, 2, -51, 46, -5, 12, -57, 59, -61, 34, 26, -10, 5, 0, -61, 66, 1, -31, -40, 69, -8, -23, 13

Offset: 1

Charles K. Layman (cklayman(AT)juno.com), Nov 25 2003

sign

If a(n) = 0 then n is a Ruth-Aaron number. - Andrew Slattery, Apr 15 2020

			a(24)=-1 because 24=2*2*2*3, 25=5*5 and (2+2+2+3)-(5+5)=-1.

Cf. A001414 (sopfr), A090341, A090342, A090343.

Ruth-Aaron numbers: A039752.

a(n) = sopfr(n) - sopfr(n+1), where sopfr = A001414. - Wesley Ivan Hurt, Aug 10 2016

A001366 Maximal number of unattacked squares with n queens on n X n board (answers for n >= 17 only probable).

Original entry on oeis.org

0, 0, 0, 1, 3, 5, 7, 11, 18, 22, 30, 36, 47, 56, 72, 82, 97, 111, 132, 145, 170, 186, 216, 240, 260, 290, 324, 360, 381, 420

Offset: 1

Mario Velucchi (mathchess(AT)velucchi.it)

Yanan Jiang and Steven J. Miller, Generalizing Ruth-Aaron Numbers, arXiv:2010.14990 [math.NT], 2020; published in The Pump Journal of Undergraduate Research, 4 (2021), 20-62. [That paper references this sequence entry, probably by mistake; cf. A039752.]
Bernard Lemaire and Pavel Vitushinkiy, Placing n non dominating queens on the n X n chessboard. Part I, French Federation of Mathematical Games.
Bernard Lemaire and Pavel Vitushinkiy, Placing n non dominating queens on the n X n chessboard. Part II, French Federation of Mathematical Games.
Steven J. Miller, Haoyu Sheng, and Daniel Turek, When rooks miss: probability through chess, Williams College (2020).
Mario Velucchi, NON-Dominating Queens Problem
Eric Weisstein's World of Mathematics, Queens Problem.

Cf. A019317, A274947.

cp's OEIS Frontend

A039753 Numbers common to A006145 and A039752.

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A054378 Sums of prime factors of A039752, including multiplicities.

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

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A090340 Difference between the sums of the prime factors, including multiplicity, of n and those of n + 1.

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