A104272 -id:A104272 - cp's OEIS frontend

A178128 Lesser of twin primes if it is a Ramanujan prime.

11, 17, 29, 41, 59, 71, 101, 107, 149, 179, 227, 239, 269, 281, 311, 347, 419, 431, 461, 569, 599, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1427, 1451, 1481, 1487, 1607, 1667, 1721, 1787, 1871, 1877, 1997

Offset: 1

Jonathan Sondow, May 20 2010

nonn

By definition, a number p is a member if p and p+2 are primes and p is a Ramanujan prime A104272.
Supersequence of A178127.
In the first 328 pairs of twin primes, more than 78% of their first members are Ramanujan primes. For a partial explanation, see "Ramanujan primes and Bertrand's postulate" Section 7.
See A001359 and A104272 for additional comments, links, and references.

			a(1) = 11 because 11 and 13 are the 1st twin primes the lesser of which is a Ramanujan prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 2009, 2010.
J. Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, 116 (2009), 630-635.
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2.

Cf. A001359 (lesser of twin primes), A104272 (Ramanujan primes), A164371 (lesser of twin prime pairs which are non-Ramanujan primes), A178127 (lesser of twin Ramanujan primes).

nn = 200; R = Table[0, {nn}]; s = 0;
Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s < nn, R[[s + 1]] = k], {k, Prime[3 nn]}];
A104272 = R + 1;
A001359 = Select[Prime[Range[2 nn]], PrimeQ[# + 2]&];
Intersection[A001359, A104272] (* Jean-François Alcover, Nov 07 2018, using T. D. Noe's code for A104262 *)

use ntheory ":all"; my @t = grep { is_prime($+2) } @{ramanujan_primes(10000)}; say "@t"; # _Dana Jacobsen, Sep 06 2015

A001359 intersect A104272.

A185004 Ramanujan modulo primes R_(3,1)(n): a(n) is the smallest number such that if x >= a(n), then pi_(3,1)(x) - pi_(3,1)(x/2) >= n, where pi_(3,1)(x) is the number of primes==1 (mod 3) <= x.

Original entry on oeis.org

7, 31, 43, 67, 97, 103, 151, 163, 181, 223, 229, 271, 331, 337, 367, 373, 409, 433, 487, 499, 571, 577, 601, 607, 631, 643, 709, 727, 751, 769, 823, 853, 883, 937, 991, 1009, 1021, 1033, 1051, 1063, 1087, 1117, 1123, 1231, 1291, 1297, 1303

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Dec 18 2012

nonn

All terms are primes==1 (mod 3).
A modular generalization of Ramanujan numbers, see Section 6 of the Shevelev-Greathouse-Moses paper.
We conjecture that for all n >= 1 a(n) <= A104272(3*n). This conjecture is based on observation that, if interval (x/2, x] contains >= 3*n primes, then at least n of them are of the form 3*k+1.
The function pi_(3,1)(n) starts 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3,... with records occurring as specified in A123365/A002476. - R. J. Mathar, Jan 10 2013

G. C. Greubel, Table of n, a(n) for n = 1..1000
Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. arXiv:1212.2785

Cf. A104272, A185005, A185006, A185007.

pimod := proc(m,n,x)
    option remember;
    a := 0 ;
    for k from n to x by m do
        if isprime(k) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
a := [seq(0,n=1..100)] ;
for x from 1 do
    pdiff := pimod(3,1,x)-pimod(3,1,x/2) ;
    if pdiff+1 <= nops(a) then
        v := x+1 ;
        n := pdiff+1 ;
        if nR. J. Mathar, Jan 10 2013

max = 100; pimod[m_, n_, x_] := pimod[m, n, x] = Module[{a = 0}, For[k = n, k <= x, k = k + m, If[PrimeQ[k], a = a + 1]]; a]; a[] = 0; For[x = 1, x <= max^2, x++, pdiff = pimod[3, 1, x] - pimod[3, 1, x/2]; If[ pdiff + 1 <= max, v = x + 1; n = pdiff + 1; If[ n < v , a[n] = v ] ] ]; Table[a[n], {n, 1, max}] (* _Jean-François Alcover, Jan 28 2013, translated and adapted from R. J. Mathar's Maple program *)

lim(a(n)/prime(4*n)) = 1 as n tends to infinity.

A194674 Positions of nonzero terms of A194659(n)-A194186(n+1), n>=1.

Original entry on oeis.org

20, 27, 73, 77, 85, 95, 106, 116, 117, 122, 125, 132, 137, 144, 145, 152, 162, 167, 168, 189, 191, 192, 193, 198, 201, 208, 213, 234, 235, 236, 243, 249, 258, 259, 265, 275, 279, 286, 287, 291, 318, 319, 321, 329, 330, 331, 340

Offset: 1

Vladimir Shevelev, Sep 01 2011

nonn

The sequence (together with A194953) 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.

V. Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes

Cf. A104272, A080359, A193507, A194186, A164368, A194598, A194658, A194659, A194953, A164288, A164294.

A212493 Let p_n=prime(n), n>=1. Then a(n) is the least prime p which differs from p_n, for which the intervals (p/2,p_n/2), (p,p_n], if pp_n, contain the same number of primes, and a(n)=0, if no such prime p exists.

Original entry on oeis.org

0, 5, 3, 3, 3, 17, 13, 23, 19, 19, 37, 31, 31, 47, 43, 59, 53, 67, 61, 0, 79, 73, 73, 73, 73, 0, 107, 103, 127, 131, 109, 113, 113, 151, 113, 139, 163, 157, 157, 179, 173, 0, 223, 197, 193, 233, 193, 191, 191, 193, 199, 0, 0, 257, 251, 251, 0, 277, 271, 271

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, May 18 2012

nonn

a(n)=0 if and only if p_n is a peculiar prime, i.e., simultaneously Ramanujan (A104272) and Labos (A080359) prime (see sequence A164554).

a(n)>p_n if and only if p_n is Labos prime but not Ramanujan prime.

			Let n=5, p_5=11; p=2 is not suitable, since in (1,5.5) we have 3 primes, while in (2,11] there are 4 primes. Consider p=3. Now in intervals (1.5,5.5) and (3,11] we have the same number (3) of primes. Therefore, a(5)=3. The same value we can obtain by the formula. Since 11 is not a Labos prime, then a(5)=A080359(5-pi(5.5))=A080359(2)=3.

Vladimir Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4.
Jonathan Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011; J. Integer Seq. 14 (2011) Article 11.6.2.

Cf. A104272, A080359, A164554.

terms = 60; nn = Prime[terms];
R = Table[0, {nn}]; s = 0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s < nn, R[[s + 1]] = k], {k, Prime[3 nn]}];
A104272 = R + 1;
t = Table[0, {nn + 1}]; s = 0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s <= nn && t[[s + 1]] == 0, t[[s + 1]] = k], {k, Prime[3 nn]}];
A080359 = Rest[t];
a[n_] := Module[{}, pn = Prime[n]; If[MemberQ[A104272, pn] && MemberQ[ A080359, pn], Return[0]]; For[p = 2, True, p = NextPrime[p], Which[ppn, If[PrimePi[p/2] - PrimePi[pn/2] == PrimePi[p] - PrimePi[pn], Return[p]]]]];
Array[a, terms] (* Jean-François Alcover, Dec 04 2018, after T. D. Noe in A104272 *)

If p_n is not a Labos prime, then a(n) = A080359(n-pi(p_n/2)).

A290394 First k-Ramanujan prime, where k = 1 + 1/n.

Original entry on oeis.org

2, 11, 11, 29, 29, 37, 37, 53, 127, 127, 127, 127, 127, 149, 149, 149, 211, 223, 223, 223, 307, 307, 331, 331, 331, 331, 331, 331, 331, 541, 541, 541, 541, 541, 541, 541, 541, 541, 541, 1361, 1361, 1361, 1361, 1361, 1361, 1361, 1361, 1361, 1361, 1361, 1361, 1361, 1361, 1361, 1361, 1361, 1361, 1361, 1361, 1361, 1361, 1361, 1361, 1361, 1361, 1361, 1361, 1361, 1361, 1693

Offset: 1

Jonathan Sondow, Jul 29 2017

nonn

For real k > 1, the first k-Ramanujan prime is the smallest integer m with pi(x) - pi(x/k) >= 1 for all real x >= m. For 0 < c < 1, the first c-Ramanujan prime is the first k-Ramanujan prime with k = 1/c.
Axler (2015, Cor. 2.4 and Prop. 2.5(ii)) and Axler and Leßmann (2017, Theorem 1) computed the first k-Ramanujan prime for all k >= 1.000040690557321. With k = 1 + 1/n, this gives 1 <= n <= 24575; in particular, a(24575) = 2898359. They also give the isolated result a(28313999) = 10726905041 on p. 646.
The Mathematica program below is based on their algorithm but uses only part of their data (compare A277719) and is valid only for 1 <= n <= 1014; in particular, a(1014) = 48731. Their algorithm uses their result that for N > 1 the N-th prime p_N is the first k-Ramanujan prime if and only if p_N > k*p_{N-1} and p_n <= k*p_{n-1} for all n > N.
See A104272 for additional comments, references, links, formulas, examples, programs, and cross-refs.

			a(1) = first 2-Ramanujan prime = first 1/2-Ramanujan prime = first Ramanujan prime = A104272(1) = 2.
a(3) = first 4/3-Ramanujan prime = first 3/4-Ramanujan prime = A193880(1) = 11.

N. Amersi, O. Beckwith, S. J. Miller, R. Ronan, J. Sondow, Generalized Ramanujan primes, Combinatorial and Additive Number Theory, Springer Proc. in Math. & Stat., CANT 2011 and 2012, Vol. 101 (2014), 1-13; arXiv:1108.0475 [math.NT], 2011.
Christian Axler, On generalized Ramanujan primes, Ramanujan J., online 30 April 2015, 1-30.
Christian Axler and Thomas Leßmann, An explicit upper bound for the first k-Ramanujan prime, arXiv:1504.05485 [math.NT], 2015.
Christian Axler and Thomas Leßmann, On the first k-Ramanujan prime, Amer. Math. Monthly, 124 (2017), 642-646; correction by J. Sondow, Editor's endnotes, Amer. Math. Monthly, 124 (2017), 985.
V. Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4.

Cf. A104272, A164952, A193761, A193880, A277718, A277719.

A = {3, 5, 7, 10, 12, 16, 31, 35, 47, 48, 63, 67, 100, 218, 264, 298, 328,  368, 430, 463, 591, 651, 739, 758, 782, 843, 891, 929, 1060, 1184, 1230, 1316, 1410, 1832, 2226, 3386, 3645, 3794, 3796, 4523, 4613, 4755, 5009, 5950}; kR1[k_] := If[k >= 5/3, 2, (m = 1;
   While[k >= Prime[A[[m]]]/Prime[A[[m]] - 1] ||
     k < Prime[A[[m + 1]]]/Prime[A[[m + 1]] - 1], m++];
   Prime[A[[m]]])]; Table[kR1[1 + 1/n], {n, 70}]

A291465 a(n) is the least m >= n for which the complete bipartite graph K_{m,n} has a prime labeling.

Original entry on oeis.org

1, 2, 4, 9, 14, 25, 36, 45, 52, 61, 62, 89, 90, 95, 98, 123, 140, 155, 162, 171, 172, 177, 216, 217, 226, 243, 244, 255, 264, 283, 318, 321, 340, 345, 374, 383, 384, 395, 400, 403, 422, 449, 456, 465, 478, 531, 546, 551, 552, 557, 562, 567, 594, 599, 604, 605

Offset: 1

Jonathan Sondow, Aug 24 2017

nonn

A prime labeling of K_{m,n} is a pair of sets A and B whose union is {1,2,...,m+n} such that |A| = m, |B| = n, and gcd(a,b) = 1 for all a in A and b in B. For an equivalent definition, the data above, and the formula below involving R_{n-1}, see Berliner, Dean, Hook, Marr, Mbirika (2016) Section 3.2.

			A = {1,3} and B = {2,4} is a prime labeling of K_{2,2}, so a(2) = 2.

Paul Tabatabai, Table of n, a(n) for n = 1..75
Adam H. Berliner, N. Dean, J. Hook, A. Marr, and A. Mbirika, Coprime and prime labelings of graphs, arXiv:1604.07698 [math.CO], 2016; Journal of Integer Sequences, Vol. 19 (2016), #16.5.8.

Cf. A104272, A213273, A213806, A284875.

n+1 <= a(n) <= R_{n-1} - n for n > 2, where R_{n-1} is a Ramanujan prime A104272.

a(14) onward from Paul Tabatabai, Apr 29 2019

A336869 Number of divisors d of n! with distinct prime multiplicities such that the quotient n!/d also has distinct prime multiplicities.

Original entry on oeis.org

1, 1, 2, 2, 6, 4, 12, 8, 20, 28, 68, 40, 80, 0, 56, 160, 256, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Aug 08 2020

nonn

Does this sequence converge to zero?
A number has distinct prime multiplicities iff its prime signature is strict.
From Edward Moody, Jan 18 2021: (Start)
a(n) = 0 for n >= 17.
Proof: 17 is the third Ramanujan prime (A104272). Therefore, for n>=17, there are at least three primes greater than n/2 and less than or equal to n. These primes must have exponent 1 in the prime factorization of n!, therefore, at least two of them must have exponent 1 in the prime factorization of either d or n!/d, so d and n!/d cannot both have distinct prime multiplicities. (End)

			The a(1) = 1 through a(7) = 8 divisors:
  1  1  2  1   3   1    5
     2  3  2   5   2    7
           3   24  5    45
           8   40  9    63
           12      16   80
           24      18   112
                   40   720
                   45   1008
                   80
                   144
                   360
                   720

A336419 is the version for superprimorials.
A336500 is the generalization to non-factorials.
A336616 is the maximum among these divisors.
A336617 is the minimum among these divisors.
A336939 has these row sums.
A000005 counts divisors.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A327498 gives the maximum divisor of n with distinct prime multiplicities.
A336414 counts divisors of n! with distinct prime multiplicities.
A336415 counts divisors of n! with equal prime multiplicities.
A336423 counts chains using A130091.
Cf. A098859, A118914, A124010, A327527, A336424, A336568, A336571, A336870.
Factorial numbers: A000142, A007489, A022559, A027423, A048656, A071626, A325272, A325273, A325617, A336416.

Table[Length[Select[Divisors[n!],UnsameQ@@Last/@FactorInteger[#]&&UnsameQ@@Last/@FactorInteger[n!/#]&]],{n,0,10}]

a(31)-a(80) from Edward Moody, Jan 19 2021

A164918 The smallest starting prime which reaches prime(n) by repeated application of the map x->A060308(x).

Original entry on oeis.org

2, 2, 2, 2, 11, 2, 17, 11, 2, 29, 17, 11, 41, 2, 47, 29, 59, 17, 67, 71, 11, 41, 2, 47, 97, 101, 29, 107, 109, 17, 127, 67, 137, 11, 149, 151, 41, 2, 167, 47, 179, 181, 191, 97, 197, 29, 107, 17, 227, 229, 233, 239, 241, 127, 67, 263, 269, 137, 11, 281, 283, 149, 307, 311, 41

Offset: 1

Vladimir Shevelev, Aug 31 2009

nonn

a(n) is the starting value of the prime chain described in A164917 which contains (touches) prime(n).

By construction, each member of this sequence here is one of the values of A164368, the head elements of all chains of this map.

			The first four values are 2 because prime(1)=2, prime(2)=3, prime(3)=5 and prime(4)=7 are all in the prime chain starting at 2.

V. Shevelev, On critical small intervals containing primes, arXiv:0908.2319 [math.NT], 2009.

Cf. A006992, A164917, A104272, A164368, A164288.

A060308 := proc(n) prevprime(2*n+1) ; end:
isA164368 := proc(p) local q ; q := nextprime(floor(p/2)) ; return (numtheory[pi](2*q) -numtheory[pi](p) >= 1); end proc:
A164368 := proc(n) option remember; local a; if n = 1 then 2; else a := nextprime( procname(n-1)) ; while not isA164368(a) do a := nextprime(a) ; end do : RETURN(a) ; end if; end proc:
A164918 := proc(n) local p, a, j, q, itr ; p := ithprime(n) ; a := 1000000000000000 ; for j from 1 do q := A164368(j) ; if q > p then break; end if; itr := 0 ; while q < p do q := A060308(q) ; itr := itr+1 ; end do; if q = p then return A164368(j) ; end if; end do: end proc:
seq(A164918(n), n=1..120) ; # R. J. Mathar, Mar 12 2010

lp[n_] := NextPrime[2n, -1];
a[n_] := For[pn = Prime[n]; p = 2, p <= pn, p = NextPrime[p], nwl = NestWhileList[lp, p, # <= Prime[n]&]; If[MemberQ[nwl, pn], Return[p]]];
Array[a, 120] (* Jean-François Alcover, Dec 01 2017 *)

Edited and extended by R. J. Mathar, Mar 12 2010

A182366 Records in A194217.

Original entry on oeis.org

8, 10, 24, 36, 60, 64, 84, 114, 124, 144, 202, 226, 228

Offset: 1

Vladimir Shevelev, Apr 26 2012

nonn

Records in A194217(n) occur at n = 2, 4, 10, 14, 43, 95, 145, 167, 287, 415, 560, 635, 982,...

Cf. A104272, A080359, A193507, A194184, A194186, A194217.

A185007 Ramanujan primes R_(4,3)(n): a(n) is the smallest number such that if x >= a(n), then pi_(4,3)(x) - pi_(4,3)(x/2) >= n, where pi_(4,3)(x) is the number of primes==3 (mod 4) <= x.

Original entry on oeis.org

7, 23, 47, 67, 71, 103, 127, 167, 179, 191, 223, 227, 263, 307, 359, 367, 431, 463, 479, 487, 491, 547, 571, 587, 599, 631, 643, 647, 719, 739, 787, 811, 823, 839, 887, 907, 1019, 1031, 1051, 1063, 1087, 1151, 1223, 1279, 1303, 1319, 1399

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Dec 18 2012

nonn

All terms are primes==3 (mod 4).
A general conception of generalized Ramanujan numbers, see in Section 6 of the Shevelev, Greathouse IV, & Moses link.
We conjecture that for all n >= 1, a(n) <= A104272(3*n). This conjecture is based on observation that, if interval (x/2, x] contains >= 3*n primes, then at least n of them are of the form 4*k+3.

Vladimir Shevelev, Charles R. Greathouse IV, Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. arXiv:1212.2785

Cf. A104272, A185004, A185005, A185006.

Table[1+NestWhile[#1-1&,A104272[[3 k]],Count[Mod[Select[Range@@{Floor[#1/2+1],#1},PrimeQ],4],3]>=k&],{k,1,10}] using the code nn=1000;A104272=Table[0,{nn}];s=0;Do[If[PrimeQ[k],s++];If[PrimeQ[k/2],s--];If[sA104272[[s+1]]=k],{k,Prime[3*nn]}];A104272=A104272+1 (* T. D. Noe, Nov 15 2010 *)

lim(a(n)/prime(4*n)) = 1 as n tends to infinity.

cp's OEIS Frontend

A178128 Lesser of twin primes if it is a Ramanujan prime.

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A185004 Ramanujan modulo primes R_(3,1)(n): a(n) is the smallest number such that if x >= a(n), then pi_(3,1)(x) - pi_(3,1)(x/2) >= n, where pi_(3,1)(x) is the number of primes==1 (mod 3) <= x.

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A194674 Positions of nonzero terms of A194659(n)-A194186(n+1), n>=1.

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A212493 Let p_n=prime(n), n>=1. Then a(n) is the least prime p which differs from p_n, for which the intervals (p/2,p_n/2), (p,p_n], if pp_n, contain the same number of primes, and a(n)=0, if no such prime p exists.

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A290394 First k-Ramanujan prime, where k = 1 + 1/n.

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A291465 a(n) is the least m >= n for which the complete bipartite graph K_{m,n} has a prime labeling.

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A336869 Number of divisors d of n! with distinct prime multiplicities such that the quotient n!/d also has distinct prime multiplicities.

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A164918 The smallest starting prime which reaches prime(n) by repeated application of the map x->A060308(x).

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