A096003 -id:A096003 - cp's OEIS frontend

A097824 Gaps associated with the arithmetic progressions of semiprimes in A096003.

12, 12, 4, 52, 52, 72, 96, 198, 198, 114, 594, 48, 354, 1860, 3942, 2574, 2574, 2574, 20910, 20910, 9600, 9600, 152250, 152250, 152250, 152250, 1887270, 4667040, 4094790

Offset: 4

Hugo Pfoertner, Aug 27 2004

The terms a(1),a(2) and a(3) are omitted to avoid the ambiguity caused by the two progressions of length 3 ending at A096003(3)=14: a(3)=5 for (4,9,14) or a(3)=4 for (6,10,14).

			a(6)=4 because the 6 semiprimes in the progression ending at A096003(6)=221 are separated by an increment of 4: 201=3*67, 205=5*41, 209=11*19, 213=3*71, 217=7*31, 221=13*17.

Cf. A096003, A093364 gaps in arithmetic progressions of primes, A082919 clusters of 8 consecutive semiprimes.

a(26)-a(30) from Hugo Pfoertner, Sep 07 2004

a(31)-a(32) from Donovan Johnson, Jun 03 2012

A005115 Let i, i+d, i+2d, ..., i+(n-1)d be an n-term arithmetic progression of primes; choose the one which minimizes the last term; then a(n) = last term i+(n-1)d.

Original entry on oeis.org

2, 3, 7, 23, 29, 157, 907, 1669, 1879, 2089, 249037, 262897, 725663, 36850999, 173471351, 198793279, 4827507229, 17010526363, 83547839407, 572945039351, 6269243827111

Offset: 1

N. J. A. Sloane

In other words, smallest prime which is at the end of an arithmetic progression of n primes.
For the corresponding values of the first term and the common difference, see A113827 and A093364. For the actual arithmetic progressions, see A133277.
One may also minimize the common difference: this leads to A033189, A033188 and A113872.
One may also specify that the first term is the n-th prime and then minimize the common difference (or, equally, the last term): this leads to A088430 and A113834.
One may also ask for n consecutive primes in arithmetic progression: this gives A006560.

			n, AP, last term
1 2 2
2 2+j 3
3 3+2j 7
4 5+6j 23
5 5+6j 29
6 7+30j 157
7 7+150j 907
8 199+210j 1669
9 199+210j 1879
10 199+210j 2089
11 110437+13860j 249037
12 110437+13860j 262897
..........................
a(11)=249037 since 110437,124297,...,235177,249037 is an arithmetic progression of 11 primes ending with 249037 and it is the least number with this property.

R. K. Guy, Unsolved Problems in Number Theory, A5.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jens Kruse Andersen, Primes in Arithmetic Progression Records [May have candidates for later terms in this sequence.]
H. Dubner and H. Nelson, Seven consecutive primes in arithmetic progression, Math. Comp., 66 (1997) 1743-1749. MR 98a:11122.
Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Mathematics 167 (2008), pp. 481-547. arXiv:math/0404188 [math.NT], 2004-2007.
Ben Green and Terence Tao, A bound for progressions of length k in the primes
Andrew Granville, Prime number patterns, American Mathematical Monthly 115 (2008), pp. 279-296.
A. Moran, P. Pritchard and A. Thyssen, Twenty-two primes in arithmetic progression, Math. Comp.64 (1995), no.211, 1337-1339.
Index entries for sequences related to primes in arithmetic progressions

For the associated gaps, see A093364. For the initial terms, see A113827. For the arithmetic progressions, see A133277.

Cf. A006560, A096003, A113830-A113834, A088430.

(* This program will generate the 4 to 12 terms to use a[n_] to generate term 13 or higher, it will have a prolonged run time. *) a[n_] := Module[{i, p, found, j, df, k}, i = 1; While[i++; p = Prime[i]; found = 0; j = 0; While[j++; df = 6*j; (p > ((n - 1)*df)) && (found == 0), found = 1; Do[If[! PrimeQ[p - k*df], found = 0], {k, 1, n - 1}]]; found == 0]; p]; Table[a[i], {i, 4, 12}]

Green & Tao prove that this sequence is infinite, and further a(n) < 2^2^2^2^2^2^2^2^O(n). Granville conjectures that a(n) <= n! + 1 for n >= 3 and give a heuristic suggesting a(n) is around (exp(1-gamma) n/2)^(n/2). - Charles R Greathouse IV, Feb 26 2013

a(11)-a(13) from Michael Somos, Mar 14 2004
a(14) and corrected version of a(7) from Hugo Pfoertner, Apr 27 2004
a(15)-a(17) from Don Reble, Apr 27 2004
a(18)-a(21) from Granville's paper, Jan 26 2006
Entry revised by N. J. A. Sloane, Jan 26 2006, Oct 17 2007

A226833 Triangle whose n-th row has the smallest n semiprimes in an arithmetic progression.

Original entry on oeis.org

4, 4, 6, 6, 10, 14, 10, 22, 34, 46, 10, 22, 34, 46, 58, 201, 205, 209, 213, 217, 221, 133, 185, 237, 289, 341, 393, 445, 133, 185, 237, 289, 341, 393, 445, 497, 635, 707, 779, 851, 923, 995, 1067, 1139, 1211, 697, 793, 889, 985, 1081, 1177, 1273, 1369, 1465, 1561

Offset: 1

T. D. Noe, Jun 28 2013

The largest term of each row is as small as possible. Although Nowicki reports on the 28th row of this triangle, those terms are too large. Sequence A096003 reports the largest terms.

			Triangle:
4,
4,   6,
6,   10,  14,
10,  22,  34,  46,
10,  22,  34,  46,  58,
201, 205, 209, 213, 217,  221,
133, 185, 237, 289, 341,  393,  445,
133, 185, 237, 289, 341,  393,  445,  497,
635, 707, 779, 851, 923,  995,  1067, 1139, 1211,
697, 793, 889, 985, 1081, 1177, 1273, 1369, 1465, 1561

T. D. Noe, Rows n = 1..32 of triangle, flattened
Andrzej Nowicki, Second numbers in arithmetic progressions, arxiv 1306.6424

Cf. A226834 (first term), A096003 (last term), A097824 (gaps).

SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])]; p2 = Select[Range[2000], SemiPrimeQ]; nn = Length[p2]; t = {}; n = 0; last = 1; While[n++; found = False; last = n; While[k = last - 1; While[d = p2[[last]] - p2[[k]]; nums = Table[p2[[last]] - i*d, {i, 0, n - 1}]; int = Intersection[nums, Take[p2, last]]; nums[[-1]] > 0 && Length[int] < n, k--]; nums[[-1]] <= 0 && last < nn, last++]; If[last < nn, AppendTo[t, Reverse[nums]]]; last < nn]; t

A373887 a(n) is the length of the longest arithmetic progression of semiprimes ending in the n-th semiprime.

Original entry on oeis.org

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

Offset: 1

Robert Israel, Aug 10 2024

nonn

a(n) is the greatest k such that there exists d > 0 such that A001358(n) - j*d is in A001358 for j = 0 .. k-1.
The first appearance of m in this sequence is at n where A001358(n) = A096003(m).
Conjectures: a(n) >= 3 for n >= 16.
Limit_{n -> oo} a(n) = oo.
If A001358(n) is divisible by A000040(m), then a(n) >= A373888(m). In particular, the conjectures above are implied by the corresponding conjectures for A373888. - Robert Israel, Aug 19 2024

			a(5) = 3 because the 5th semiprime is A001358(5) = 14 and there is an arithmetic progression of 3 semiprimes ending in 14, namely 4, 9, 14, and no such arithmetic progression of 4 semiprimes.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001358, A096583, A373888.

S:= select(t -> numtheory:-bigomega(t)=2, [$1..10^5]):
f:= proc(n) local s,i,m,d,j;
  m:= 1;
  s:= S[n];
  for i from n-1 to 1 by -1 do
    d:= s - S[i];
    if s - m*d < 4 then return m fi;
    for j from 2 while ListTools:-BinarySearch(S,s-j*d) <> 0 do od;
    m:= max(m, j);
  od;
m;
end proc:
map(f, [$1..100]);

A090548 Least number that ends an arithmetic progression of n numbers with the same number of divisors.

Original entry on oeis.org

1, 3, 7, 23, 29, 157, 215, 1139, 1211, 2089, 5161, 5293, 6347, 10717, 14233, 28213, 31451, 72965, 119029, 121603, 124177, 611261, 632171, 2003171, 2012771

Offset: 1

Amarnath Murthy, Dec 09 2003

nonn

In the following triangle the n-th row contains the least set of n numbers in arithmetic progression with the same number of divisors. By "least" we mean that the largest term is minimized. Sequence contains the leading diagonal. In other words, largest of n numbers in arithmetic progression with the same tau function, or 0 if no such number exists.

			Triangle (A113470) begins:
1
2 3
3 5 7
5 11 17 23
5 11 17 23 29 ...

Leading diagonal of A113470.

Cf. A087309, A096003, A113470, A087308, A090547, A090549.

a(n)=A090547(n)+(n-1)*A090549(n). - R. J. Mathar, Apr 28 2007

Corrected and extended by R. J. Mathar, Apr 28 2007
More terms from David Wasserman, Jan 08 2006, May 11 2007
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 11 2007

A376109 a(n) is the length of the longest arithmetic progression ending at n consisting of numbers with the same number of prime factors as n, counted with multiplicity.

Original entry on oeis.org

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

Offset: 1

Robert Israel, Sep 10 2024

nonn

a(n) is the greatest k such that there exists d >= 1 with A001222(n-i*d) = A001222(n) for 0 <= i < k.
If m divides n, then a(n) >= a(m).
a(n) = 1 if and only if n is a power of 2.

			a(7) = 3 because 7 is prime and there is an arithmetic progression of 3 primes, namely 3, 5, 7, ending with 7 but no such arithmetic progression of 4 primes.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001222, A005115, A096003, A373887, A373888.

M:= Array(1..10):
for n from 2 to 100 do
  v:= numtheory:-bigomega(n);
  if M[v] = 0 then M[v]:= n else M[v]:= M[v],n fi;
od:
for i from 1 to 10 do M[i]:= [M[i]] od:
f:= proc(s) local n,i,m,d,v,j;
   m:= 1;
   v:= numtheory:-bigomega(s);
   member(s,M[v],n);
   for i from n-1 to 1 by -1 do
     d:= s - M[v][i];
     if s - m*d < M[v][1] then return m fi;
     for j from 2 while ListTools:-BinarySearch(M[v],s-j*d) <> 0 do od:
     m:= max(m,j);
   od;
  m;
end proc:
f(1):= 1:
map(f, [$1..100]);

A226834 Smallest semiprime (A001358) which is at the beginning of an arithmetic progression of n semiprimes whose largest term is as small as possible.

Original entry on oeis.org

4, 4, 6, 10, 10, 201, 133, 133, 635, 697, 2215, 2215, 4979, 2995, 13561, 22903, 1691, 5951, 72697, 72697, 72697, 172151, 172151, 1782371, 1782371, 3660743, 3660743, 3660743, 3660743, 13298267, 2235913, 41963249

Offset: 1

T. D. Noe, Jun 28 2013

Smallest number in row A226833(n).

Cf. A096003 (largest semiprime in row), A097824 (gaps).

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A097824 Gaps associated with the arithmetic progressions of semiprimes in A096003.

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A005115 Let i, i+d, i+2d, ..., i+(n-1)d be an n-term arithmetic progression of primes; choose the one which minimizes the last term; then a(n) = last term i+(n-1)d.

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A226833 Triangle whose n-th row has the smallest n semiprimes in an arithmetic progression.

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A373887 a(n) is the length of the longest arithmetic progression of semiprimes ending in the n-th semiprime.

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A090548 Least number that ends an arithmetic progression of n numbers with the same number of divisors.

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A376109 a(n) is the length of the longest arithmetic progression ending at n consisting of numbers with the same number of prime factors as n, counted with multiplicity.

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Maple

A226834 Smallest semiprime (A001358) which is at the beginning of an arithmetic progression of n semiprimes whose largest term is as small as possible.

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