A019565 -id:A019565 - cp's OEIS frontend

A353954 a(0) = 1; a(n) = A019565(A109812(n)).

1, 2, 3, 5, 6, 7, 10, 21, 11, 15, 14, 33, 35, 22, 105, 13, 30, 77, 26, 55, 42, 65, 66, 91, 110, 39, 70, 143, 210, 17, 165, 182, 51, 154, 195, 34, 231, 130, 119, 330, 221, 462, 85, 78, 385, 102, 455, 187, 390, 1309, 19, 770, 663, 38, 1155, 442, 57, 910, 561, 95, 273

Offset: 0

Michael De Vlieger, May 12 2022

nonn

Interpretation of A109812(n) written in binary instead as if written in "multiplicity notation", that is, as if we write 1 if divisible by prime(k+1), otherwise 0 in the k-th place. Example, decimal 12 is written in binary as 1100 = 2^2 + 2^3, and take exponents 2 and 3 and instead construe them as prime(2+1) * prime(3+1) = 5*7 = 35.

Permutation of squarefree numbers A005117.

			Table showing n, A109812(n), and b(n), the binary expansion of A109812(n) writing "." for zeros for clarity. a(n) interprets 1's in the k-th place of b(n) as prime(k+1) and thereafter takes the product. We find a(n) = A005117(j). Note that A109812(0) is not defined.
   n A109812(n) b(n)  a(n)   j
  ----------------------------
   0     -        .     1    1
   1     1        1     2    2
   2     2       1.     3    3
   3     4      1..     5    4
   4     3       11     6    5
   5     8     1...     7    6
   6     5      1.1    10    7
   7    10     1.1.    21   14
   8    16    1....    11    8
   9     6      11.    15   11
  10     9     1..1    14   10
  11    18    1..1.    33   21
  12    12     11..    35   23
  13    17    1...1    22   15
  14    14     111.   105   65
  15    32   1.....    13    9
  16     7      111    30   19
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Michael De Vlieger, Annotated log log scatterplot of a(n), n = 1..2^14, with records in red and local minima in blue, highlighting primes in green and fixed points in gold.

Cf. A005117, A019565, A109812.

Clear[c, a]; nn = 60; c[_] = 0; a[0] = c[1] = j = 1; a[1] = u = 2; Do[k = u; While[Nand[c[k] == 0, BitAnd[j, k] == 0], k++]; If[k == u, While[c[u] > 0, u++]]; j = k; Set[{a[i], c[k]}, {Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, Reverse@ IntegerDigits[k, 2]], i}], {i, 2, nn}]; Array[a, nn + 1, 0]

a(0) = 1; a(n) = Product p_k where A109812(n) = Sum 2^(k-1) for n > 0.

A353955 a(n) = A019565(A353709(n)).

Original entry on oeis.org

1, 2, 3, 5, 7, 6, 11, 35, 13, 22, 15, 91, 17, 10, 21, 143, 34, 105, 19, 26, 33, 85, 14, 39, 55, 119, 78, 95, 77, 51, 65, 154, 57, 221, 70, 209, 663, 23, 110, 273, 323, 46, 165, 1547, 38, 69, 385, 442, 437, 231, 130, 391, 133, 30, 187, 247, 42, 935, 299, 114, 595

Offset: 0

Michael De Vlieger, May 12 2022

nonn

Interpretation of A353709(n) written in binary instead as if written in "multiplicity notation", that is, as if we write 1 if divisible by prime(k+1), otherwise 0 in the k-th place. Example, decimal 12 is written in binary as 1100 = 2^2 + 2^3, and take exponents 2 and 3 and instead construe them as prime(2+1) * prime(3+1) = 5*7 = 35.

If A353709 is a permutation of nonnegative numbers, then this sequence is a permutation of squarefree numbers A005117.

			Table showing n, A353709(n), and b(n), the binary expansion of A353709(n) writing "." for zeros for clarity. a(n) interprets 1's in the k-th place of b(n) as prime(k+1) and thereafter takes the product. We find a(n) = A005117(j).
   n A353709(n) b(n)  a(n)   j
  ----------------------------
   1    0         .     1    1
   2    1         1     2    2
   3    2        1.     3    3
   4    4       1..     5    4
   5    8      1...     7    6
   6    3        11     6    5
   7   16     1....    11    8
   8   12      11..    35   23
   9   32    1.....    13    9
  10   17     1...1    22   15
  11    6       11.    15   11
  12   40    1.1...    91   57
  13   64   1......    17   12
  14    5       1.1    10    7
  15   10      1.1.    21   14
  16   48    11....   143   89
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Michael De Vlieger, Annotated log log scatterplot of a(n), n = 1..2^14, with records in red and local minima in blue, highlighting primes in green and fixed points in gold.

Cf. A005117, A019565, A353709.

nn = 2^7; c[_] = -1; c[0] = i = 0; a[0] = c[1] = j = 1; a[1] = u = 2; Do[k = u; While[Nand[c[k] == -1, BitAnd[i, k] == 0, BitAnd[j, k] == 0], k++]; If[k == u, While[c[u] > -1, u++]]; i = j; j = k; Set[{a[n], c[k]}, {Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, Reverse@ IntegerDigits[k, 2]], n}], {n, 2, nn}]; Array[a, nn + 1, 0]

a(n) = Product p_k where A353709(n) = Sum 2^(k-1).

A354825 Dirichlet inverse of A293442, where A293442 is multiplicative with a(p^e) = A019565(e).

Original entry on oeis.org

1, -2, -2, 1, -2, 4, -2, -2, 1, 4, -2, -2, -2, 4, 4, 8, -2, -2, -2, -2, 4, 4, -2, 4, 1, 4, -2, -2, -2, -8, -2, -16, 4, 4, 4, 1, -2, 4, 4, 4, -2, -8, -2, -2, -2, 4, -2, -16, 1, -2, 4, -2, -2, 4, 4, 4, 4, 4, -2, 4, -2, 4, -2, 20, 4, -8, -2, -2, 4, -8, -2, -2, -2, 4, -2, -2, 4, -8, -2, -16, 8, 4, -2, 4, 4, 4, 4, 4

Offset: 1

Antti Karttunen, Jun 09 2022

Multiplicative because A293442 is.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A019565, A293442.

A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A293442(n) = factorback(apply(e -> A019565(e),factor(n)[,2]));
memoA354825 = Map();
A354825(n) = if(1==n,1,my(v); if(mapisdefined(memoA354825,n,&v), v, v = -sumdiv(n,d,if(dA293442(n/d)*A354825(d),0)); mapput(memoA354825,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA293442(n/d) * a(d).

A364919 a(0) = 1; a(n) is the smallest number m not already in the sequence such that rad(m) divides A019565(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 6, 7, 14, 21, 12, 25, 10, 15, 16, 11, 22, 27, 18, 55, 20, 33, 24, 49, 28, 63, 32, 35, 40, 45, 30, 13, 26, 39, 36, 65, 50, 75, 48, 91, 52, 81, 42, 125, 56, 105, 54, 121, 44, 99, 64, 143, 80, 117, 60, 77, 88, 147, 66, 169, 70, 135, 72, 17, 34

Offset: 0

Michael De Vlieger, Aug 30 2023

nonn

Let k be a squarefree number and define R_k to be the set of numbers m such that rad(m) | k.
For n > 0, a(n) is the smallest m in R_k such that a(j) != m, j < n.
Conjecture: permutation of natural numbers.

			Let b(n) = A019565(n).
a(1) = 2 since b(1) = 2. Since 2 is prime, we find the first number in the prime power range of 2 that is not in the sequence and that is 2.
a(3) = 4 since b(3) = 6, and the smallest number m such that rad(m) | 6 that has not already appeared is 4.
a(5) = 8 since b(5) = 10. R_10 begins {1, 2, 4, 5, 8, 10, 16, ...} and the smallest number m in that list that is not already in the sequence is 8.
a(6) = 9 since b(6) = 15. R_15 begins {1, 3, 5, 9, 15, 25, ...} and the smallest m in that list not already in the sequence is 9, etc.

Michael De Vlieger, Table of n, a(n) for n = 0..16384
Michael De Vlieger, Log log scatterplot of a(n), n = 0..2^14, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue. We accentuate numbers in A001694 that are not prime powers with large light blue points.
Michael De Vlieger, Plot p(k)^e(k) at (x,y) = (n,k), n = 0..2^11, with a color function representing e(k) = 1 in black, e(k) = 2 in red, e(k) = 3 in orange, etc., and the highest e(k) in magenta. The bar at bottom indicates a(n) in a color code similar to the scatterplot above.
Michael De Vlieger, Fan style binary tree showing a(n), n = 0..2^12-1, with a color code similar to the scatterplot above.

Cf. A005117, A007947, A019565, A289280.

nn = 120; rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
f[x_] := Times @@ Prime@ Position[Reverse@ IntegerDigits[x, 2], 1][[All, 1]];
c[] := False; c[1] = True; q[] := 1; a[0] = 1; r[_] := 1;
Do[If[PrimeQ[#],
  While[c[Set[k, #^q[#]]], q[#]++],
  While[Or[c[r[#]], ! Divisible[#, rad[r[#]]]], r[#]++]; k = r[#] ] &[f[i]]; Set[{a[i], c[k]}, {k, True}], {i, nn}];
Array[a, nn + 1, 0]

a(2^k) = prime(k+1).

A381500 a(n) = A019565(A187769(n)).

Original entry on oeis.org

1, 2, 3, 6, 5, 10, 15, 30, 7, 14, 21, 35, 42, 70, 105, 210, 11, 22, 33, 55, 77, 66, 110, 165, 154, 231, 385, 330, 462, 770, 1155, 2310, 13, 26, 39, 65, 91, 143, 78, 130, 195, 182, 273, 455, 286, 429, 715, 1001, 390, 546, 910, 1365, 858, 1430, 2145, 2002, 3003

Offset: 0

Michael De Vlieger, Peter Munn, and Antti Karttunen, Feb 27 2025

The squarefree numbers, ordered first by largest prime factor (dividing the sequence into rows), then by number of prime factors, then lexicographically by their prime factors (written in descending order).

We index (a(n)) from offset 0, matching the choice for A019565 and similar sequences.

			Table begins:
  Row 0:  1;
  Row 1:  2;
  Row 2:  3,  6;
  Row 3:  5, 10, 15, 30;
  Row 4:  7, 14, 21, 35, 42, 70, 105, 210;
  Row 5: 11, 22, 33, 55, 77, 66, 110, 165, 154, 231, 385, 330, 462, 770, 1155, 2310;
  ...
Table of a(n) for n = 0..31, demonstrating relationship of this sequence with s = A187769:
          <-factors                    <-factors
   n  a(n)  2 3 5 7  s(n)  |   n   a(n)  2 3 5 7 11 s(n)
  -------------------------|----------------------------
   0    1   .          0   |  16    11   . . . . x   16
   1    2   x          1   |  17    22   x . . . x   17
   2    3   . x        2   |  18    33   . x . . x   18
   3    6   x x        3   |  19    55   . . x . x   20
   4    5   . . x      4   |  20    77   . . . x x   24
   5   10   x . x      5   |  21    66   x x . . x   19
   6   15   . x x      6   |  22   110   x . x . x   21
   7   30   x x x      7   |  23   165   . x x . x   22
   8    7   . . . x    8   |  24   154   x . . x x   25
   9   14   x . . x    9   |  25   231   . x . x x   26
  10   21   . x . x   10   |  26   385   . . x x x   28
  11   35   . . x x   12   |  27   330   x x x . x   23
  12   42   x x . x   11   |  28   462   x x . x x   27
  13   70   x . x x   13   |  29   770   x . x x x   29
  14  105   . x x x   14   |  30  1155   . x x x x   30
  15  210   x x x x   15   |  31  2310   x x x x x   31
  -------------------------|----------------------------
            1 2 4 8  s(n)  |             1 2 4 8 16 s(n)
             bits->                         bits->

Michael De Vlieger, Table of n, a(n) for n = 0..16383
Michael De Vlieger, Log log scatterplot of a(n), n = 0..2^16-1.
Michael De Vlieger, Plot p | a(n) at (x,y) = (n, pi(p)), n = 0..2047, 12X vertical exaggeration.
Michael De Vlieger, Fan style binary tree showing a(n), n = 0..2048, with a color function showing smallest and greatest terms in each row in green and red, respectively.

Cf. A019565, A119416, A163866, A187769, A253550, A294648, A344085.

A002110, A098012 are subsequences.

a187769 = {{0}}~Join~Table[SortBy[Range[2^n, 2^(n + 1) - 1], DigitCount[#, 2, 1] &], {n, 0, 8}] // Flatten; a019565[x_] := Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[x, 2]; Map[a019565, a187769]

a(n) = A019565(A187769(n)).
As an irregular triangle T(n,k), where row 0 = {1}:
For n > 1, omega(T(n,1)) = 1, omega(T(n, 2^(n-1))) = n, thus row n is divided into n segments S such that with S, omega(T(n,k)) = m, where m = 1..n. (See A187769 for the lengths of segments associated with Pascal's triangle A007318.)
S(-1,-1) = (1).
For n >= 0:
S(n-1, n) = (); S(n, -1) = ();
for 0 <= m <= n, S(n,m) = ( A253550'(S(n-1, m)), A119416'(S(n-1, m-1)) ), where Axxx'((i_1, i_2, ..., i_j)) denotes Axxx(i_1), Axxx(i_2), ..., Axxx(i_j).
a(A163866(n)) = A098012(n).

A383180 Irregular table T(n,k) = A010846(A019565(2^n + k)).

Original entry on oeis.org

1, 2, 2, 5, 2, 6, 5, 18, 2, 6, 5, 19, 5, 20, 16, 68, 2, 7, 6, 22, 5, 21, 18, 77, 5, 22, 17, 79, 16, 74, 60, 283, 2, 7, 6, 23, 5, 23, 18, 80, 5, 22, 18, 82, 16, 78, 62, 295, 5, 24, 19, 87, 16, 82, 64, 315, 15, 80, 62, 316, 55, 290, 226, 1161

Offset: 0

Michael De Vlieger, May 09 2025

			Triangle begins:
  0: 1;
  1: 2;
  2: 2, 5;
  3: 2, 6, 5, 18;
  4: 2, 6, 5, 19, 5, 20, 16, 68;
  5: 2, 7, 6, 22, 5, 21, 18, 77, 5, 22, 17, 79, 16, 74, 60, 283;
   ...

Michael De Vlieger, Table of n, a(n) for n = 0..16384 (rows n = 0..14, flattened)
Michael De Vlieger, Log log scatterplot of a(n), n = 0..2^14.
Michael De Vlieger, Fan style binary tree showing a(n) for n = 0..14, with a color function showing smaller numbers in reds and oranges, larger numbers in blues and magentas.

Cf. A005117, A010846, A019565, A363061.

(* Load the "theta" program at the Mathematica link in A369609, then: *)
f[x_] := Times @@ Prime@ Position[Reverse@ IntegerDigits[x, 2], 1][[All, 1]]; Table[theta[f[2^n + k] ], {n, 0, 7}, {k, 0, 2^n - 1}]

T(0,0) = 1.
T(n,0) = 2.
T(n,2^(n-1)-1) = A363061(n).

A103788 a(n) = number of ks that make primorial P(n)/A019565(k)-A019565(k) prime.

Original entry on oeis.org

0, 1, 3, 6, 13, 28, 39, 78, 138, 207, 437, 865, 1423, 2750, 4904, 8861, 16201, 33346, 58534, 111878, 208914, 397522

Offset: 1

Lei Zhou, Feb 16 2005

			P(2)/A(0)-A(0)=6-1=5 is prime, so a(2)=1;
P(4)/A(k)-A(k): 210/2-2=103; 210/3-3=67; 210/6-6=29; 210/5-5=37; 210/10-10=11; 210/7-7=23; so a(4)=6;

Cf. A019565, A002110, A103785, A103786, A103787.

npd = 1; Do[npd = npd*Prime[n]; tn = 0; tt = 1; cp = npd/tt - tt; ct = 0; While[IntegerQ[cp], If[(cp > 0) && PrimeQ[cp], ct = ct + 1]; tn = tn + 1; tt = 1; k1 = tn; o = 1; While[k1 > 0, k2 = Mod[k1, 2]; If[k2 == 1, tt = tt*Prime[o]]; k1 = ( k1 - k2)/2; o = o + 1]; cp = npd/tt - tt]; Print[ct], {n, 1, 22}]

A103792 Index k of the first occurrence of A019565(2n-1) as the smallest term that makes prime(k)-A019565(2n-1) prime.

Original entry on oeis.org

3, 5, 13, 25, 67, 79, 140, 127, 345, 129, 222, 206, 479, 1008, 1577, 766, 2583, 869, 1406, 3427, 5367, 4215, 4141, 9716, 23067, 5030, 13586, 7502, 17340, 19211, 14991, 30961, 27008, 82915, 84387, 91387, 92294, 32886, 30890, 70886, 271430, 131908

Offset: 1

Lei Zhou, Feb 28 2005

			n=1: A019565(2n-1)=2; Prime(3)-2=3 is prime, so a(1)=3;
Prime(4)-A019565(1)=5 is prime, not counted;
n=2: A019565(2n-1)=6; Prime(5)-A019565(1)=9 is not prime; ... Prime(5)-6=5 is prime, so a(2)=5;
Prime(6)-A019565(1)=11 is prime, not counted;
...
Prime(12)-A019565(3)=31 is prime, not counted;
n=3; A019565(2n-1)=10; Prime(13)-2=39, Prime(13)-6=35; Prime(13)-10=31 is prime, so a(3)=13.

Cf. A103790, A103791.

A019565 = Function[tn, k1 = tn; o = 1; tt = 1; While[k1 > 0, k2 = Mod[k1, 2]; If[k2 == 1, tt = tt*Prime[o]]; k1 = (k1 - k2)/2; o = o + 1]; tt]; Array[fa, {1, 500}]; Do[fa[n] = 0, {n, 1, 500}]; n = 2; npd = Prime[n]; ct = 1; wt = 1; While[wt < 200, cr = (ct + 1)/2; If[fa[cr] == 0, fa[cr] = n; While[fa[wt] > 0, Print[fa[wt]]; wt = wt + 1]]; n = n + 1; npd = Prime[n]; ct = 1; tt = ct; cp = npd - A019565[tt]; While[ ! (PrimeQ[cp]), ct = ct + 1; tt = ct; cp = npd - A019565[tt]]]

A103801 Indices n such that A019565(n)+/-2 are both primes.

Original entry on oeis.org

4, 6, 10, 14, 22, 26, 34, 38, 62, 78, 106, 138, 142, 162, 190, 194, 230, 258, 274, 278, 302, 402, 430, 458, 494, 498, 534, 550, 618, 670, 674, 690, 694, 742, 770, 814, 822, 902, 934, 1014, 1030, 1038, 1062, 1090, 1102, 1106, 1142, 1146, 1182, 1222, 1234

Offset: 1

Lei Zhou, Feb 22 2005

			A019565(4)=5, 3 and 7 are both primes, so a(1)=4;
A019565(6)=15, 13 and 17 are both primes, so a(2)=6;

Cf. A019565, A103799, A103800.

A019565 = Function[tn, k1 = tn; o = 1; tt = 1; While[k1 > 0, k2 = Mod[k1, 2]; If[k2 == 1, tt = tt*Prime[o]]; k1 = (k1 - k2)/2; o = o + 1]; tt]; Do[cp1 = A019565[n] - 2; cp2 = cp1 + 4; If[PrimeQ[cp1] && PrimeQ[cp2], Print[n]], {n, 0, 20000}]

A109163 a(n) = A019565(n-th prime).

Original entry on oeis.org

3, 6, 10, 30, 42, 70, 22, 66, 330, 770, 2310, 130, 182, 546, 2730, 1430, 6006, 10010, 102, 510, 238, 3570, 1122, 2618, 442, 2210, 6630, 9282, 15470, 4862, 510510, 114, 266, 798, 2090, 6270, 14630, 1482, 7410, 17290, 16302, 27170, 570570, 646, 3230

Offset: 1

Leroy Quet, Aug 18 2005

nonn

			The 5th prime is 11 (decimal), which is 1011 in binary. So a(5) is the product of the primes corresponding to the 1's of 1011, 2*3*7 = 42.

Cf. A019565.

Table[m = 1; o = 1; k1 = Prime[k]; While[k1 > 0, k2 = Mod[k1, 2]; If[k2 == 1, m = m*Prime[o]]; k1 = (k1 - k2)/2; o = o + 1]; m, {k, 1, 55}] (* Stefan Steinerberger, Mar 19 2006 *)

More terms from Stefan Steinerberger, Mar 19 2006

cp's OEIS Frontend

A353954 a(0) = 1; a(n) = A019565(A109812(n)).

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A353955 a(n) = A019565(A353709(n)).

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A354825 Dirichlet inverse of A293442, where A293442 is multiplicative with a(p^e) = A019565(e).

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A364919 a(0) = 1; a(n) is the smallest number m not already in the sequence such that rad(m) divides A019565(n).

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A381500 a(n) = A019565(A187769(n)).

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A383180 Irregular table T(n,k) = A010846(A019565(2^n + k)).

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A103788 a(n) = number of ks that make primorial P(n)/A019565(k)-A019565(k) prime.

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A103792 Index k of the first occurrence of A019565(2n-1) as the smallest term that makes prime(k)-A019565(2n-1) prime.

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