cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A233341 Least number m for which n applications of the mapping r(k) = k - (greatest prime divisor of k) map m to 0.

Original entry on oeis.org

1, 4, 8, 12, 25, 30, 32, 48, 63, 70, 75, 80, 165, 176, 189, 192, 289, 306, 315, 320, 385, 392, 507, 520, 575, 598, 621, 644, 841, 858, 957, 968, 1015, 1044, 1071, 1088, 1105, 1122, 1425, 1444, 1463, 1470, 1771, 1782, 1935, 1978, 2145, 2156, 2303, 2350, 2397
Offset: 1

Views

Author

Clark Kimberling, Dec 07 2013

Keywords

Comments

r(m) = 1 if and only if m = 1 or m is a prime. Conjecture: Every positive integer divides infinitely many terms of this sequence.
Sequence is empirically observed to be strictly increasing for n <= 1000, in contrast to similar map in A050710. - Christian N. K. Anderson, May 05 2023
Observe that for ~4/7 of the first thousand terms, r(a(n)) = a(n-1); e.g., a(12)=80, r(80)=75=a(11) -> 70=a(10) -> 63=a(9). However, the other ~3/7 take a different route to zero; e.g., a(9)=63 decreases by 7 at all 9 steps. Contrast A048133, where every term's iteration ends when r(k)=5. - Christian N. K. Anderson, May 05 2023

Examples

			r(8) = 8 - 2 = 6; r(6) = 6 - 3 = 3; r(3) = 3 - 3 = 0. Thus 3 applications of r map 8 to 0, whereas 1 or 2 applications suffice for n < 8. Therefore, a(3) = 8.

Links

Crossrefs

Cf. A233342.

Programs

  • Mathematica
    z = 10000; h[n_] := h[n] = n - FactorInteger[n][[-1, 1]]; t[n_] := Drop[FixedPointList[h, n], -2]; Table[t[n], {n, 1, z}]; a = Table[Length[t[n]], {n, 1, z}]; f[n_] := First[Flatten[Position[a, n]]]; Table[f[n], {n, 1, 80}]

Formula

For 228 <= n <= 1000, a(n) ~ 0.8526*n^2.023 to within 4% (empirical observation). - Christian N. K. Anderson, May 05 2023

A050766 Primes occurring in A050765.

Original entry on oeis.org

38431, 59333, 89753, 93683, 156007, 356441, 682141, 1226959, 1468387, 1648697, 2592043, 2995249, 3595589, 3933731, 5845309, 5967433, 8861059, 8953631, 9748537, 11351399, 12819329, 13227023, 13712551, 16193579, 18375053, 19294523, 21206117, 25259593, 28297559
Offset: 1

Views

Author

Patrick De Geest, Sep 15 1999

Keywords

Examples

			Prime 19294523 = composite 19294436 + sopf 2+2+7+7+7+7+7+7+41 = 19294436 + 87 = nextprime(19294436).

Links

Crossrefs

Cf. A050765, A050703, A050710.

Programs

  • Mathematica
    Reap[For[n = 4, n <= 2*10^7, n = If[PrimeQ[n+1], n+2, n+1], p = NextPrime[n]; If[n + Total[Times @@@ FactorInteger[n] ] == p, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Oct 18 2013 *)

Extensions

Offset corrected by Donovan Johnson, Oct 18 2013

A050778 Primes that are ending primes after the iterated procedure of 'composite added to the sum of its prime factors reaches a prime'.

Original entry on oeis.org

11, 17, 19, 23, 29, 31, 41, 47, 53, 59, 71, 79, 83, 89, 103, 107, 109, 113, 127, 131, 149, 151, 167, 179, 191, 227, 229, 239, 241, 251, 263, 269, 271, 293, 311, 313, 317, 331, 337, 359, 373, 383, 401, 419, 431, 439, 443, 449, 461, 467, 479, 491, 503, 509, 521
Offset: 1

Views

Author

Patrick De Geest, Sep 15 1999

Keywords

Examples

			383 is ending prime for the following composites 182, 204, 218, 226, 228, 254, 329 and 341. For instance 341 (=11*31) -> 341 + (11+31) = 341 + 42 = 383.

Crossrefs

Cf. A050779, A050703-A050710.

Programs

  • Mathematica
    a[n_]:=NestWhile[#+Total[Times@@@FactorInteger[#]]&,n,!PrimeQ[#]&]; t={}; Do[If[!PrimeQ[n],AppendTo[t,a[n]]],{n,4,nn=522}]; Select[Union[t],#Jayanta Basu, Jun 01 2013 *)

A053093 Smallest composite which when sum of prime factors is repeatedly subtracted reaches a prime after n iterations.

Original entry on oeis.org

8, 16, 27, 58, 104, 150, 186, 256, 324, 378, 576, 636, 805, 1470, 1520, 1564, 1612, 3478, 3864, 8022, 9996, 10548, 11573, 11946, 12006, 13587, 13640, 15156, 20504, 21204, 21252, 24017, 24080, 25823, 27989, 30067, 30866, 31165, 31212, 47628
Offset: 1

Views

Author

Enoch Haga, Feb 28 2000

Keywords

Examples

			a(5)=104. Iteration 1: 104=2*2*2*13 and sum of factors is 19; 104-19=85. Iteration 2: 85=5*17; 85-22=63. Iterations 3,4,5 produce 63-13=50, 50-12=38 and 38-21=17, a prime.

Links

Crossrefs

A050710, A053094.

A050767 Let f(m) = smallest composite number that takes m steps of "add prime factors to number" to reach a prime and g(m) be the prime that is reached. Sequence gives values of g(m), sorted and duplicates removed.

Original entry on oeis.org

11, 23, 83, 149, 167, 251, 269, 881, 2039, 5021, 9311, 35111, 82219, 89809, 181421, 289099, 420899, 885839, 996157, 1963799, 2554439, 2843903, 3269039, 3632701, 4648519, 6448103, 9748579, 35200439, 71927183, 82216501, 283709077
Offset: 0

Views

Author

Patrick De Geest, Sep 15 1999

Keywords

Examples

			23 is included as it is both g(2) (with f(2)=8) and g(3) (with f(3)=4): '8' + (2+2+2) => 14 + (2+7) => 23; '4' + (2+2) => 8 + (2+2+2) => 14 + (2+7) => 23.

Links

Crossrefs

Cf. A050703, A050710 (for f(n)).

A050768 Iterated procedure 'composite k added to sum of its prime factors reaches a prime' yields 1 skipped prime.

Original entry on oeis.org

6, 20, 48, 112, 320, 1326, 1400, 4165, 4374, 10395, 12852, 15827, 20412, 23232, 24300, 24990, 25000, 27200, 27300, 31407, 33660, 34965, 38480, 41553, 42525, 50688, 53508, 65450, 66000, 68400, 69498
Offset: 0

Views

Author

Patrick De Geest, Sep 15 1999

Keywords

Examples

			a(2)=20 + (2+2+5) = ending prime 29. Between 20 and 29 lies exactly one prime 23.

Crossrefs

Cf. A050703, A050710.

Programs

  • Mathematica
    aQ[n_]:=NextPrime[NestWhile[#+Total[Times@@@FactorInteger[#]]&,n,!PrimeQ[#]&],-1]==NextPrime[n]; Select[Range[70000],!PrimeQ[#]&&aQ[#]&] (* Jayanta Basu, May 31 2013 *)

A050769 Iterated procedure 'composite k added to sum of its prime factors reaches a prime' yields 2 skipped primes.

Original entry on oeis.org

10, 12, 14, 15, 21, 44, 90, 135, 210, 252, 294, 384, 468, 504, 513, 686, 704, 720, 768, 832, 864, 972, 1155, 1368, 1701, 1890, 2176, 2184, 2352, 2400, 2880, 3080, 3400, 3640, 3888, 4032, 4536, 4725, 5200, 6174, 6384, 8750, 9548, 10350, 10400, 10500
Offset: 0

Views

Author

Patrick De Geest, Sep 15 1999

Keywords

Examples

			a(6) = 44 + (2 + 2 + 11) = ending prime 59. Between 44 and 59 there are 2 primes: 47 and 53.

Links

Crossrefs

Cf. A050703, A050710.

Programs

  • Mathematica
    ckpgQ[n_]:=Module[{c=n+Total[Flatten[Table[#[[1]],{#[[2]]}]&/@ FactorInteger[ n]]]},CompositeQ[n]&&PrimeQ[c]&&PrimePi[c]-PrimePi[n] == 3]; Select[Range[11000],ckpgQ] (* Harvey P. Dale, Nov 29 2014 *)

A050772 Iterated procedure 'composite k added to sum of its prime factors reaches a prime' yields 5 skipped primes.

Original entry on oeis.org

18, 24, 25, 46, 57, 161, 203, 209, 288, 319, 323, 391, 736, 798, 837, 858, 928, 930, 1035, 1088, 1089, 1218, 1300, 1376, 1690, 2254, 2418, 2478, 2673, 2842, 2871, 3045, 3220, 3325, 3458, 3510, 3588, 4186, 4508, 4617, 4824, 5054, 5180, 5248, 5472, 6069
Offset: 1

Views

Author

Patrick De Geest, Sep 15 1999

Keywords

Examples

			18 is a term because 18 + (2+3+3) = 26 + (2+13) = ending prime 41. Between 18 and 41 one finds 5 primes 19, 23, 29, 31 and 37.

Links

Crossrefs

Cf. A050703, A050710.

Programs

  • Maple
    filter:= proc(n) local r, s, t;
  if isprime(n) then return false fi;
  t:= 0: s:= n;
  do
   r:= s;
   s:= s + add(p[1]*p[2],p=ifactors(s)[2]);
   t:= t + numtheory:-pi(s-1) - numtheory:-pi(r);
   if isprime(s) then return t=5 fi;
   if t > 5 then return false fi;
  od;
end proc:
select(filter, [$2..10000]); # Robert Israel, May 08 2020
  • Mathematica
    ok[n_] := CompositeQ[n] && Block[{k=n, p = NextPrime[n, 6]}, While[k < p, k += Total[ Times @@@ FactorInteger[k]]]; k == p]; Select[Range@ 6069, ok] (* Giovanni Resta, May 08 2020 *)

Extensions

Offset changed to 1 by Robert Israel, May 08 2020

A050776 Iterated procedure 'composite k added to sum of its prime factors reaches a prime' yields 9 skipped primes.

Original entry on oeis.org

30, 42, 99, 174, 188, 212, 216, 295, 329, 348, 371, 620, 627, 629, 649, 851, 901, 925, 957, 1081, 1189, 1248, 1353, 1363, 1696, 1830, 1880, 2133, 2173, 2491, 2664, 3660, 3843, 4020, 4420, 5056, 5688, 6327, 6700, 7540, 7626, 7808, 7888, 7900, 8295, 8778
Offset: 1

Views

Author

Patrick De Geest, Sep 15 1999

Keywords

Examples

			a(3)=99 + (3+3+11) = 116 + (2+2+29) = ending prime 149. Between 99 and 149 one finds 9 primes 101, 103, 107, 109, 113, 127, 131, 137 and 139.

Crossrefs

Cf. A050703, A050710.

Extensions

Offset corrected by Sean A. Irvine, Aug 18 2021

A050779 Primes that are not ending primes after the iterated procedure of 'composite added to the sum of its prime factors reaches a prime'.

Original entry on oeis.org

2, 3, 5, 7, 13, 37, 43, 61, 67, 73, 97, 101, 137, 139, 157, 163, 173, 181, 193, 197, 199, 211, 223, 233, 257, 277, 281, 283, 307, 347, 349, 353, 367, 379, 389, 397, 409, 421, 433, 457, 463, 487, 499, 547, 557, 563, 577, 601, 613, 617, 641, 643, 661, 673, 677
Offset: 1

Views

Author

Patrick De Geest, Sep 15 1999

Keywords

Crossrefs

Cf. A050778, A050703-A050710.

Programs

  • Mathematica
    a[n_]:=NestWhile[#+Total[Times@@@FactorInteger[#]]&,n,!PrimeQ[#]&]; t={}; Do[If[!PrimeQ[n],AppendTo[t,a[n]]],{n,4,nn=678}]; Complement[Prime[Range[PrimePi[nn]]],Select[Union[t],#Jayanta Basu, Jun 01 2013 *)
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