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.

Showing 1-5 of 5 results.

A003508 a(1) = 1; for n>1, a(n) = a(n-1) + 1 + sum of distinct prime factors of a(n-1) that are < a(n-1).

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 11, 12, 18, 24, 30, 41, 42, 55, 72, 78, 97, 98, 108, 114, 139, 140, 155, 192, 198, 215, 264, 281, 282, 335, 408, 431, 432, 438, 517, 576, 582, 685, 828, 857, 858, 888, 931, 958, 1440, 1451, 1452, 1469, 1596, 1628, 1679, 1776, 1819, 1944
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

R. K. Guy reports, Apr 14 2005: In Math. Mag. 48 (1975) 301 one finds "C. W. Trigg, C. C. Oursler and R. Cormier & J. L. Selfridge have sent calculations on Problem 886 [Nov 1973] for which we had received only partial results [Jan 1975]. Cormier and Selfridge sent the following results: There appear to be five sequences beginning with integers less than 1000 which do not merge. These sequences were carried out to 10^8 or more." The five sequences are A003508, A105210-A105213.
This suggests that there may be infinitely many different (non-merging) sequences obtained by choosing different starting values.
All terms of these five sequences are distinct up to least 10^30. - T. D. Noe, Oct 19 2007

Examples

			a(6)=8, so a(7) = 8 + 1 + 2 = 11.

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A096460, A105221, A105233.
Cf. A027748, A105210, A105211, A105212, A105213.

Programs

  • Haskell
    a003508 n = a003508_list !! (n-1)
a003508_list = 1 : map
      (\x -> x + 1 + sum (takeWhile (< x) $ a027748_row x)) a003508_list
-- Reinhard Zumkeller, Jan 15 2015
  • Mathematica
    a[1] = 1; a[n_] := a[n] = a[n - 1] + 1 + Plus @@ Select[ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[ a[n - 1]]], # < a[n - 1] &]; Table[ a[n], {n, 54}] (* Robert G. Wilson v, Apr 13 2005 *)
nxt[n_]:=n+1+Total[Select[Transpose[FactorInteger[n]][[1]],#Harvey P. Dale, Jul 19 2015 *)

Extensions

More terms from Henry Bottomley, May 09 2000

A105210 a(1) = 393; for n > 1, a(n) = a(n-1) + 1 + sum of distinct prime factors of a(n-1) that are < a(n-1). edit.

Original entry on oeis.org

393, 528, 545, 660, 682, 727, 728, 751, 752, 802, 1206, 1279, 1280, 1288, 1321, 1322, 1986, 2323, 2448, 2471, 2832, 2897, 2898, 2934, 3103, 3240, 3251, 3252, 3529, 3530, 3891, 5192, 5265, 5287, 5616, 5635, 5671, 5832, 5838, 5990, 6597, 7334, 7549, 7550
Offset: 1

Views

Author

R. K. Guy, Apr 14 2005

Keywords

Comments

In Math. Mag. 48 (1975) 301 one finds "C. W. Trigg, C. C. Oursler and R. Cormier and J. L. Selfridge have sent calculations on Problem 886 [Nov 1973] for which we had received only partial results [Jan 1975]. Cormier and Selfridge sent the following results: There appear to be five sequences beginning with integers less than 1000 which do not merge. These sequences were carried out to 10^8 or more." The five sequences are A003508, A105210-A105213.
This suggests that there may be infinitely many different (non-merging) sequences obtained by choosing different starting values.

Examples

			a(2)=528 because a(1)=393, the distinct prime factors of a(1) are 3 and 131; finally, 1 + 393 + 3 + 131 = 528.

Links

Crossrefs

Cf. A003508, A027748, A105211, A105212, A105213.

Programs

  • Haskell
    a105210 n = a105210_list !! (n-1)
a105210_list = 393 : map
      (\x -> x + 1 + sum (takeWhile (< x) $ a027748_row x)) a105210_list
-- Reinhard Zumkeller, Jan 15 2015
  • Maple
    with(numtheory): p:=proc(n) local nn,ct,s: if isprime(n)=true then s:=0 else nn:=convert(factorset(n),list): ct:=nops(nn): s:=sum(nn[j],j=1..ct):fi: end: a[1]:=393: for n from 2 to 50 do a[n]:=1+a[n-1]+p(a[n-1]) od:seq(a[n],n=1..50); # Emeric Deutsch, Apr 14 2005
  • Mathematica
    a[1] = 393; a[n_] := a[n] = a[n - 1] + 1 + Plus @@ Select[ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[a[n - 1]]], # < a[n - 1] &]; Table[a[n], {n, 44}] (* Robert G. Wilson v, Apr 14 2005 *)
a[1] = 412; a[n_] := a[n] = a[n - 1] + 1 + Plus @@ Select[ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[a[n - 1]]], # < a[n - 1] &]; Table[a[n], {n, 43}] (* Robert G. Wilson v, Apr 14 2005 *)
a[1] = 668; a[n_] := a[n] = a[n - 1] + 1 + Plus @@ Select[ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[a[n - 1]]], # < a[n - 1] &]; Table[a[n], {n, 40}] (* Robert G. Wilson v, Apr 14 2005 *)
a[1] = 932; a[n_] := a[n] = a[n - 1] + 1 + Plus @@ Select[ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[a[n - 1]]], # < a[n - 1] &]; Table[a[n], {n, 40}] (* Robert G. Wilson v, Apr 14 2005 *)
nxt[n_]:=n+1+Total[Select[FactorInteger[n][[All,1]],#Harvey P. Dale, Mar 02 2019 *)

Extensions

More terms from Robert G. Wilson v and Emeric Deutsch, Apr 14 2005

A105211 a(1) = 412; for n > 1, a(n) = a(n-1) + 1 + sum of distinct prime factors of a(n-1) that are < a(n-1).

Original entry on oeis.org

412, 518, 565, 684, 709, 710, 789, 1056, 1073, 1140, 1170, 1194, 1399, 1400, 1415, 1704, 1781, 1932, 1968, 2015, 2065, 2137, 2138, 3210, 3328, 3344, 3377, 3696, 3720, 3762, 3798, 4015, 4105, 4932, 5075, 5117, 5185, 5269, 5760, 5771, 6000, 6011, 6012
Offset: 1

Views

Author

R. K. Guy, Apr 14 2005

Keywords

Comments

In Math. Mag. 48 (1975) 301 one finds "C. W. Trigg, C. C. Oursler and R. Cormier and J. L. Selfridge have sent calculations on Problem 886 [Nov 1973] for which we had received only partial results [Jan 1975]. Cormier and Selfridge sent the following results: There appear to be five sequences beginning with integers less than 1000 which do not merge. These sequences were carried out to 10^8 or more." The five sequences are A003508, A105210-A105213.

Examples

			a(2)=518 because a(1)=412, the distinct prime factors of a(1) are 2 and 103; finally, 1 + 412 + 2 + 103 = 518.

Links

Crossrefs

Cf. A003508, A027748, A105210, A105212, A105213.

Programs

  • Haskell
    a105211 n = a105211_list !! (n-1)
a105211_list = 412 : map
      (\x -> x + 1 + sum (takeWhile (< x) $ a027748_row x)) a105211_list
-- Reinhard Zumkeller, Jan 15 2015
  • Maple
    with(numtheory): p:=proc(n) local nn,ct,s: if isprime(n)=true then s:=0 else nn:=convert(factorset(n),list): ct:=nops(nn): s:=sum(nn[j],j=1..ct):fi: end: a[1]:=412: for n from 2 to 50 do a[n]:=1+a[n-1]+p(a[n-1]) od:seq(a[n],n=1..50); # Emeric Deutsch, Apr 14 2005
  • Mathematica
    nxt[n_]:=n+1+Total[Select[Transpose[FactorInteger[n]][[1]],#Harvey P. Dale, Sep 13 2013 *)

Extensions

More terms from Robert G. Wilson v and Emeric Deutsch, Apr 14 2005

A105213 a(1) = 932; for n > 1, a(n) = a(n-1) + 1 + sum of distinct prime factors of a(n-1) that are < a(n-1).

Original entry on oeis.org

932, 1168, 1244, 1558, 1621, 1622, 2436, 2478, 2550, 2578, 3870, 3924, 4039, 4624, 4644, 4693, 4726, 4885, 5868, 6037, 6038, 9060, 9222, 9310, 9344, 9420, 9588, 9658, 10111, 10112, 10194, 11899, 12136, 12217, 12880, 12918, 15077, 15078, 15450
Offset: 1

Views

Author

R. K. Guy, Apr 14 2005

Keywords

Comments

In Math. Mag. 48 (1975) 301 one finds "C. W. Trigg, C. C. Oursler and R. Cormier and J. L. Selfridge have sent calculations on Problem 886 [Nov 1973] for which we had received only partial results [Jan 1975]. Cormier and Selfridge sent the following results: There appear to be five sequences beginning with integers less than 1000 which do not merge. These sequences were carried out to 10^8 or more." The five sequences are A003508, A105210-A105213.

Examples

			a(2)=1168 because a(1)=932, the distinct prime factors of a(1) are 2 and 233; finally, 1 + 932 + 2 + 233 = 1168.

Links

Crossrefs

Cf. A027748, A003508, A105210, A105211, A105212.

Programs

  • Haskell
    a105213 n = a105213_list !! (n-1)
a105213_list = 932 : map
      (\x -> x + 1 + sum (takeWhile (< x) $ a027748_row x)) a105213_list
-- Reinhard Zumkeller, Jan 15 2015
  • Maple
    with(numtheory): p:=proc(n) local nn,ct,s: if isprime(n)=true then s:=0 else nn:=convert(factorset(n),list): ct:=nops(nn): s:=sum(nn[j],j=1..ct):fi: end: a[1]:=932: for n from 2 to 46 do a[n]:=1+a[n-1]+p(a[n-1]) od:seq(a[n],n=1..46); # Emeric Deutsch, Apr 14 2005
  • Mathematica
    nx[n_]:=n+1+Total[Select[Transpose[FactorInteger[n]][[1]],#Harvey P. Dale, Jul 24 2011 *)

Extensions

More terms from Robert G. Wilson v and Emeric Deutsch, Apr 14 2005

A105233 Conjectured numbers n such that the trajectory of n as defined in A003508 is unique.

Original entry on oeis.org

1, 393, 412, 668, 932, 1096, 1425, 1676, 1706, 1959, 2258, 2476, 2590, 3819, 4162, 4359, 4363, 4569, 4707, 5314, 5462, 5503, 5547, 5949, 6002, 6110, 6207, 6393, 6429, 6484, 6500, 7226, 7706, 8151, 8654, 9566, 9586, 9759, 10085, 10141, 10455, 10774
Offset: 1

Views

Author

R. K. Guy and Robert G. Wilson v, Apr 14 2005

Keywords

Comments

The trajectory in A003508, etc., is defined as a(1)=n, for n>1, a(n) = a(n-1) + 1 + sum of distinct prime factors of a(n-1) that are < a(n-1).
If n is a term of this sequence then by definition all later terms in the trajectory of n are excluded.

Crossrefs

Cf. A003508, A105210, A105211, A105212 and A105213.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = a[n - 1] + 1 + Plus @@ Select[ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[ a[n - 1]]], # < a[n - 1] &]; t = Table[ a[n], {n, 1200}]; f[n_] := Module[{b, k = 1}, b[1] = n; b[m_] := b[m] = b[m - 1] + 1 + Plus @@ Select[ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[ b[m - 1]]], # < b[m - 1] &]; While[ Position[t, b[k]] == {} && k < 1000, k++ ]; t = Select[ Union[ Join[t, Table[ b[i], {i, 2, k}]]], # > n &]; If[k == 1000, -1, k - 1]]; lst = {1}; Do[ If[ f[n] == -1, AppendTo[lst, n]], {n, 12500}]; lst
Showing 1-5 of 5 results.

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