A039945 -id:A039945 - cp's OEIS frontend

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).

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

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

cp's OEIS Frontend

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).

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