cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 96 results. Next

A349995 Record gaps between odd squarefree semiprimes (A046388).

Original entry on oeis.org

6, 12, 16, 20, 22, 24, 26, 28, 32, 36, 38, 40, 44, 50, 52, 60, 64, 70, 74, 84, 90, 92, 100, 102, 116, 118, 120, 132, 136, 138, 140, 142, 146, 152, 154, 156, 164, 170, 184, 186, 210
Offset: 1

Views

Author

Hugo Pfoertner, Dec 25 2021

Keywords

Examples

			  n  A350098(n)  A350099(n)  a(n)
  1      15          21        6
  2      21          33       12
  3      95         111       16
  4     267         287       20
  5    2369        2391       22

Links

Crossrefs

Records in A341828.
Cf. A350098 lower ends of the record gaps, A350099 upper ends of the record gaps.
Cf. A000101, A002386, A005250, A085809, A114057.

Extensions

a(35)-a(41) from Lucas A. Brown, Feb 29 2024

A350098 a(n) is the lower end of a record gap A349995(n) between consecutive odd squarefree semiprimes (A046388).

Original entry on oeis.org

15, 21, 95, 267, 2369, 6559, 8817, 13705, 15261, 21583, 35981, 66921, 113009, 340891, 783757, 872219, 3058853, 3586843, 5835191, 12345473, 108994623, 248706917, 268749691, 679956119, 709239621, 3648864859, 3790337723, 4171420481, 33955869693, 34279038379, 34840796369
Offset: 1

Views

Author

Hugo Pfoertner, Dec 26 2021

Keywords

Examples

			See A349995.

Links

Crossrefs

Cf. A046388, A085809, A341828, A349995, A350099.
Starting at a(3)=95 the terms coincide with the known terms of A114057.

Formula

a(n) = A350099(n) - A349995(n).

A350099 a(n) is the upper end of a record gap A349995(n) between consecutive odd squarefree semiprimes (A046388).

Original entry on oeis.org

21, 33, 111, 287, 2391, 6583, 8843, 13733, 15293, 21619, 36019, 66961, 113053, 340941, 783809, 872279, 3058917, 3586913, 5835265, 12345557, 108994713, 248707009, 268749791, 679956221, 709239737, 3648864977, 3790337843, 4171420613, 33955869829, 34279038517, 34840796509
Offset: 1

Views

Author

Hugo Pfoertner, Dec 26 2021

Keywords

Examples

			See A349995.

Links

Crossrefs

Cf. A046388, A085809, A114057, A341828, A349995, A350098.

Formula

a(n) = A350098(n) + A349995(n).

A146168 Number of odd squarefree semiprimes (A046388) < 2^n.

Original entry on oeis.org

0, 0, 0, 1, 2, 8, 20, 46, 96, 197, 404, 798, 1599, 3134, 6169, 12093, 23640, 46199, 90180, 176198, 343927, 671783, 1312304, 2564485, 5012807, 9803883, 19181677, 37545265, 73524262, 144038812, 282313035, 553557959, 1085860455, 2130904274, 4183364732, 8215861037
Offset: 1

Views

Author

Washington Bomfim, Oct 27 2008

Keywords

Examples

			a(5) = 2. The odd squarefree semiprimes less than 2^5 are 15 and 21. The formula gives 10 - pi(5) - pi(2^4) + 1 = 2.

Links

Crossrefs

Cf. A046388, A001358 (semiprimes), A000720 (pi(n), the number of primes <= n), A007053 (number of primes <= 2^n), A060967, A125527 (number of semiprimes <= 2^n).

Programs

  • Mathematica
    Table[lim=2^n; Sum[PrimePi[lim/p]-PrimePi[p], {p, Prime[Range[2,PrimePi[Sqrt[lim]]]]}], {n,20}]

Formula

a(n) = A125527(n) - A060967(n) - A007053(n-1) + 1, for n > 1.

Extensions

a(34) onwards from Amiram Eldar, Sep 05 2024

A350095 a(n) is the smaller of 2 consecutive primes bounding an interval containing a record number A350097(n) of odd squarefree semiprimes (A046388).

Original entry on oeis.org

13, 31, 89, 199, 211, 887, 1129, 1327, 9973, 15683, 19609, 44293, 155921, 370261, 396733, 492113, 1357201, 1671781, 3826019, 17836409, 20831323, 47465267, 107534587, 122164747, 434865437, 436273009, 2300942549, 4302407359, 10726904659, 25056082087, 42652618343
Offset: 1

Views

Author

Hugo Pfoertner, Dec 25 2021

Keywords

Examples

			a(1) = 13: semiprime 15 < 17 = nextprime(a(1)) = A350096(1);
a(2) = 31: semiprimes 33, 35 < 37 = A350096(2);
a(6) = 887: semiprimes 889, 893, 895, 899, 901, 905 < 907 = A350096(6);
a(7) = 1129: semiprimes 1133, 1135, 1137, 1139, 1141, 1145, 1147, 1149 < 1151 = A350096(7);
a(8) = 1327: semiprimes 1329, 1333, 1337, 1339, 1343, 1345, 1347, 1349, 1351, 1355, 1357 < 1361 = A350096(8).

Links

Crossrefs

A350096 are the upper ends of the intervals, A350097 are the corresponding counts of odd squarefree semiprimes in the intervals.
Cf. A000101, A005250, A046388.

Formula

A350096(n) = nextprime(a(n)).

Extensions

a(29)-a(31) from Martin Ehrenstein, Dec 28 2021
a(32) from Lucas A. Brown, Mar 21 2024

A350096 a(n) is the larger of 2 consecutive primes bounding an interval containing a record number A350097(n) of odd squarefree semiprimes (A046388).

Original entry on oeis.org

17, 37, 97, 211, 223, 907, 1151, 1361, 10007, 15727, 19661, 44351, 156007, 370373, 396833, 492227, 1357333, 1671907, 3826157, 17836561, 20831533, 47465443, 107534789, 122164969, 434865671, 436273291, 2300942869, 4302407713, 10726905041, 25056082543, 42652618807
Offset: 1

Views

Author

Hugo Pfoertner, Dec 25 2021

Keywords

Examples

			See A350095.

Links

Crossrefs

A350097 gives the corresponding counts.
Cf. A000101, A002386, A005250, A046388, A349995, A350095.

Formula

a(n) = nextprime(A350095(n)).

Extensions

a(29)-a(31) from Martin Ehrenstein, Dec 28 2021
a(32) from Lucas A. Brown, Mar 21 2024

A350097 Record numbers of counts of odd squarefree semiprimes (A046388) in an interval between 2 consecutive primes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 11, 12, 13, 17, 18, 23, 24, 25, 27, 29, 30, 34, 36, 41, 42, 43, 45, 49, 54, 58, 59, 71, 74, 81, 99
Offset: 1

Views

Author

Hugo Pfoertner, Dec 25 2021

Keywords

Examples

			See A350095.

Links

Crossrefs

A350095 and A350096 are the primes delimiting the corresponding intervals.
Cf. A046388.

Extensions

a(29)-a(31) from Martin Ehrenstein, Dec 28 2021
a(32) from Lucas A. Brown, Mar 21 2024

A350101 Numbers k such that 2*k-1 and 2*k+1 are squarefree semiprimes (A046388).

Original entry on oeis.org

17, 28, 43, 46, 47, 71, 72, 80, 92, 93, 101, 102, 107, 108, 109, 110, 118, 124, 133, 150, 151, 152, 160, 161, 164, 170, 196, 197, 206, 207, 208, 223, 226, 235, 236, 258, 259, 267, 268, 272, 276, 290, 291, 295, 317, 334, 335, 340, 343, 344, 348, 349, 361, 377, 390
Offset: 1

Views

Author

Hugo Pfoertner, Dec 14 2021

Keywords

Examples

			a(1) = 17: 2*17 - 1 = 33 = 3*11 and 2*17 + 1 = 35 = 5*7 are both in A046388.

Links

Crossrefs

Cf. A046388.
Intersection of A234093 and A234096.

Programs

  • Maple
    N:= 1000: # for terms <= N
P:= select(isprime,[seq(i,i=3..2*N/3,2)]):
S:= NULL:
for i from 1 to nops(P) do
  for j from 1 to i-1 while P[i]*P[j] <= 2*N+1 do S:= S,P[i]*P[j] od
od:
S:= {S}:
T:= S intersect map(`-`,S,2):
sort(convert(map(t -> (t+1)/2, T),list)); # Robert Israel, Nov 11 2022
  • Mathematica
    semiQ[n_] := FactorInteger[n][[;; , 2]] == {1, 1}; Select[Range[400], AllTrue[2*# + {-1, 1}, semiQ] &] (* Amiram Eldar, Dec 14 2021 *)
  • PARI
    a350101(limit) = {my(sp(k)=omega(k)==2&&bigomega(k)==2);forstep(k=2,2*limit,2, if(sp(k-1)&&sp(k+1),print1(k/2,", ")))};
a350101(390)

A379915 a(n) is the deficiency of the odd squarefree semiprime A046388(n), divided by 2.

Original entry on oeis.org

3, 5, 9, 11, 11, 15, 19, 17, 23, 21, 29, 31, 27, 35, 29, 35, 35, 43, 47, 39, 41, 53, 45, 59, 55, 59, 51, 65, 57, 59, 71, 79, 65, 83, 79, 89, 69, 83, 89, 71, 95, 91, 77, 107, 81, 109, 107, 103, 87, 119, 95, 115, 131, 125, 99, 119, 101, 139, 105, 143, 107, 137, 131
Offset: 1

Views

Author

Hugo Pfoertner, Jan 06 2025

Keywords

Links

Crossrefs

Cf. A000203, A033879, A046388, A379916, A379917.

Programs

  • PARI
    \\ uses function a379915_17 from A379917
a379915_17(300,2)

Formula

a(n) = (2*A046388(n) - sigma(A046388(n)))/2, where sigma is A000203.
a(n) = A033879(A046388(n))/2.

A146167 Number of odd squarefree semiprimes (A046388) <= n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 11
Offset: 1

Views

Author

Washington Bomfim, Oct 27 2008

Keywords

Comments

A346622 is a different although very similar sequence. - N. J. A. Sloane, Aug 23 2021

Examples

			a(33)= 3. The semiprimes <=33 are 15, 21 and 33. Formula gives 11-pi(5)-pi(16)+1 = 3.

Crossrefs

Cf. A046388, A001358 (semiprimes), A072000 (Number of semiprimes <= n), A000720 (pi(n), the number of primes <= n).
Cf. also A346622.

Programs

  • Mathematica
    Accumulate[Table[If[OddQ[n]&&SquareFreeQ[n]&&PrimeOmega[n]==2,1,0],{n,0,100}]] (* Harvey P. Dale, Feb 08 2016 *)
  • Python
    from math import isqrt
from sympy import prime, primepi
def A146167(n): return int(sum(primepi(n//prime(k))-k+1 for k in range(2,primepi(isqrt(n))+1)))-primepi(isqrt(n))+1 if n>3 else 0 # Chai Wah Wu, Jul 23 2024

Formula

a(n) = A072000(n) - A000720(floor(sqrt(n))) - A000720(floor(n/2)) + 1.
Showing 1-10 of 96 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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