A124652 -id:A124652 - cp's OEIS frontend

A372111 Partial sums of A124652.

1, 3, 6, 10, 15, 24, 30, 38, 54, 66, 77, 84, 98, 126, 144, 168, 189, 216, 248, 279, 360, 370, 390, 403, 572, 594, 627, 646, 663, 702, 728, 777, 814, 858, 894, 942, 996, 1060, 1085, 1120, 1160, 1189, 1230, 1245, 1290, 1320, 1370, 1450, 1508, 1560, 1620, 1692, 1739

Offset: 1

Michael De Vlieger, Apr 25 2024

nonn

Analogous to A109735 with respect to A109890.

A007947(A124652(n+1)) | a(n) for n > 2.

			See A124652.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007947, A109890, A109735, A124652.

nn = 54; c[_] := False; a[1] = 1; a[2] = 2; s = u = 3; c[1] = c[2] = True;
f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]];
{1}~Join~Reap[Do[Sow[s]; k = u; While[Or[Mod[s, f[k]] != 0, c[k]], k++];
    Set[{a[n], c[k]}, {k, True}];
    s += k; If[k == u, While[c[u], u++]], {n, 3, nn}] ][[-1, 1]]

A372009 Indices k such that A124652(k) is prime.

Original entry on oeis.org

2, 3, 5, 11, 12, 20, 24, 28, 29, 33, 42, 43, 53, 58, 67, 78, 93, 98, 104, 105, 109, 112, 118, 125, 126, 137, 141, 145, 146, 162, 174, 182, 185, 187, 188, 195, 200, 223, 224, 231, 232, 239, 246, 249, 252, 255, 259, 264, 271, 275, 283, 286, 287, 296, 298, 300, 326

Offset: 1

Michael De Vlieger, Apr 29 2024

nonn

Analogous to A111238, a sequence which instead pertains to A109890.

			Let b(x) = A124652(x).
Table of first terms.
   n  a(n)  b(a(n))
  -----------------
   1    2      2
   2    3      3
   3    5      5
   4   11     11
   5   12      7
   6   20     31
   7   24     13
   8   28     19
   9   29     17
  10   33     37
  11   42     29
  12   43     41
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..2500
Michael De Vlieger, Log log scatterplot of A124652(n), n = 1..25000, showing primes in red.

Cf. A124652, A372028, A372284.

nn = 300; c[_] := False;
rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
f[x_] := Select[Range[x], Divisible[x, rad[#]] &];
Array[Set[{a[#], c[#]}, {#, True}] &, 2]; s = a[1] + a[2];
{2}~Join~Reap[Do[r = f[s]; k = SelectFirst[r, ! c[#] &];
    If[PrimeQ[k], Sow[i]]; c[k] = True;
    s += k, {i, 3, nn}] ][[-1, 1]]

Proper subset of A372028.

A372028 Numbers k such that A124652(k) divides A372111(k-1).

Original entry on oeis.org

3, 5, 7, 11, 12, 13, 15, 16, 17, 18, 20, 22, 24, 26, 27, 28, 29, 30, 31, 33, 40, 41, 42, 43, 44, 46, 49, 50, 51, 53, 55, 58, 59, 60, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 73, 78, 79, 80, 92, 93, 95, 98, 101, 102, 103, 104, 105, 107, 109, 111, 112, 115, 116, 117

Offset: 1

Michael De Vlieger, May 05 2024

nonn

Contains A372009(m), m > 1.

For k in this sequence, A124652(k) has the same relationship with A372111(k-1) as A109890(i) has with A109735(i-1) for i > 2.

			Let b(x) = A124652(x) and s(x) = A372111(x), where A372111 contains partial sums of A124652.
a(1) = 3 since b(3) = 3, a divisor of s(2) = 3.
a(2) = 5 since b(5) = 5, a divisor of s(4) = 10.
a(3) = 7 since b(7) = 6, a divisor of s(6) = 24, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of A124652(n), n = 1..10^5, showing A124652(a(n)) in green, but A124652(a(n)) that are prime in red.

Cf. A007947, A027750, A109735, A109890, A124652, A372009, A372111, A372399.

nn = 120; c[_] := False;
rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
f[x_] := Select[Range[x], Divisible[x, rad[#]] &];
Array[Set[{a[#], c[#]}, {#, True}] &, 2]; s = a[1] + a[2];
Reap[Do[r = f[s]; k = SelectFirst[r, ! c[#] &];
  If[Divisible[s, k], Sow[i]]; c[k] = True;
  s += k, {i, 3, nn}] ][[-1, 1]]

A124652(a(n)) is a number in row A372111(a(n)-1) of A027750.

A372284 Primes in the order in which they appear in A124652.

Original entry on oeis.org

2, 3, 5, 11, 7, 31, 13, 19, 17, 37, 29, 41, 47, 59, 23, 97, 53, 829, 71, 149, 101, 167, 79, 151, 43, 103, 157, 113, 139, 109, 137, 197, 353, 89, 277, 293, 269, 229, 73, 61, 571, 83, 191, 691, 251, 179, 127, 257, 193, 173, 239, 163, 331, 613, 617, 311, 67, 523

Offset: 1

Michael De Vlieger, Apr 29 2024

nonn

Analogous to A111239, a sequence which instead pertains to A109890.

			Let b(x) = A372009(x).
Table of first terms:
   n  b(n) a(n)
  -------------
   1    2    2
   2    3    3
   3    5    5
   4   11   11
   5   12    7
   6   20   31
   7   24   13
   8   28   19
   9   29   17
  10   33   37
  11   42   29
  12   43   41
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..2500

Cf. A124652, A372009.

nn = 300; c[_] := False;
rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
f[x_] := Select[Range[x], Divisible[x, rad[#]] &];
Array[Set[{a[#], c[#]}, {#, True}] &, 2]; s = a[1] + a[2];
{2}~Join~Reap[Do[r = f[s]; k = SelectFirst[r, ! c[#] &];
    If[PrimeQ[k], Sow[k]]; c[k] = True;
    s += k, {i, 3, nn}] ][[-1, 1]]

A372323 A124652(n) is the a(n)-th term in row A372111(n-1) of irregular triangle A162306.

Original entry on oeis.org

2, 4, 4, 4, 5, 7, 5, 8, 8, 2, 10, 8, 12, 11, 13, 6, 13, 6, 6, 9, 8, 11, 4, 8, 16, 5, 6, 7, 13, 12, 7, 10, 19, 15, 16, 17, 9, 6, 15, 10, 3, 11, 8, 18, 28, 14, 14, 10, 30, 28, 15, 4, 20, 33, 13, 12, 6, 22, 18, 21, 12, 11, 29, 12, 11, 8, 24, 18, 8, 14, 17, 32, 33

Offset: 3

Michael De Vlieger, May 05 2024

nonn

Let b(x) = A124652(x) and let s(x) = A372111(x), where A372111 contains partial sums of A124652.
Let r(x) = A010846(x), the number of m <= x such that rad(m) | x, where rad = A007947.
Let row k of A162306 contain { m : rad(m) | k, m <= k }. Thus r(k) is the length of row k of A162306.
Let T(k,j) represent the j-th term in row k of irregular triangle A162306.
a(n) = j is the position of b(n) in row s(n-1) of A162306.
b(n) = T(s(n-1), a(n)).
Analogous to A371910, which instead regards A109890 and A109735.

			Let b(x) = A124652(x) and let s(x) = A372111(x), where A372111 contains partial sums of A124652.
a(3) = 2 since b(3) = 3 is the 2nd term in row s(3) = 3 of A162306, {1, [3]}.
a(4) = 4 since b(4) = 4 is the 4th term in row s(4) = 6 of A162306, {1, 2, 3, [4], 6}.
a(5) = 4 since b(5) = 5 is T(s(n-1), 4) = T(10, 4), {1, 2, 4, [5], 8, 10}.
a(6) = 4 since b(6) = 9 is T(s(n-1), 4) = T(15, 4), {1, 3, 5, [9], 15}.
a(7) = 5 since b(7) = 6 is T(s(n-1), 5) = T(24, 5), {1, 2, 3, 4, [6], 8, 9, 12, 16, 18, 24}, etc.
Table relating this sequence to b = A124652, s = A372111, r = A372322, and A162306.
   n b(n) s(n-1) a(n) r(n) row s(n-1) of A162306
  ---------------------------------------------------------------------
   3    3    3    2    2   {1, [3]}
   4    4    6    4    5   {1, 2, 3, [4], 6}
   5    5   10    4    6   {1, 2, 4, [5], 8, 10}
   6    9   15    4    5   {1, 3, 5, [9], 15}
   7    6   24    5   11   {1, 2, 3, 4, [6], ..., 24}
   8    8   30    7   18   {1, 2, 3, 4, 5, 6, [8], ..., 30}
   9   16   38    5    8   {1, 2, 4, 8, [16], 19, 32, 38}
  10   12   54    8   16   {1, 2, 3, 4, 6, 8, 9, [12], ..., 54}
  11   11   66    8   22   {1, 2, 3, 4, 6, 8, 9, [11], ..., 66}
  12    7   77    2    5   {1, [7], 11, 49, 77}
  13   14   84   10   28   {1, 2, 3, 4, ..., 12, [14], ..., 84}
  14   28   98    8   13   {1, 2, 4, 7, ..., 16, [28], ..., 98}

Michael De Vlieger, Table of n, a(n) for n = 3..10000
Michael De Vlieger, Bar chart showing a(n)/A372322(n-1) for n = 3..1024. This chart illustrates the "depth" of A124652(n) among the terms of the A372111(n-1)-th row of A162306.

Cf. A007947, A010846, A124652, A162306, A371910, A372111, A372322.

nn = 75; c[_] := False;
rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
f[x_] := Select[Range[x], Divisible[x, rad[#]] &];
Array[Set[{a[#], c[#]}, {#, True}] &, 2]; s = a[1] + a[2];
Reap[Do[r = f[s]; k = SelectFirst[r, ! c[#] &];
  Sow[FirstPosition[r, k][[1]]]; c[k] = True;
  s += k, {i, 3, nn}] ][[-1, 1]]

A372399 Numbers k such that A124652(k) does not divide A372111(k-1).

Original entry on oeis.org

2, 4, 6, 8, 9, 10, 14, 19, 21, 23, 25, 32, 34, 35, 36, 37, 38, 39, 45, 47, 48, 52, 54, 56, 57, 61, 65, 74, 75, 76, 77, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 94, 96, 97, 99, 100, 106, 108, 110, 113, 114, 122, 123, 130, 136, 142, 153, 157, 158, 159, 170, 171

Offset: 1

Michael De Vlieger, May 05 2024

nonn

Let b(x) = A124652(x) and let s(x) = A372111(x), where A372111 contains partial sums of A124652.
For n > 2, 1 < gcd(b(a(n)), s(a(n)-1)) < b(a(n)).
For n > 2, both b(a(n)) and s(a(n)-1) are necessarily composite, since prime p either divides or is coprime to n. Furthermore, both b(a(n)) and s(a(n)-1) have at least 2 distinct prime factors.
Indices of records in A124652 except {1, 2, 3, 5} are in this sequence.

			a(1) = 2 since b(2) = 2 does not divide s(1) = 1.
a(2) = 4 since b(4) = 4 does not divide s(3) = 6.
a(3) = 6 since b(6) = 9 does not divide s(5) = 15.
a(4) = 8 since b(8) = 8 does not divide s(7) = 30.
a(5) = 9 since b(9) = 16 does not divide s(8) = 38, etc.
Table of b(k) and s(k-1), where k = a(n), n = 2..12. Asterisked k denote terms such that rad(b(k)) | rad(s(k-1)); k = 73 and k = 4316 are the only other known indices where the terms have this quality.
     k      b(k)                        s(k-1)
    ----------------------------------------------------------
     4      4 =  2^2                    6 =  2 * 3
     6      9 =  3^2                   15 =  3 * 5
     8      8 =  2^3                   30 =  2 * 3 * 5
     9     16 =  2^4                   38 =  2 * 19
    10*    12 =  2^2 * 3               54 =  2 * 3^3
    14*    28 =  2^2 * 7               98 =  2 * 7^2
    19     32 =  2^5                  216 =  2^3 * 3^3
    21     81 =  3^4                  279 =  3^2 * 31
    23     20 =  2^2 * 5              370 =  2 * 5 * 37
    25    169 = 13^2                  403 = 13 * 31
    32     49 =  7^2                  728 =  2^3 * 7 * 13
    ...
    73*   100 =  2^2 * 5^2           4800 =  2^6 * 3 * 5^2
    ...
  4316*  4720 =  2^4 * 5 * 59    30806850 =  2 * 3 * 5^2 * 59^3

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of A124652(n), n = 1..10^5, showing A124652(a(n)) in red.

Cf. A007947, A124652, A272618, A372111.

nn = 120; c[_] := False;
rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
f[x_] := Select[Range[x], Divisible[x, rad[#]] &];
Array[Set[{a[#], c[#]}, {#, True}] &, 2]; s = a[1] + a[2];
{2}~Join~Reap[Do[
  r = f[s]; k = SelectFirst[r, ! c[#] &];
  If[! Divisible[s, k], Sow[i]];
  c[k] = True;
  s += k, {i, 3, nn}] ][[-1, 1]]

A124652(a(n)) is a number in row A372111(a(n)-1) of A272618.

A372322 a(n) = A010846(A372111(n)).

Original entry on oeis.org

1, 2, 5, 6, 5, 11, 18, 8, 16, 22, 5, 28, 13, 33, 23, 38, 11, 26, 12, 9, 58, 28, 80, 5, 30, 55, 19, 27, 19, 56, 37, 21, 27, 87, 44, 44, 48, 38, 18, 58, 42, 5, 110, 26, 112, 140, 38, 45, 32, 144, 102, 59, 5, 139, 225, 39, 44, 22, 180, 86, 114, 34, 23, 133, 41, 115

Offset: 1

Michael De Vlieger, May 05 2024

nonn

Let r(x) = A010846(x), the number of m <= x such that rad(m) | x, where rad = A007947.
Let row k of A162306 contain { m : rad(m) | k, m <= k }. Thus r(k) is the length of row k of A162306.
a(n) is the length of row A372111(n) of A162306.
Analogous to A371909, which instead regards A109890 and A109735.

			Let s(x) = A372111(x) and let r(x) = A010846(x).
a(1) = 1 since r(s(1)) = r(1) = 1.
a(2) = 2 since r(s(2)) = r(3) = 2. For prime p, r(p) = card({1, p}) = 2.
a(3) = 5 since r(s(3)) = r(6) = 5. r(6) = card({1, 2, 3, 4, 6}) = 5.
a(4) = 6 since r(s(4)) = r(10) = 6. r(10) = card({1, 2, 4, 5, 8, 10}) = 6.
a(5) = 5 since r(s(5)) = r(15) = 5. r(15) = card({1, 3, 5, 9, 15}) = 5.
a(6) = 11 since r(s(6)) = r(24) = 11. r(24) = card({1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24}) = 11, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007947, A010846, A124652, A162306, A371909, A372111, A372323.

nn = 68; c[_] := False;
rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
f[x_] := Select[Range[x], Divisible[x, rad[#]] &];
Array[Set[{a[#], c[#]}, {#, True}] &, 2]; s = a[1] + a[2];
{1}~Join~Reap[Do[r = f[s]; k = SelectFirst[r, ! c[#] &];
  Sow[Length[r]]; c[k] = True;
  s += k, {i, 3, nn}] ][[-1, 1]]

cp's OEIS Frontend

A372111 Partial sums of A124652.

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A372009 Indices k such that A124652(k) is prime.

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A372028 Numbers k such that A124652(k) divides A372111(k-1).

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A372284 Primes in the order in which they appear in A124652.

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A372323 A124652(n) is the a(n)-th term in row A372111(n-1) of irregular triangle A162306.

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A372399 Numbers k such that A124652(k) does not divide A372111(k-1).

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A372322 a(n) = A010846(A372111(n)).

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