A003508 -id:A003508 - cp's OEIS frontend

A105593 Number of steps for n to join the main line of A003508.

0, 0, 0, 0, 2, 1, 0, 0, 3, 1, 0, 0, 2, 1, 1, 7, 1, 0, 6, 5, 7, 2, 1, 0, 8, 1, 8, 4, 1, 0, 7, 6, 5, 4, 5, 1, 4, 3, 5, 5, 0, 0, 12, 11, 4, 1, 5, 4, 11, 11, 1, 10, 4, 3, 0, 4, 10, 10, 3, 2, 10, 9, 2, 11, 2, 3, 10, 9, 9, 2, 1, 0, 2, 1, 2, 1, 9, 0, 10, 9, 2, 2, 2, 1, 1, 5, 7, 8, 10, 9, 23, 4, 7, 4, 7, 8, 0, 0, 1

Offset: 1

R. K. Guy and Robert G. Wilson v, Apr 13 2005

nonn

a(n)=0 if n is a member of A003508. First term whose value is in question is 393.

			a(5)=2 because 5=>6=>12 and 12 is the eighth term of A003508.

Cf. A003508.

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, 100}]; f[n_] := Module[{b, k = 1, t = Table[a[i], {i, 100}]}, 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++ ]; k - 1]; Table[ f[n], {n, 105}]

A105600 Assume the conjectured terms of A105594 are the correct beginnings of the trajectories described in A003508. a(n) is a record length of b(n) iterations to arrive at the collected trajectories. This sequence cites the a(n)'s.

Original entry on oeis.org

1, 5, 9, 16, 25, 43, 91, 105, 427, 463, 484, 4085, 4306, 4413, 5583, 6273, 10172, 18105, 24946, 31686, 31886

Offset: 0

R. K. Guy and Robert G. Wilson v, Apr 15 2005

nonn

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).

Cf. A105593, the b(n)'s are in A105600.

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, 1500}]; 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++ ]; If[ k == 1000, t = Select[ Union[ Join[t, Table[ b[i], {i, 2, k}]]], # > n &]; -1, k - 1]]; lst = {{1, 0}}; Do[d = f[n]; If[d > lst[[ -1, 2]], AppendTo[lst, {n, d}]], {n, 60000}]; Transpose[ lst][[1]]

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

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

nonn

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.

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

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

A105601 Assume the conjectured terms of A105594 are the correct beginnings of the trajectories described in A003508. a(n) is a record length of b(n) iterations to arrive at the collected trajectories. This sequence cites the b(n)'s.

Original entry on oeis.org

0, 2, 3, 7, 8, 12, 23, 40, 53, 54, 56, 72, 82, 113, 124, 129, 213, 214, 215, 216, 220

Offset: 0

R. K. Guy and Robert G. Wilson v, Apr 15 2005

nonn

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).

Cf. A105593, the a(n)'s are in A105600.

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, 1500}]; 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++ ]; If[ k == 1000, t = Select[ Union[ Join[t, Table[ b[i], {i, 2, k}]]], # > n &]; -1, k - 1]]; lst = {{1, 0}}; Do[d = f[n]; If[d > lst[[ -1, 2]], AppendTo[lst, {n, d}]], {n, 60000}]; Transpose[ lst][[2]]

A105221 a(n) is the sum of n's distinct prime factors below n.

Original entry on oeis.org

0, 0, 0, 2, 0, 5, 0, 2, 3, 7, 0, 5, 0, 9, 8, 2, 0, 5, 0, 7, 10, 13, 0, 5, 5, 15, 3, 9, 0, 10, 0, 2, 14, 19, 12, 5, 0, 21, 16, 7, 0, 12, 0, 13, 8, 25, 0, 5, 7, 7, 20, 15, 0, 5, 16, 9, 22, 31, 0, 10, 0, 33, 10, 2, 18, 16, 0, 19, 26, 14, 0, 5, 0, 39, 8, 21, 18, 18, 0, 7, 3, 43, 0, 12, 22, 45

Offset: 1

Alexandre Wajnberg, Apr 13 2005

			a(12)=5 because 12's distinct prime factors 2 and 3 sum to 5.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A003508, A008472, A010051.

a105221 n = a008472 n - n * fromIntegral (a010051 n)
-- Reinhard Zumkeller, Apr 05 2013

f:= n -> convert(numtheory:-factorset(n) minus {n}, `+`):
map(f, [$1..100]); # Robert Israel, Sep 18 2023

Table[Total@Select[Join@@Union@*Table@@@FactorInteger@k,#Giorgos Kalogeropoulos, Nov 21 2021 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, if (f[k,1]Michel Marcus, Nov 21 2021

from sympy import primefactors
def A105221(n): return sum(p for p in primefactors(n) if p < n) # Chai Wah Wu, Sep 18 2023

a(n) = A008472(n) - A010051(n) * n. - Reinhard Zumkeller, Apr 05 2013

G.f.: Sum_{k>=1} prime(k) * x^(2*prime(k)) / (1 - x^prime(k)). - Ilya Gutkovskiy, Apr 13 2021

Edited by Don Reble, Nov 17 2005

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

R. K. Guy, Apr 14 2005

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.

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

T. D. Noe, Table of n, a(n) for n = 1..2000
Doug Engel, Problem 886, Math. Mag., 48 (1975), 57-58.

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

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

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

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 *)

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

R. K. Guy, Apr 14 2005

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.

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

T. D. Noe, Table of n, a(n) for n = 1..2000
Doug Engel, Problem 886, Math. Mag., 48 (1975), 57-58.

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

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

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

nxt[n_]:=n+1+Total[Select[Transpose[FactorInteger[n]][[1]],#Harvey P. Dale, Sep 13 2013 *)

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

A105212 a(1) = 668; 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

668, 838, 1260, 1278, 1355, 1632, 1655, 1992, 2081, 2082, 2435, 2928, 2995, 3600, 3611, 3792, 3877, 3878, 4165, 4195, 5040, 5058, 5345, 6420, 6538, 7015, 7105, 7147, 8176, 8259, 11016, 11039, 11149, 11150, 11381, 12000, 12011, 12012, 12049, 12050

Offset: 1

R. K. Guy, Apr 14 2005

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.

			a(2)=838 because a(1)=668, the distinct prime factors of a(1) are 2 and 167; finally, 1 + 668 + 2 + 167 = 838.

T. D. Noe, Table of n, a(n) for n = 1..2000
Doug Engel, Problem 886, Math. Mag., 48 (1975), 57-58.

Cf. A003508, A027748, A105210, A105211, A105213.

a105212 n = a105212_list !! (n-1)
a105212_list = 668 : map
      (\x -> x + 1 + sum (takeWhile (< x) $ a027748_row x)) a105212_list
-- Reinhard Zumkeller, Jan 15 2015

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]:=668: 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

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

R. K. Guy, Apr 14 2005

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.

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

T. D. Noe, Table of n, a(n) for n = 1..2000
Doug Engel, Problem 886, Math. Mag., 48 (1975), 57-58.

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

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

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

nx[n_]:=n+1+Total[Select[Transpose[FactorInteger[n]][[1]],#Harvey P. Dale, Jul 24 2011 *)

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

cp's OEIS Frontend

A105593 Number of steps for n to join the main line of A003508.

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A105600 Assume the conjectured terms of A105594 are the correct beginnings of the trajectories described in A003508. a(n) is a record length of b(n) iterations to arrive at the collected trajectories. This sequence cites the a(n)'s.

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

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A105601 Assume the conjectured terms of A105594 are the correct beginnings of the trajectories described in A003508. a(n) is a record length of b(n) iterations to arrive at the collected trajectories. This sequence cites the b(n)'s.

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A105221 a(n) is the sum of n's distinct prime factors below n.

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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.

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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).

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A105212 a(1) = 668; for n > 1, a(n) = a(n-1) + 1 + sum of distinct prime factors of a(n-1) that are < a(n-1).

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