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.

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A186945 The smallest integer x > 0 such that the number of terms of A050376 in (x/2,x] equals n.

Original entry on oeis.org

2, 3, 5, 13, 17, 25, 31, 49, 61, 71, 73, 81, 103, 109, 113, 131, 139, 157, 173, 181, 191, 193, 199, 239, 241, 257, 269, 271, 283, 289, 293, 313, 353, 361, 373, 379, 409, 419, 421, 433, 439, 443, 463, 491, 499, 509, 571, 577, 599, 601, 607, 613, 619, 643, 647
Offset: 1

Views

Author

Vladimir Shevelev, Aug 30 2013

Keywords

Comments

The sequence is an analog of Labos primes (A080359) in Fermi-Dirac arithmetic, since in this arithmetic terms of A050376 play role of primes (see comments in A050376).

Links

Crossrefs

Cf. A080359, A228520, A228592.

Formula

a(n) <= A228520(n).

Extensions

More terms from Peter J. C. Moses

A164958 Primes p with the property that if p/3 is in the interval (p_m, p_(m+1)), where p_m>=3 and p_k is the k-th prime, then the interval (3p_m, p) contains a prime.

Original entry on oeis.org

2, 3, 5, 13, 19, 29, 31, 43, 47, 61, 67, 73, 79, 83, 101, 103, 107, 109, 137, 139, 151, 157, 167, 173, 181, 193, 197, 199, 229, 233, 241, 257, 263, 271, 277, 281, 283, 313, 317, 349, 353, 359, 367, 373, 379, 389, 401, 409, 431, 433, 439, 443, 461, 463, 467, 487, 499
Offset: 1

Views

Author

Vladimir Shevelev, Sep 02 2009

Keywords

Comments

For k>1 (not necessarily integer), we call a Labos k-prime L_n^(k) the prime a_k(n) which is the smallest number such that pi(a_k(n)) - pi(a_k(n)/k)= n. Note that, the sequence of all primes corresponds to the case of "k=oo". Let p be a k-Labos prime, such that p/k is in the interval (p_m, p_(m+1)), where p_m>=3 and p_n is the n-th prime. Then the interval (k*p_(m), p) contains a prime. Conjecture. For every k>1 there exist non-k-Labos primes, which possess the latter property. For example, for k=2, the smallest such prime is 131. Problem. For every k>1 to estimate the smallest non-k-Labos prime, which possess the latter property. [From Vladimir Shevelev, Sep 02 2009]
All 3-Labos primes are in this sequence.

Examples

			If p=61, the p/3 is in the interval (19, 23); we see that the interval (57, 61) contains a prime (59). Thus 61 is in the sequence.

Crossrefs

Cf. A104272, A080359, A164952, A164368, A164288, A164294

Programs

  • Mathematica
    nn=1000; t=Table[0, {nn+1}]; s=0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/3], s--]; If[s<=nn && t[[s+1]]==0, t[[s+1]]=k], {k, Prime[3*nn]}]; Rest[t]

Extensions

Extended by T. D. Noe, Nov 23 2010

A166574 If p, q are successive primes, and there is a number k with p < k <= q such that r = p+k is a prime, then r is in the sequence.

Original entry on oeis.org

5, 7, 11, 17, 23, 29, 41, 47, 59, 67, 83, 89, 97, 107, 109, 127, 137, 149, 151, 167, 179, 181, 197, 227, 229, 233, 239, 257, 263, 281, 283, 307, 317, 337, 347, 349, 359, 367, 383, 389, 401, 409, 431, 433, 449, 461, 467, 479, 487, 491
Offset: 1

Views

Author

Vladimir Shevelev, Oct 17 2009

Keywords

Comments

The old definition was: Primes p>=5 with the property: if Prime(k)

If A(x) is the counting function of a(n) not exceeding x, then, in view of the symmetry, it is natural to conjecture that A(x)~pi(x)/2.

Examples

			Taking p=2, q=3, k=3 we get r=2+3=5, the first term.
Taking p=3, q=5, k=4 we get r=3+4=7, the second term.
From p=89, q=97 we can take both k=90 and k=92, getting the terms 89+90=179 and 89+92=181. - _Art Baker_, Mar 16 2019

Links

Crossrefs

Cf. A182365, A166307, A166252, A166251, A164368, A104272, A080359, A164333, A164288, A164294, A164554

Programs

  • Mathematica
    Reap[Do[p=Prime[n]; k=PrimePi[p/2]; If[p<=Prime[k]+Prime[k+1], Sow[p]], {n,3,PrimePi[1000]}]][[2,1]]
Select[#[[1]]+Range[#[[1]]+1,#[[2]]],PrimeQ]&/@Partition[Prime[Range[60]],2,1]//Flatten (* Harvey P. Dale, Jul 02 2024 *)

Extensions

Extended by T. D. Noe, Dec 01 2010
Edited with simpler definition based on a suggestion from Art Baker. -N. J. A. Sloane, Mar 16 2019

A080361 a(n) is the difference between the largest and the smallest positive integers x such that the number of unitary-prime-divisors of x! equals n. Same as the difference between the largest and the smallest positive integers x such that the number of primes in (x/2,x] equals n.

Original entry on oeis.org

8, 13, 15, 21, 15, 15, 13, 9, 25, 27, 5, 23, 39, 37, 27, 21, 7, 45, 37, 39, 39, 1, 21, 17, 11, 35, 27, 53, 35, 17, 19, 27, 29, 9, 11, 11, 5, 21, 43, 27, 11, 69, 61, 63, 15, 5, 1, 5, 33, 29, 27, 5, 5, 17, 57, 43, 47, 17, 25, 51, 47, 11, 3, 25, 27, 23, 77, 57, 35, 19, 29, 37, 27, 23, 9
Offset: 1

Views

Author

Labos Elemer, Feb 21 2003

Keywords

Links

Crossrefs

Cf. A056171, A056169, A000720, A000142, A080359, A080360.
Cf. A104272 Ramanujan primes. [From Jonathan Sondow, Aug 10 2008]

Formula

a(n)=Max{x; Pi[x]-Pi[x/2]=n}-Min{x; Pi[x]-Pi[x/2]=n}=A080360[n]-A080359[n].

Extensions

Definition corrected by Jonathan Sondow, Aug 10 2008
Typo in formula corrected by Daniel Forgues, Aug 06 2009

A080362 a(n) is the number of positive integers x such that the number of unitary-prime-divisors of x! equals n. Same as the number of positive integers x such that the number of primes in (x/2,x] equals n.

Original entry on oeis.org

4, 10, 7, 14, 7, 10, 12, 5, 14, 16, 3, 10, 18, 16, 15, 11, 7, 16, 19, 14, 9, 2, 14, 14, 8, 11, 18, 19, 24, 10, 14, 16, 20, 10, 11, 3, 6, 13, 18, 21, 9, 31, 37, 10, 15, 6, 2, 6, 21, 12, 7, 6, 6, 16, 15, 34, 14, 10, 15, 29, 22, 9, 4, 14, 16, 17, 25, 36, 12, 15, 13, 19, 19, 8, 10, 5, 12
Offset: 1

Views

Author

Labos Elemer, Feb 21 2003

Keywords

Examples

			n=5,a(5)=7 because in 7 factorials 5 primes arise with exponent 1: in factorials of 31,32,33,37,41,46; e.g. in 37! these are {19,23,29,31,37}, or 10 numbers x, exist such ones that number of unitary prime divisors of x! equals 2, namely in factorials of {3,5,7,8,9,11,12,13,15,16}.

Links

Crossrefs

Cf. A056171, A056169, A000720, A000142, A080359, A080360, A080361.
Cf. A104272 Ramanujan primes. [From Jonathan Sondow, Aug 10 2008]

Formula

a(n)=Card{x; Pi[x]-Pi[x/2]=n}, where Pi()=A000720().

Extensions

Definition corrected by Jonathan Sondow, Aug 10 2008

A164962 a(n) is the least prime from the union {2,3} and A164333, beginning with which the n-th prime p_n is obtained by some number of iterations of the S operator g(see A164960).

Original entry on oeis.org

2, 3, 2, 3, 2, 13, 3, 19, 2, 13, 31, 3, 19, 43, 2, 53, 13, 61, 31, 71, 73, 3, 19, 43, 2, 101, 103, 53, 109, 113, 13, 131, 31
Offset: 1

Views

Author

Vladimir Shevelev, Sep 02 2009

Keywords

Comments

The sequence is connected with our sieve selecting the primes of the union {2,3} and A164333 from all primes. Note that a(n)=n iff p_n is in the considered union, which corresponds to 0's iterations of g.

Links

Crossrefs

Cf. A164333, A055496, A164960, A104272, A080359, A164918, A006992, A164917, A164368, A164288.

A186946 The smallest integer x > 0 such that the number of prime powers p^k (k>=1) in (x/2,x] equals n.

Original entry on oeis.org

2, 3, 5, 9, 13, 25, 29, 31, 43, 49, 71, 73, 81, 103, 109, 113, 127, 131, 139, 157, 173, 181, 191, 193, 199, 239, 241, 269, 271, 283, 289, 293, 313, 349, 353, 361, 373, 379, 409, 419, 421, 433, 439, 443, 463, 499, 509, 523, 571, 577, 599, 601, 607, 613, 619
Offset: 1

Views

Author

Vladimir Shevelev, Aug 30 2013

Keywords

Comments

An analog of Labos primes (A080359) on prime powers > 1 (A000961).

Links

Crossrefs

Cf. A080459, A186945, A228520, A228592.

Programs

  • Mathematica
    a000961Q[n_]:=(Length[FactorInteger[n]]==1) && IntegerQ[n]; nn=99; t=Table[0,{nn+1}]; s=0; Do[If[a000961Q[k], s++]; If[a000961Q[k/2], s--]; If[s<=nn && t[[s+1]]==0, t[[s+1]]=k], {k, 2, Prime[3*nn]}]; Prepend[Rest[t],2] (* after T. D. Noe's code at A080359 *) (* Peter J. C. Moses, Sep 11 2013 *)

Formula

a(n) <= A186945(n).

Extensions

More terms from Peter J. C. Moses, Aug 30 2013

A194953 Nonzero values of |A194659(n)-A194186(n+1)|.

Original entry on oeis.org

2, 6, 2, 4, 4, 4, 2, 2, 8, 2, 2, 4, 4, 4, 6, 2, 2, 4, 2, 2, 10, 6, 6, 2, 2, 2, 6, 2, 8, 8, 4, 6, 4, 2, 8, 4, 8, 4, 4, 6, 4, 2, 4, 2, 4, 2, 2, 22, 2, 2, 6, 4, 4, 8, 2, 2, 10, 2, 2, 2, 2, 4, 4, 4, 2, 2, 2, 2, 2, 10, 2, 2, 8, 18, 2, 2, 4, 4, 2, 12, 6, 6, 8, 20
Offset: 1

Views

Author

Vladimir Shevelev, Sep 06 2011

Keywords

Comments

The sequence (together with A194674) characterizes a right-left symmetry in the distribution of primes over intervals (2*p_n, 2*p_(n+1)), n=1,2,..., where p_n is the n-th prime.

Crossrefs

Cf. A104272, A080359, A193507, A194186, A164368, A194598, A194658, A194659, A194674, A164288, A164294.

A247558 Smallest integer x > 0 such that the number of semiprimes in the interval (x/2, x] equals n.

Original entry on oeis.org

4, 6, 10, 15, 25, 26, 35, 38, 39, 57, 58, 62, 65, 86, 87, 91, 94, 95, 121, 122, 123, 134, 142, 143, 145, 146, 159, 161, 169, 202, 203, 205, 206, 209, 214, 215, 217, 218, 219, 221, 262, 265, 278, 299, 301, 302, 303, 305, 309, 326, 327, 329, 335, 341, 346, 361, 362, 365, 382, 386, 393, 394, 395, 398
Offset: 1

Views

Author

Jonathan Vos Post and Robert G. Wilson v, Sep 19 2014

Keywords

Comments

Analogous to A080359: the Labos Elemer primes.

Examples

			a(6) = 26 because in the interval, (13, 26], {14, 15, 21, 22, 25, 26} are six semiprimes.

Links

Crossrefs

Cf. A001358, A072000, A080359.

Programs

  • Mathematica
    SemiPrimeQ[n_] := PrimeOmega[n] == 2; mx = 1000; t = Table[0, {mx + 1}]; s = 0; Do[ If[ SemiPrimeQ[k], s++]; If[ SemiPrimeQ[k/2], s--]; If[s <= mx && t[[s + 1]] == 0, t[[s + 1]] = k], {k, 8*mx}]; Rest[t]

A164966 Primes which are obtained at least by two ways using the iterations of the S operator (see A164960) beginning with primes of the union of {2,3} and A164333.

Original entry on oeis.org

127, 149, 211, 223, 257, 307, 431, 449
Offset: 1

Views

Author

Vladimir Shevelev, Sep 02 2009

Keywords

Comments

The sequence is connected with our sieve selecting the primes of the union of {2,3} and A164333 from all primes.

Links

Crossrefs

Cf. A055496, A164333 A164960, A164962, A164920, A006992, A164917, A164918, A104272, A080359, A164368, A164288.
Previous Showing 31-40 of 43 results. Next

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