A006145 -id:A006145 - cp's OEIS frontend

A064111 Numbers k such that sopf(k) + 1 = sopf(k+1), where sopf(k) = A008472(k).

2, 8, 120, 168, 175, 247, 860, 1044, 1444, 1659, 1849, 3626, 3834, 4233, 4300, 4345, 4814, 6867, 8240, 14905, 23287, 24476, 28919, 29087, 29464, 30457, 30650, 33725, 34945, 35585, 37214, 49468, 52206, 54900, 58113, 62049, 63440, 65631, 68264

Offset: 1

Jason Earls, Sep 08 2001

nonn

Also k such that z(k) = z(k+1) where z(k) = k - sopf(k).

Prime factors counted without multiplicity. - Harvey P. Dale, Dec 26 2015

			sopf(8) + 1 = 3, sopf(8 + 1) = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A006145, A008472.

Flatten[Position[Partition[Table[Total[Transpose[FactorInteger[n]] [[1]]], {n, 2,70000}],2,1],?(#[[1]]+1==#[[2]]&),{1},Heads->False]]+1 (* _Harvey P. Dale, Dec 26 2015 *)

sopf(n,s,fac,i)=fac=factor(n); for(i=1,matsize(fac)[1],s=s+fac[i,1]); return(s);
j=[]; for(n=1,100000, if(sopf(n)+1==sopf(n+1), j=concat(j,n))); j

z(n)= { local(f,s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); return(n - s) }
{ n=0; zm=z(1); for (m=1, 10^9, zp=z(m + 1); if (zm==zp, write("b064111.txt", n++, " ", m); if (n==1000, break)); zm=zp ) } \\ Harry J. Smith, Sep 07 2009

A129320 a(n) = A129319(n)/A129318(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 5, 15, 2, 5, 1, 3, 11, 2, 11, 1, 5, 5, 1, 4, 4, 1, 5, 2, 16, 10, 1, 6, 13, 4, 3, 9, 1, 1, 6, 2, 26, 2, 3, 23, 4, 1, 1, 6, 2, 3, 1, 11, 2, 1, 8, 14, 28, 8, 1, 7, 1, 23, 8, 20, 7, 2

Offset: 1

Walter Kehowski, Apr 09 2007

			a(6)=2 since A129319(n)/A129318(n)=30/15=2.

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

Cf. A001414, A006145, A039752, A129316, A129317, A129318, A129319.

A330999 Infinitary Ruth-Aaron numbers: numbers k such that A181894(k) = A181894(k+1).

Original entry on oeis.org

5, 77, 714, 948, 2431, 2491, 2996, 3450, 4293, 5405, 5560, 5885, 5959, 11124, 13869, 14587, 16932, 17080, 17346, 18468, 19551, 26642, 31931, 33019, 37925, 42250, 47544, 48635, 49240, 52554, 53192, 60048, 79248, 80837, 89979, 95709, 98119, 98644, 99163, 108458

Offset: 1

Amiram Eldar, Jan 05 2020

nonn

A variation of Ruth-Aaron numbers with "Fermi-Dirac primes" (or infinitary components) instead of prime divisors.

			5 is a term since A181894(5) = A181894(6) = 5.

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

Cf. A006145, A039752, A050376, A181894, A331000.

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); s[1] = 0; s[n] := Plus @@ (Flatten @ (f @@@ FactorInteger[n])); seq ={}; s1 = 0; Do[s2 = s[n]; If[s1 == s2, AppendTo[seq, n-1]]; s1 = s2, {n, 2, 10^5}]; seq

A331000 Unitary Ruth-Aaron numbers: numbers k such that A008475(k) = A008475(k+1).

Original entry on oeis.org

5, 77, 714, 948, 2491, 2996, 3450, 4293, 5405, 6669, 9125, 10807, 13869, 14587, 16932, 17346, 19511, 19967, 23323, 26642, 27104, 31931, 33019, 37925, 41124, 43616, 48635, 52554, 55499, 58077, 58695, 79248, 80837, 86088, 89979, 95709, 98644, 99163, 108458, 117467

Offset: 1

Amiram Eldar, Jan 05 2020

nonn

A variation of Ruth-Aaron numbers with unitary prime-power divisors instead of prime divisors.

			5 is a term since A008475(5) = A008475(6) = 5.

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

Cf. A006145, A039752, A008475, A330999.

s[1] = 0; s[n_] := Plus @@ (Power @@@ FactorInteger[n]); seq = {}; s1 = 0; Do[s2 = s[n]; If[s1 == s2, AppendTo[seq, n - 1]]; s1 = s2, {n, 2, 10^5}]; seq

A063968 Numbers k such that sopf(k) = sopf(k+2), where sopf(k) = A008472(k).

Original entry on oeis.org

2, 340, 845, 950, 1340, 3724, 5694, 6102, 7657, 8991, 9331, 9709, 10323, 11388, 11390, 12460, 15870, 18912, 19778, 20882, 21715, 24732, 26978, 29052, 29632, 32428, 33596, 35028, 38178, 42718, 43068, 45750, 46102, 50396, 53251, 61408

Offset: 1

Jason Earls, Sep 05 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A008472, A006145.

sopf(n,s,fac,i)=fac=factor(n); for(i=1,matsize(fac)[1],s=s+fac[i,1]); return(s);
j=[]; for(n=1,100000, if(sopf(n)==sopf(n+2),j=concat(j,n))); j

sopf(n)= { local(f,s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); return(s) }
{ n=0; r=sopf(1); s=sopf(2); for (m=1, 10^9, t=sopf(m + 2); if(r==t, write("b063968.txt", n++, " ", m); if (n==1000, break)); r=s; s=t ) } \\ Harry J. Smith, Sep 04 2009

A156787 Composite integers n such that 2^{n-1}=1 mod s(n), where s(n) is the sum of the distinct prime factors of n.

Original entry on oeis.org

9, 10, 25, 27, 40, 49, 81, 100, 105, 116, 121, 125, 160, 169, 243, 250, 289, 343, 361, 400, 525, 529, 561, 568, 625, 640, 729, 805, 841, 945, 961, 1000, 1001, 1018, 1045, 1105, 1309, 1331, 1369, 1596, 1600, 1681, 1729, 1849, 1856, 1881, 2001, 2187, 2197, 2205

Offset: 1

Florian Luca (fluca(AT)matmor.unam.mx), Feb 15 2009

nonn

			For n=2, the second number is a(2)=10 because s(10)=2+5=7 divides 2^{10-1}-1=2^9-1=511.

Amiram Eldar, Table of n, a(n) for n = 1..10000
F. Luca and V. Tipu, On positive integers n with a certain divisibility property, J. Integer Sequences 12 (2009), 11 pp.

Cf. A006145.

B := {}; for n from 2 to 1000 do A := (numtheory[factorset])(n); b := add(a, `in`(a, A)); if `and`(b < n, `mod`(2^(n-1), b) = 1) then B := [op(B), n] else end if end do; print(c := 2);

Select[Range[2, 2300], CompositeQ[#] && PowerMod[2, #-1, Total[First /@ FactorInteger[#]]] == 1 &] (* Amiram Eldar, Nov 20 2019 *)

is(n)=if(isprime(n),0,my(f=factor(n)[,1]);Mod(2, sum(i=1, #f, f[i]))^(n-1)==1) \\ Charles R Greathouse IV, Feb 01 2013

n log n << a(n) << n^(1+e) for any e > 0. See Luca & Tipu for more precise results. - Charles R Greathouse IV, Feb 01 2013

More terms from Amiram Eldar, Nov 20 2019

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

Original entry on oeis.org

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.

A192283 Sum of prime anti-divisors of n = sum of prime anti-divisors of n+1 with n > 1.

Original entry on oeis.org

237, 4019, 7401, 14178, 14339, 18435, 19146, 21405, 54562, 56348, 60125, 82967, 98447, 99347, 109157, 113391, 125333, 132096, 132386, 145063, 173399, 195213, 260288, 278271, 343848, 384169, 396813, 434375, 460758, 474105, 477707, 528845, 550400, 587211

Offset: 1

Paolo P. Lava, Jul 27 2011

nonn

Like A006145 but using anti-divisors.

			Anti-divisors of 7401 are 2, 6, 19, 41, 113, 131, 361, 779, 4934. The primes are 2, 19, 41, 113 and 131 whose sum is 306.
Anti-divisors of 7402 are 3, 4, 5, 7, 9, 15, 21, 35, 45, 47, 63, 105, 113, 131, 141, 235, 315, 329, 423, 705, 987, 1645, 2115, 2961, 4935. The primes are 3, 5, 7, 47, 113 and 131 whose sum is 306.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A002961, A006145, A066272, A192282

with(numtheory);
P:=proc(n)
local a,b,i,k;
b:=2;
for i from 4 to n do
  a:=0;
  for k from 2 to i-1 do
    if abs((i mod k)- k/2) < 1 then if isprime(k) then a:=a+k; fi; fi;
  od;
  if a=b then print(i-1); fi;
  b:=a;
od;
end:
P(200000);

A193314 The smallest k such that the product k*(k+1) is divisible by the first n primes and no others.

Original entry on oeis.org

1, 2, 5, 14, 384, 1715, 714, 633555

Offset: 1

Robert G. Wilson v, Aug 17 2011

nonn

a(9)-a(21) do not exist. It seems unlikely that a(n) exists for larger n. [Charles R Greathouse IV, Aug 18 2011]

If a term beyond a(8) exists, it is larger than 2.29*10^25. - Giovanni Resta, Nov 30 2019

			n  smallest k   k*(k+1) prime factorization
1  1            2
2  2            2*3
3  5            2*3*5
4  14           2*3*5*7
5  384          2^7*3*5*7*11
6  1715         2^2*3*7^3*11*13
7  714          2*3*5*7*11*13*17
8  633555       2^2*3^3*5*7*11^3*13*17*19^2

Carlos Rivera, Puzzle 358. Ruth-Aaron pairs revisited, The Prime Puzzles & Problems Connection.

Cf. A006145, A039945.

Cf. A002110, A002378, A006530, A118478.

a193314 n = head [k | k <- [1..], let kk' = a002378 k,
                      mod kk' (a002110 n) == 0, a006530 kk' == a000040 n]
-- Reinhard Zumkeller, Jun 14 2015

f[n_] := Block[{k = 1, p = Fold[ Times, 1, Prime@ Range@ n], tst = Prime@ Range@ n},While[ First@ Transpose@ FactorInteger[ k*p]!=tst || IntegerQ@ Sqrt[ 4k*p+1], k++]; Floor@ Sqrt[k*p]]; Array[f, 8]
(* the search for a(9), I also used *) lst = {}; p = Prime@ Range@ 9; Do[ q = {a, b, c, d, e, f, g, h, i}; If[ IntegerQ[ Sqrt[4Times @@ (p^q) + 1]], r = Floor@ Sqrt@ Times @@ (p^q); Print@ r; AppendTo[lst, r]], {i, 9}, {h, 9}, {g, 9}, {f, 10}, {e, 11}, {d, 14}, {c, 16}, {b, 24}, {a, 8}]

a(n)={
  my(v=[Mod(0,1)],u,P=1,t,g,k);
  forprime(p=2,prime(n),
    P*=p;
    u=List();
    for(i=1,#v,
      listput(u,chinese(v[i],Mod(-1,p)));
      listput(u,chinese(v[i],Mod(0,p)))
    );
    v=0;v=Vec(u)
  );
  v=vecsort(lift(v));
  while(1,
    for(i=1,#v,
      t=(v[i]+k)*(v[i]+k+1)/P;
      if(!t,next);
      while((g=gcd(P,t))>1, t/=g);
        if (t==1, return(v[i]+k))
    );
    k += P
  )
}; \\ Charles R Greathouse IV, Aug 18 2011

A237929 Numbers n such that (i) the sum of prime divisors of n (with repetition) is one less than the sum of prime divisors (with repetition) of n+1, and (ii) n and n+1 have the same number of prime divisors (with repetition).

Original entry on oeis.org

2, 9, 98, 170, 1274, 4233, 4345, 7105, 7625, 14905, 21385, 30457, 34945, 66585, 69874, 77314, 82946, 98841, 175354, 177122, 233090, 236282, 238017, 263145, 265225, 295274, 298082, 322234, 335793, 336106

Offset: 1

Abhiram R Devesh, Feb 16 2014

nonn

The first term a(1)=2 is the only prime number in this sequence.

			For n=98: prime factors = 2,7,7; sum of prime factors = 16; number of prime divisors = 3
For n+1=99: prime factors = 3,3,11; sum of prime factors = 17; number of prime divisors=3.

Abhiram R Devesh, Table of n, a(n) for n = 1..96049

Cf. A001414, A006145 Ruth-Aaron numbers (1): sum of prime divisors of n = sum of prime divisors of n+1.
Cf. A228126 Sum of prime divisors of n (with repetition) is one less than the sum of prime divisors (with repetition) of n+1.
Cf. A045920 Numbers n such that factorizations of n and n+1 have same number of primes (including multiplicities).

Select[Partition[Table[{n,PrimeOmega[n],Total[Times@@@FactorInteger[n]]},{n,34*10^4}],2,1],#[[1,2]]==#[[2,2]]&&#[[1,3]]+1==#[[2,3]]&][[;;,1,1]] (* Harvey P. Dale, May 03 2024 *)

from sympy import primeomega
def is_A237929(n): return A001414(n) == A001414(n+1)-1 and primeomega(n) == primeomega(n+1) # David Radcliffe, Aug 08 2025

cp's OEIS Frontend

A064111 Numbers k such that sopf(k) + 1 = sopf(k+1), where sopf(k) = A008472(k).

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A129320 a(n) = A129319(n)/A129318(n).

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A330999 Infinitary Ruth-Aaron numbers: numbers k such that A181894(k) = A181894(k+1).

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A331000 Unitary Ruth-Aaron numbers: numbers k such that A008475(k) = A008475(k+1).

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A063968 Numbers k such that sopf(k) = sopf(k+2), where sopf(k) = A008472(k).

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A156787 Composite integers n such that 2^{n-1}=1 mod s(n), where s(n) is the sum of the distinct prime factors of n.

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

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A192283 Sum of prime anti-divisors of n = sum of prime anti-divisors of n+1 with n > 1.

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A193314 The smallest k such that the product k*(k+1) is divisible by the first n primes and no others.