A125624 -id:A125624 - cp's OEIS frontend

A328637 Change the array in A125624 by putting a 1 at the top of the first column and pushing that column down by one place; a(n) is the sum of the terms in the modified array down to and including n, in the column that contains n.

1, 3, 6, 4, 11, 10, 18, 8, 17, 20, 29, 12, 42, 34, 32, 28, 59, 18, 78, 48, 53, 56, 101, 24, 43, 82, 27, 76, 130, 54, 161, 75, 86, 116, 110, 36, 198, 154, 125, 67, 239, 96, 282, 120, 81, 200, 329, 48, 116, 98, 176, 172, 382, 54, 165, 137, 233, 258, 441

Offset: 1

J. Stauduhar, Oct 22 2019

			The modified version of the array A125624 is:
1,  4,  8, 16, 32, 64, 128, 256, 512, ...
2,  6,  9, 12, 18, 24,  27,  36,  48, ...
3, 10, 15, 20, 25, 30,  40,  45,  50, ...
5, 14, 21, 28, 35, 42,  49,  56,  63, ...
7, 22, 33, 44, 55, 66,  77,  88,  99, ...
11, 26, 39, 52, 65, 78,  91, 104, 117, ...
...
Then:
a(1) = 1,
a(2) = 1+2,
a(3) = 1+2+3,
a(4) = 4,
a(5) = 1+2+3+5,
a(6) = 4+6,
...

J. Stauduhar, Python program

Cf. A125624, A328638.

Edited by N. J. A. Sloane, Dec 05 2019

A083140 Sieve of Eratosthenes arranged as an array and read by antidiagonals in the up direction; n-th row has property that smallest prime factor is prime(n).

Original entry on oeis.org

2, 3, 4, 5, 9, 6, 7, 25, 15, 8, 11, 49, 35, 21, 10, 13, 121, 77, 55, 27, 12, 17, 169, 143, 91, 65, 33, 14, 19, 289, 221, 187, 119, 85, 39, 16, 23, 361, 323, 247, 209, 133, 95, 45, 18, 29, 529, 437, 391, 299, 253, 161, 115, 51, 20, 31, 841, 667, 551, 493, 377, 319, 203, 125, 57, 22

Offset: 2

Yasutoshi Kohmoto, Jun 05 2003

A permutation of natural numbers >= 2.
The proportion of integers in the n-th row of the array is given by A005867(n-1)/A002110(n) = A038110(n)/A038111(n). - Peter Kagey, Jun 03 2019, based on comments by Jamie Morken and discussion with Tom Hanlon.
The proportion of the integers after the n-th row of the array is given by A005867(n)/A002110(n). - Tom Hanlon, Jun 08 2019

			Array begins:
   2   4   6   8  10  12  14  16  18  20  22  24 .... (A005843 \ {0})
   3   9  15  21  27  33  39  45  51  57  63  69 .... (A016945)
   5  25  35  55  65  85  95 115 125 145 155 175 .... (A084967)
   7  49  77  91 119 133 161 203 217 259 287 301 .... (A084968)
  11 121 143 187 209 253 319 341 407 451 473 517 .... (A084969)
  13 169 221 247 299 377 403 481 533 559 611 689 .... (A084970)

Cf. A083141 (main diagonal), A083221 (transpose), A004280, A038179, A084967, A084968, A084969, A084970, A084971.
Arrays of integers grouped into rows by various criteria:
by greatest prime factor: A125624,
by lowest prime factor: this sequence (upward antidiagonals), A083221 (downward antidiagonals),
by number of distinct prime factors: A125666,
by number of prime factors counted with multiplicity: A078840,
by prime signature: A095904,
by ordered prime signature: A096153,
by number of divisors: A119586,
by number of 1's in binary expansion: A066884 (upward), A067576 (downward),
by distance to next prime: A192179.
Cf. A002110, A005867, A038110, A038111.

a = Join[ {Table[2n, {n, 1, 12}]}, Table[ Take[ Prime[n]*Select[ Range[100], GCD[ Prime[n] #, Product[ Prime[i], {i, 1, n - 1}]] == 1 &], 12], {n, 2, 12}]]; Flatten[ Table[ a[[i, n - i]], {n, 2, 12}, {i, n - 1, 1, -1}]]
(* second program: *)
rows = 12; Clear[T]; Do[For[m = p = Prime[n]; k = 1, k <= rows, m += p, If[ FactorInteger[m][[1, 1]] == p, T[n, k++] = m]], {n, rows}]; Table[T[n - k + 1, k], {n, rows}, {k, n}] // Flatten (* Jean-François Alcover, Mar 08 2016 *)

More terms from Hugo Pfoertner and Robert G. Wilson v, Jun 13 2003

A289662 a(n) is the smallest m such that p = n-th popular prime = A385503(n) is uniquely popular on the interval [2,m] or -1 if p is never uniquely popular.

Original entry on oeis.org

2, 12, 80, 196, 1638, 4864, 22425, 46500, 109779, 158625, 603564, 2552416, 2620142, 2627250, 41163747, 237398795, 1966466950, 13690729828, 64322158656, 79838739611, 220355987735, 232268774850, 618745972214, 1882062476406, 9607713772982, 19364051829855, 26393150937218, 37636607806688, 114514665167797, 173974642809066, 228013323978930, 259772858868378, 284600479573629, 644741545246282

Offset: 1

N. J. A. Sloane, Jul 25 2017

nonn

See A385503 for further links and information.

Nathan McNew, Popular values of the largest prime divisor function (corrected version), page 16, November 2015.

Cf. A006530, A385503, A125624, A289661-A289665.

Deleted "first" from definition. - N. J. A. Sloane, Oct 03 2019

Edited, and terms updated from revised version of McNew, by Peter Munn, Jul 04 2025

A298268 a(1) = 1, and for any n > 1, if n is the k-th number with greatest prime factor p, then a(n) is the k-th number with least prime factor p.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 7, 6, 15, 25, 11, 21, 13, 49, 35, 8, 17, 27, 19, 55, 77, 121, 23, 33, 65, 169, 39, 91, 29, 85, 31, 10, 143, 289, 119, 45, 37, 361, 221, 95, 41, 133, 43, 187, 115, 529, 47, 51, 161, 125, 323, 247, 53, 57, 209, 203, 437, 841, 59, 145, 61, 961

Offset: 1

Rémy Sigrist, Jan 27 2018

nonn

This sequence is a permutation of the natural numbers, with inverse A298882.
For any prime p and k > 0:
- if s_p(k) is the k-th p-smooth number and r_p(k) is the k-th p-rough number,
- then a(p * s_p(k)) = p * r_p(k),
- for example: a(11 * A051038(k)) = 11 * A008364(k).

			The first terms, alongside A006530(n), are:
  n     a(n)   gpf(n)
  --    ----   ------
   1      1      1
   2      2      2
   3      3      3
   4      4      2
   5      5      5
   6      9      3
   7      7      7
   8      6      2
   9     15      3
  10     25      5
  11     11     11
  12     21      3
  13     13     13
  14     49      7
  15     35      5
  16      8      2
  17     17     17
  18     27      3
  19     19     19
  20     55      5

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (with prime and semiprime values highlighted)
Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (where the color is function of A006530(n))
Rémy Sigrist, Colored logarithmic scatterplot of the first 100000 terms (where the color is function of A176506(k) when a(n) is the k-th semiprime)
Rémy Sigrist, PARI program for A298268
Index entries for sequences that are permutations of the natural numbers

Cf. A006530, A008364, A046022, A051038, A061395, A078899, A083140, A125624, A151800, A176506, A298268, A298882 (inverse).

PARI
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See Links section.
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a(1) = 1.
a(A125624(n, k)) = A083140(n, k) for any n > 0 and k > 0.
a(n) = A083140(A061395(n), A078899(n)) for any n > 1.
Empirically:
- a(n) = n iff n belongs to A046022,
- a(2^k) = 2 * k for any k > 0,
- a(2 * p) = p^2 for any prime p,
- a(3 * p) = p * A151800(p) for any odd prime p.

A207802 Permutation of positive numbers. See comments.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 6, 16, 7, 32, 9, 64, 10, 128, 12, 256, 11, 512, 18, 1024, 15, 2048, 24, 4096, 14, 8192, 27, 16384, 20, 32768, 36, 65536, 13, 131072, 48, 262144, 25, 524288, 54, 1048576, 21, 2097152, 72, 4194304, 30, 8388608, 81, 16777216, 22, 33554432, 96, 67108864, 40, 134217728, 108

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Feb 20 2012

nonn

a(1)=1; on places 2,4,6,8,... we put 2^m, m=1,2,3,..., i.e., numbers with the greatest prime divisor p_1=2; on places 3,7,11,15,... we put numbers with the greatest prime divisor p_2=3; on places 5,13,21,29,... we put numbers with the greatest prime divisor p_3=5; etc.

For general description of the order, see the comment in A207790.

Cf. A207790, A207800, A207801.

For n>1, a(n) = A125624(A209268(n-1)). - Ivan Neretin, Apr 30 2016

a(30) fixed by Ivan Neretin, Apr 30 2016

A298882 a(1) = 1, and for any n > 1, if n is the k-th number with least prime factor p, then a(n) is the k-th number with greatest prime factor p.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 16, 6, 32, 11, 64, 13, 128, 9, 256, 17, 512, 19, 1024, 12, 2048, 23, 4096, 10, 8192, 18, 16384, 29, 32768, 31, 65536, 24, 131072, 15, 262144, 37, 524288, 27, 1048576, 41, 2097152, 43, 4194304, 36, 8388608, 47, 16777216, 14, 33554432, 48

Offset: 1

Rémy Sigrist, Jan 28 2018

nonn

This sequence is a permutation of the natural numbers, with inverse A298268.
For any prime p and k > 0:
- if s_p(k) is the k-th p-smooth number and r_p(k) is the k-th p-rough number,
- then a(p * r_p(k)) = p * s_p(k),
- for example: a(11 * A008364(k)) = 11 * A051038(k).

			The first terms, alongside A020639(n), are:
  n     a(n)    lpf(n)
  --    ----    ------
   1       1      1
   2       2      2
   3       3      3
   4       4      2
   5       5      5
   6       8      2
   7       7      7
   8      16      2
   9       6      3
  10      32      2
  11      11     11
  12      64      2
  13      13     13
  14     128      2
  15       9      3
  16     256      2
  17      17     17
  18     512      2
  19      19     19
  20    1024      2

Index entries for sequences that are permutations of the natural numbers

Cf. A008364, A020639, A046022, A051038, A055396, A078898, A083140, A125624, A298268 (inverse).

a(1) = 1.
a(A083140(n, k)) = A125624(n, k) for any n > 0 and k > 0.
a(n) = A125624(A055396(n), A078898(n)) for any n > 1.
Empirically:
- a(n) = n iff n belongs to A046022,
- a(2 * k) = 2^k for any k > 0,
- a(p^2) = 2 * p for any prime p,
- a(p * q) = 3 * p for any pair of consecutive odd primes (p, q).

A328638 a(n) is the sum of m such that 2 <= m <= n and gpf(m) = gpf(n), where gpf(i) is the greatest prime factor of i (A006530), with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 6, 5, 9, 7, 14, 18, 15, 11, 30, 13, 21, 30, 30, 17, 48, 19, 50, 42, 33, 23, 72, 75, 39, 99, 70, 29, 105, 31, 62, 66, 51, 105, 135, 37, 57, 78, 145, 41, 147, 43, 110, 190, 69, 47, 183, 196, 240, 102, 130, 53, 237, 165, 252, 114, 87, 59, 300

Offset: 1

J. Stauduhar, Oct 22 2019

For n >= 2, a(n) is the sum of the terms that precede n on the row, of the A125624 array, that contains n.

			5, 10, 15 and 20 have same gpf as 20, so a(20) = 5 + 10 + 15 + 20 = 50.

Cf. A006530 (gpf), A125624, A078899, A328637.

gpf(n)={my(f=factor(n)[,1]); f[#f]}
a(n)={if(n<=1, n==1, my(t=gpf(n)); sum(i=2, n, if(gpf(i)==t, i, 0)))} \\ Andrew Howroyd, Oct 28 2019

cp's OEIS Frontend

A328637 Change the array in A125624 by putting a 1 at the top of the first column and pushing that column down by one place; a(n) is the sum of the terms in the modified array down to and including n, in the column that contains n.

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A083140 Sieve of Eratosthenes arranged as an array and read by antidiagonals in the up direction; n-th row has property that smallest prime factor is prime(n).

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Mathematica

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A289662 a(n) is the smallest m such that p = n-th popular prime = A385503(n) is uniquely popular on the interval [2,m] or -1 if p is never uniquely popular.

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Extensions

A298268 a(1) = 1, and for any n > 1, if n is the k-th number with greatest prime factor p, then a(n) is the k-th number with least prime factor p.

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Comments

Examples

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Crossrefs

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PARI

Formula

A207802 Permutation of positive numbers. See comments.

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Extensions

A298882 a(1) = 1, and for any n > 1, if n is the k-th number with least prime factor p, then a(n) is the k-th number with greatest prime factor p.

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Author

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Examples

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Formula

A328638 a(n) is the sum of m such that 2 <= m <= n and gpf(m) = gpf(n), where gpf(i) is the greatest prime factor of i (A006530), with a(1) = 1.

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Author

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PARI