A083140 -id:A083140 - cp's OEIS frontend

A228098 Number of primes p > prime(n) and such that prime(n)*p < prime(n+1)^2.

1, 2, 1, 3, 1, 2, 1, 1, 2, 1, 3, 2, 1, 1, 2, 2, 1, 3, 2, 1, 2, 1, 1, 3, 2, 1, 2, 1, 1, 4, 1, 2, 1, 3, 1, 2, 2, 1, 2, 2, 1, 4, 1, 2, 1, 2, 4, 2, 1, 1, 2, 1, 2, 2, 2, 2, 1, 3, 2, 1, 1, 4, 2, 1, 1, 2, 1, 3, 1, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 1, 3

Offset: 1

Jean-Christophe Hervé, Oct 26 2013

For n > 1, a(n)+1 is the number of composite numbers < prime(n+1)^2 and removed at the n-th step of Eratosthenes's sieve. The exception for n=1 comes from prime(1)^3 = 2^3 = 8 < prime(2)^2 = 9. This does not occur any more because prime(n)^3 > prime(n+1)^2 for all n > 1.
a(n) is related to the distribution of primes around prime(n+1). High values correspond to a large gap before prime(n+1) followed by several small gaps after prime(n+1).
a(n) >= 1 for all n, because prime(n+1) always trivially satisfies the condition. The sequence tends to alternate high and low values, and takes its minimum value 1 about half the time.
a(n) is >= and almost always equal to a'(n), defined as the number of primes between prime(n+1) (inclusive) and prime(n+1) + gap(n) (inclusive), with gap(n) = prime(n+1) - prime(n) = A001223(n).
An exception is 7, for which a(7) = 3, while the following prime is 11, thus gap(7) = 4, and there are only two primes between 11 and 11 + 4 = 15. It is probably the only one, as it is easily seen that a(n) = a'(n) if gap(n) <= sqrt(2*prime(n)), which is a condition a little stronger than Andrica's Conjecture: gap(n) < 2*sqrt(prime(n))+1. 7 is probably a record for the ratio gap(n)/sqrt(prime(n)), and the only prime for which it is > sqrt(2) (see A079296 for an ordering of primes according to Andrica's conjecture).

			a(4)=3 because prime(4)=7, prime(5)=11, 11^2=121, and 7*11 < 7*13 < 7*17 < 121 < 7*19.

Jean-Christophe Hervé, Table of n, a(n) for n = 1..9999
C. K. Caldwell, Gaps between primes.
Eric W. Weisstein, Andrica's Conjecture
Wikipedia, Andrica's conjecture
Marek Wolf, A note on the Andrica conjecture, arXiv:1010.3945 [math.NT], 2010.

Cf. A000040, A001223, A083140, A079296.

Table[PrimePi[Prime[n + 1]^2/Prime[n]] - n, {n, 100}] (* T. D. Noe, Oct 29 2013 *)

P = Primes()
def a(n):
    p=P.unrank(n-1)
    p1=P.unrank(n)
    L=[q for q in [p+1..p1^2] if q in Primes() and p*qTom Edgar, Oct 29 2013

A257252 Transpose of square array A257251.

Original entry on oeis.org

2, 6, 2, 20, 6, 2, 42, 10, 6, 2, 110, 28, 20, 6, 2, 156, 22, 14, 10, 6, 2, 272, 52, 44, 28, 20, 6, 2, 342, 34, 26, 22, 14, 10, 6, 2, 506, 76, 68, 52, 44, 28, 20, 6, 2, 812, 138, 114, 102, 78, 66, 42, 10, 6, 2, 930, 58, 46, 38, 34, 26, 22, 14, 20, 6, 2, 1332, 186, 174, 138, 114, 102, 78, 66, 42, 10, 6, 2

Offset: 1

Antti Karttunen, Apr 29 2015

See A257251.

			The top left corner of the array:
  2, 6, 20, 42, 110, 156, 272, 342, 506, 812, 930, 1332
  2, 6, 10, 28,  22,  52,  34,  76, 138,  58, 186,  148
  2, 6, 20, 14,  44,  26,  68, 114,  46, 174, 124,   74
  2, 6, 10, 28,  22,  52, 102,  38, 138, 116,  62,  148
  2, 6, 20, 14,  44,  78,  34, 114,  92,  58, 124,  222
  2, 6, 10, 28,  66,  26, 102,  76,  46, 116, 186,  222
  2, 6, 20, 42,  22,  78,  68,  38,  92, 174, 186,   74
  2, 6, 10, 14,  66,  52,  34,  76, 138, 174,  62,  222
  2, 6, 20, 42,  44,  26,  68, 114, 138,  58, 186,  148
  2, 6, 10, 28,  22,  52, 102, 114,  46, 174, 124,   74
  2, 6, 20, 14,  44,  78, 102,  38, 138, 116,  62,  222
  2, 6, 10, 28,  66,  78,  34, 114,  92,  58, 186,  148
  2, 6, 20, 14,  66,  26, 102,  76,  46, 174, 124,  222
  2, 6, 10, 28,  22,  78,  68,  38, 138, 116, 186,  296
  2, 6, 20, 42,  66,  52,  34, 114,  92, 174, 248,  148
  2, 6, 10, 14,  44,  26, 102,  76, 138, 232, 124,   74
  ...

Antti Karttunen, Table of n, a(n) for n = 1..3321; the first 81 antidiagonals of the array

Transpose: A257251.
Row 1: A036689.
Cf. also A083140, A257254 (same array but with terms divided by 2).

(define (A257252 n) (A257251bi (A004736 n) (A002260 n))) ;; Other code as in A257251.

A251717 a(n) = smallest positive integer k such that A083221(k, n) has at most two prime factors (is a prime or semiprime).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 3, 2, 2, 1, 2, 3, 4, 2, 3, 1, 2, 1, 5, 3, 2, 3, 2, 1, 3, 6, 2, 1, 2, 1, 3, 2, 3, 1, 4, 2, 3, 2, 2, 1, 2, 2, 3, 2, 3, 1, 3, 1, 4, 4, 2, 3, 2, 1, 5, 2, 2, 1, 4, 1, 4, 2, 2, 3, 3, 1, 3, 3, 2, 1, 2, 4, 3, 2, 3, 1, 2, 2, 5, 3, 3, 3, 2, 1, 3, 2, 2, 1, 4, 1, 3, 3, 2, 1, 5, 1, 4, 3, 2, 1, 2, 2, 3, 2, 3, 4, 2

Offset: 1

Antti Karttunen, Dec 15 2014

nonn

Records occur at 1, 4, 8, 26, 32, 39, 238, 462, 1075, 1763, ... with record values 1, 2, 3, 4, 5, 6, 8, 9, 11, 13, ...

New distinct values occur at 1, 4, 8, 26, 32, 39, 238, 306, 462, 1075, 1106, 1763, ... with the values 1, 2, 3, 4, 5, 6, 8, 7, 9, 11, 10, 13, ...

Antti Karttunen, Table of n, a(n) for n = 1..2518

Variant: A251718.
The positions of ones: A008578.
Cf. A001222, A083221 (A083140), A251719.
a(n+1) differs from A057217(n-1) for the first time at n=19, where a(20) = 3, while A057217(18) = 4.

(define (A251717 n) (let loop ((i 1)) (if (<= (A001222 (A083221bi i n)) 2) i (loop (+ i 1))))) ;; Code for A083221bi given in A083221.

For all n, a(n) <= A251718(n) <= A251719(n).

A251718 a(n) = smallest positive integer k such that both A083221(k, n) and A083221(k+1, n) have at most two prime factors (are primes or semiprimes).

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 1, 3, 4, 2, 3, 4, 3, 5, 2, 2, 4, 3, 1, 3, 2, 4, 3, 2, 5, 4, 2, 3, 4, 2, 1, 5, 6, 2, 3, 2, 1, 3, 6, 4, 4, 4, 6, 3, 4, 5, 3, 6, 4, 6, 2, 4, 3, 4, 2, 5, 8, 5, 6, 3, 3, 4, 4, 2, 3, 2, 3, 7, 6, 5, 3, 4, 4, 6, 2, 2, 5, 7, 1, 5, 5, 4, 6, 2, 4, 6, 5, 5, 4, 2, 2, 5, 3, 3, 3, 4, 1, 3, 5, 7, 5, 4, 3, 3, 5, 2, 4, 5, 7, 4, 7, 4, 3, 7, 4, 3, 2, 3, 4, 2

Offset: 1

Antti Karttunen, Dec 15 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..509

Variant: A251717.
A005382 gives the positions of 1 after the initial a(1)=1.
Cf. A001222, A083221 (A083140), A251719.

(define (A251718 n) (let loop ((i 1)) (if (and (<= (A001222 (A083221bi i n)) 2) (<= (A001222 (A083221bi (+ i 1) n)) 2)) i (loop (+ i 1))))) ;; Code for A083221bi given in A083221.

For all n, A251717(n) <= a(n) <= A251719(n).

A257254 Transpose of square array A257253.

Original entry on oeis.org

1, 3, 1, 10, 3, 1, 21, 5, 3, 1, 55, 14, 10, 3, 1, 78, 11, 7, 5, 3, 1, 136, 26, 22, 14, 10, 3, 1, 171, 17, 13, 11, 7, 5, 3, 1, 253, 38, 34, 26, 22, 14, 10, 3, 1, 406, 69, 57, 51, 39, 33, 21, 5, 3, 1, 465, 29, 23, 19, 17, 13, 11, 7, 10, 3, 1, 666, 93, 87, 69, 57, 51, 39, 33, 21, 5, 3, 1

Offset: 1

Antti Karttunen, Apr 29 2015

See A257253.

			The top left corner of the array:
  1, 3, 10, 21, 55, 78, 136, 171, 253, 406, 465, 666, 820, 903
  1, 3,  5, 14, 11, 26,  17,  38,  69,  29,  93,  74,  41,  86
  1, 3, 10,  7, 22, 13,  34,  57,  23,  87,  62,  37,  82, 129
  1, 3,  5, 14, 11, 26,  51,  19,  69,  58,  31,  74, 123, 129
  1, 3, 10,  7, 22, 39,  17,  57,  46,  29,  62, 111, 123,  43
  1, 3,  5, 14, 33, 13,  51,  38,  23,  58,  93, 111,  41, 129
  1, 3, 10, 21, 11, 39,  34,  19,  46,  87,  93,  37, 123,  86
  1, 3,  5,  7, 33, 26,  17,  38,  69,  87,  31, 111,  82,  43
  1, 3, 10, 21, 22, 13,  34,  57,  69,  29,  93,  74,  41, 129
  1, 3,  5, 14, 11, 26,  51,  57,  23,  87,  62,  37, 123,  86
  1, 3, 10,  7, 22, 39,  51,  19,  69,  58,  31, 111,  82, 129
  1, 3,  5, 14, 33, 39,  17,  57,  46,  29,  93,  74, 123, 172
  1, 3, 10,  7, 33, 13,  51,  38,  23,  87,  62, 111, 164,  86
  1, 3,  5, 14, 11, 39,  34,  19,  69,  58,  93, 148,  82,  43
  1, 3, 10, 21, 33, 26,  17,  57,  46,  87, 124,  74,  41,  86
  1, 3,  5,  7, 22, 13,  51,  38,  69, 116,  62,  37,  82,  43
  ...

Antti Karttunen, Table of n, a(n) for n = 1..3321; the first 81 antidiagonals of the array

Transpose: A257253.
Row 1: A008837.
Cf. A083140, A083221, A257252 (same array but with terms multiplied by 2).

(define (A257254 n) (A257253bi (A004736 n) (A002260 n))) ;; Code for A257253bi given in A257253.

A279864 Irregular triangle read by rows: the n-th row corresponds to the natural numbers not exceeding A002110(n) and divisible by the n-th prime but not by a smaller prime.

Original entry on oeis.org

2, 3, 5, 25, 7, 49, 77, 91, 119, 133, 161, 203, 11, 121, 143, 187, 209, 253, 319, 341, 407, 451, 473, 517, 583, 649, 671, 737, 781, 803, 869, 913, 979, 1067, 1111, 1133, 1177, 1199, 1243, 1331, 1397, 1441, 1507, 1529, 1573, 1639, 1661, 1727, 1793, 1837, 1859

Offset: 1

Rémy Sigrist, Dec 21 2016

The n-th row has A005867(n-1) terms.
The n-th row starts with the n-th prime.
The terms of this sequence appear, in that order, while applying the sieve of Eratosthenes; the n-th rows matches the first A005867(n-1) terms of the n-th row of A083140.
Any number n>1 can be uniquely written as n = T(i,j)+k*A002110(i) (with k>=0); in that case:
- i = A055396(n),
- k = floor( (n-1)/A002110(A055396(n)) ).
This sequence corresponds to the numbers n>1 such that n <= A002110(A055396(n)).
Let S(i,j) = { T(i,j)+k*A002110(i) with k>=0 }, then:
- For any n>0, { S(n,j) } is a partition of the numbers divisible by the n-th prime but not by a smaller prime,
- For any n>0, { S(i,j) such that i<=n } is a partition of the numbers divisible by the n-th prime,
- { S(i,j) } is a partition of the numbers > 1.

			From _M. F. Hasler_, May 16 2017: (Start)
The triangle starts
2;
3;
5, 25;
7, 49, 77, 91, 119, 133, 161, 203;
11, 121, 143, 187, 209, 253, 319, 341, 407, 451, 473, 517, 583, 649, 671, 737, 781, 803, 869, 913, 979, 1067, 1111, 1133, 1177, 1199, 1243, 1331, 1397, 1441, 1507, 1529, 1573, 1639, 1661, 1727, 1793, 1837, 1859, 1903, 1969, 1991, 2057, 2101, 2123, 2167, 2189, 2299;
... (End)

Rémy Sigrist, Table of n, a(n) for n = 1..6300 [First 7 rows]

Cf. A002110, A005867, A083140, A055396, A000040.

Table[Function[p, Select[Range[Times @@ p], Function[k, And[Divisible[k, Last@ p], Total@ Boole@ Divisible[k, Most@ p] == 0]]]]@ Prime@ Range@ n, {n, 5}] // Flatten (* Michael De Vlieger, Dec 21 2016 *)
a[1] = {2}; a[2] = {3}; t[2] = {1, 5}; a[n_] := a[n] = Prime[n]*t[n - 1]; t[x_] := t[x] = Complement[Flatten[Table[k*Product[Prime[j], {j, x - 1}] + t[x - 1], {k, 0, Prime[x] - 1}]], a[x]]; Flatten[Table[a[n], {n, 6}]] (* L. Edson Jeffery, May 16 2017 *)

pp=1; for (r=1, 5, forstep(n=prime(r), pp*prime(r), prime(r), if (gcd(n,pp)==1, print1 (n ", "))); pp *= prime(r); print(""))

A279864_row(r,p=prime(r),P=prod(i=1,r-1,prime(i)))=select(n->gcd(n,P)==1,p*[1..P])  \\ M. F. Hasler, May 16 2017

T(n,1) = A000040(n) for any n>0.

T(n,k) = A083140(n,k) for any n>0 and k<=A005867(n-1).

A114883 Obtained from A114882 by deleting row 1 and dividing by 2.

Original entry on oeis.org

1, 2, 4, 3, 12, 7, 5, 24, 17, 10, 6, 60, 38, 27, 13, 8, 84, 71, 45, 32, 16, 9, 144, 110, 93, 59, 42, 19, 11, 180, 161, 123, 104, 66, 47, 22, 14, 264, 218, 195, 149, 126, 80, 57, 28

Offset: 1

Clark Kimberling, Jan 03 2006

A permutation of the natural numbers. Row 1 consists of those congruent to 1 mod 3. (One can then delete row 1 of A114883, leaving only numbers congruent to 0 or 2 mod 3 and then replace each by its rank, thus obtaining another permutation of the natural numbers.)

Also numbers from the array A083140, without the first row, minus 1 divided by 2.

			Start with the northwest corner of A114881:
  1  3  5  7   9
  2  8 14 20  26
  4 24 34 54  64
  6 48 76 90 118
Delete row 1, leaving all the even positive integers; divide them by 2:
  1  4  7 10 13
  2 12 17 27 32
  3 24 38 45 59

Cf. A083140, A114881.

A207801 Permutation of positive numbers. See comments.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 15, 12, 25, 14, 21, 16, 11, 18, 27, 20, 35, 22, 33, 24, 49, 26, 39, 28, 55, 30, 45, 32, 13, 34, 51, 36, 65, 38, 57, 40, 77, 42, 63, 44, 85, 46, 69, 48, 121, 50, 75, 52, 95, 54, 81, 56, 91, 58, 87, 60, 115, 62, 93, 64, 17, 66, 99, 68, 125, 70, 105, 72, 119, 74

Offset: 1

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

nonn

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

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

Ivan Neretin, Table of n, a(n) for n = 1..8192

Cf. A207790, A207800.

a = Array[1 &, mx = 74]; cnt = mx - 1; offs = Table[2^(i - 1) + 1, {i, 1, mx}]; n = 1; While[cnt > 0, n++; idx = PrimePi[FactorInteger[n][[1, 1]]]; pos = offs[[idx]]; If[pos > mx, Continue[]]; offs[[idx]] += 2^idx; a[[pos]] = n; cnt--]; a (* Ivan Neretin, May 06 2015 *)

For n>1, a(n) = A083140(1+A209268(n-1)). Equivalently, a(n) = A083221(1+A249725(n-1)). - Ivan Neretin, Apr 30 2016

A249730 Odd nonprime terms in A250251.

Original entry on oeis.org

1, 9, 15, 25, 35, 49, 77, 121, 143, 147, 153, 169, 221, 289, 323, 361, 437, 529, 667, 841, 899, 961, 1147, 1369, 1517, 1681, 1763, 1849, 2021, 2209, 2491, 2809, 3127, 3481, 3599

Offset: 1

Antti Karttunen, Nov 18 2014

nonn

This sequence lists the numbers which together with all their multiples with the powers of 2 give the starting values for the prime recurrences whose union is A250251, the fixed points of permutations A250249 and A250250.

After 1, contains the terms from columns 2 and 3 of the Sieve of Eratosthenes: A083140 (A083221), but only from those rows r for which A055396(r) (the index of smallest dividing prime r) is fixed by A250249 and A250250, i.e., is in A250251. The first r for which this is not the case is 73, which is in A249729 instead. However, because there are infinitely many primes in A250251, and especially because 2 and 3 are among them, this sequence is infinite.

Subsequence of A250251.

Cf. A083140, A083221, A249821, A250249, A250250.

A257514 Transpose of square array A257513.

Original entry on oeis.org

1, 2, 5, 2, 16, 9, 4, 24, 20, 13, 2, 72, 42, 34, 17, 4, 48, 66, 36, 38, 21, 2, 120, 78, 96, 54, 52, 25, 4, 72, 102, 60, 90, 48, 56, 29, 6, 168, 114, 144, 90, 120, 66, 70, 33, 2, 312, 230, 160, 194, 124, 158, 88, 74, 37, 6, 120, 232, 162, 96, 150, 84, 138, 92, 88, 41, 4, 408, 248, 360, 262, 176, 226, 140, 190, 114, 92, 45

Offset: 1

Antti Karttunen, May 01 2015

The array is read by downwards antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

See A257513.

			The top left corner of the array:
   1,   2,   2,   4,   2,   4,   2,   4,   6,   2,   6,   4,   2,   4,   6
   5,  16,  24,  72,  48, 120,  72, 168, 312, 120, 408, 312, 168, 360, 600
   9,  20,  42,  66,  78, 102, 114, 230, 232, 248, 370, 246, 258, 470, 636
  13,  34,  36,  96,  60, 144, 160, 162, 360, 198, 320, 336, 352, 494, 460
  17,  38,  54,  90,  90, 194,  96, 262, 338, 144, 406, 434, 364, 330, 684
  21,  52,  48, 120, 124, 150, 176, 240, 304, 210, 504, 458, 204, 526, 614
  25,  56,  66, 158,  84, 226, 150, 210, 374, 280, 540, 318, 380, 456, 532
  29,  70,  88, 138, 140, 216, 120, 264, 456, 292, 428, 490, 306, 378, 756
  33,  74,  92, 190, 126, 198, 162, 326, 492, 180, 588, 432, 228, 574, 686
  37,  88, 114, 192, 108, 240, 208, 350, 412, 308, 550, 366, 404, 504, 816
  41,  92, 132, 186, 138, 290, 220, 282, 540, 258, 500, 538, 330, 614, 958
  45, 106, 126, 216, 172, 314, 156, 382, 518, 204, 660, 480, 424, 732, 794
  49, 110, 144, 254, 184, 270, 236, 360, 484, 332, 622, 578, 522, 576, 712
  53, 124, 138, 306, 144, 346, 210, 330, 612, 282, 720, 684, 366, 498, 830
  57, 128, 156, 300, 200, 336, 180, 430, 590, 352, 830, 552, 288, 600, 748
  ...

Transpose of A257513.
Column 1: A016813.
Row 1: A001223, Row 2: A069482, Row 3: A109805, Row 4: A226502 (apart from the first term).
Cf. also A083140, A083221, A257252.

(define (A257514 n) (A257513bi (A004736 n) (A002260 n))) ;; A257513bi given in A257513.

cp's OEIS Frontend

A228098 Number of primes p > prime(n) and such that prime(n)*p < prime(n+1)^2.

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Sage

A257252 Transpose of square array A257251.

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A251717 a(n) = smallest positive integer k such that A083221(k, n) has at most two prime factors (is a prime or semiprime).

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A251718 a(n) = smallest positive integer k such that both A083221(k, n) and A083221(k+1, n) have at most two prime factors (are primes or semiprimes).

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A257254 Transpose of square array A257253.

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A279864 Irregular triangle read by rows: the n-th row corresponds to the natural numbers not exceeding A002110(n) and divisible by the n-th prime but not by a smaller prime.

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PARI

PARI

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A114883 Obtained from A114882 by deleting row 1 and dividing by 2.

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A207801 Permutation of positive numbers. See comments.

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