A000720 -id:A000720 - cp's OEIS frontend

A237284 Number of ordered ways to write 2*n = p + q with p, q and A000720(p) all prime.

0, 0, 1, 2, 2, 1, 2, 3, 2, 2, 4, 3, 1, 3, 2, 1, 5, 3, 1, 3, 3, 3, 4, 5, 2, 3, 4, 1, 4, 3, 3, 6, 2, 1, 6, 6, 3, 4, 7, 1, 4, 6, 3, 5, 6, 2, 4, 4, 2, 6, 5, 3, 5, 4, 3, 7, 8, 2, 4, 8, 1, 4, 5, 3, 6, 5, 4, 2, 7, 5, 6, 6, 3, 4, 6, 2, 5, 7, 2, 4

Offset: 1

Zhi-Wei Sun, Feb 06 2014

nonn

Conjecture: a(n) > 0 for all n > 2, and a(n) = 1 only for n = 3, 6, 13, 16, 19, 28, 34, 40, 61, 166, 278.
This is stronger than Goldbach's conjecture.
The conjecture is true for n <= 5*10^8. - Dmitry Kamenetsky, Mar 13 2020

			a(13) = 1 since 2*13 = 3 + 23 with 3, 23 and A000720(3) = 2 all prime.
a(278) = 1 since 2*278 = 509 + 47 with 509, 47 and A000720(509) = 97 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Carlos Rivera, Conjecture 85. Conjectures stricter that the Goldbach ones, Prime Puzzles
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2016.

Cf. A000040, A000720, A002372, A002375, A006450, A236566.

a[n_]:=Sum[If[PrimeQ[2n-Prime[Prime[k]]],1,0],{k,1,PrimePi[PrimePi[2n-1]]}]
Table[a[n],{n,1,80}]

A291440 a(n) = pi(n^2) - pi(n)^2, where pi(n) = A000720(n).

Original entry on oeis.org

0, 1, 0, 2, 0, 2, -1, 2, 6, 9, 5, 9, 3, 8, 12, 18, 12, 17, 8, 14, 21, 28, 18, 24, 33, 41, 48, 56, 46, 54, 41, 51, 60, 70, 79, 89, 75, 84, 96, 107, 94, 105, 87, 99, 110, 123, 104, 117, 132, 142, 153, 168, 153, 165, 178, 189, 201, 218, 198, 214, 195, 208, 225, 240, 254, 270, 248, 263, 280, 293, 275, 290, 264, 281, 298, 316, 338, 352, 327, 350

Offset: 1

Jonathan Sondow, Aug 23 2017

sign

The only zero values are a(1) = a(3) = a(5) = 0. The only negative value is a(7) = -1. In particular, pi(n^2) > pi(n)^2 for n > 7. These can be proved by the PNT with error term for large n and computation for smaller n.
For prime(n)^2 - prime(n^2), see A123914.
For pi(n^3) - pi(n)^3, see A291538.
Mincu and Panaitopol (2008) prove that pi(m*n) >= pi(m)*pi(n) for all positive m and n except for m = 5, n = 7; m = 7, n = 5; and m = n = 7. This implies for m = n that a(n) >= 0 if n <> 7. - Jonathan Sondow, Nov 03 2017
Diagonal of the triangular array A294508. - Jonathan Sondow and Robert G. Wilson v, Nov 08 2017

			a(7) = pi(7^2) - pi(7)^2 = 15 - 4^2 = -1.

Robert Israel, Table of n, a(n) for n = 1..10000
Gabriel Mincu and Laurentiu Panaitopol, Properties of some functions connected to prime numbers, J. Inequal. Pure Appl. Math., 9 No. 1 (2008), Art. 12.

Cf. A000720, A123914, A262199, A291538, A291539, A291540, A291541, A291542.

[#PrimesUpTo(n^2)-#PrimesUpTo(n)^2: n in [1..80]]; // Vincenzo Librandi, Aug 26 2017

seq(numtheory:-pi(n^2)-numtheory:-pi(n)^2, n=1..100); # Robert Israel, Aug 25 2017

Table[PrimePi[n^2] - PrimePi[n]^2, {n, 80}]

a(n) = primepi(n^2) - primepi(n)^2; \\ Michel Marcus, Sep 10 2017

a(n) = A000720(n^2) - A000720(n)^2.
a(n) ~ (n^2 / log(n))*(1/2 - 1/log(n)) as n tends to infinity, by the PNT.
From Jonathan Sondow and Robert G. Wilson v, Nov 08 2017: (Start)
a(n) = A294508(n*(n+1)/2).
a(n) >= A294509(n). (End)

A332808 Fully multiplicative with a(p) = A332806(A000720(p)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 12, 11, 14, 15, 16, 17, 18, 19, 20, 21, 26, 29, 24, 25, 22, 27, 28, 23, 30, 37, 32, 39, 34, 35, 36, 31, 38, 33, 40, 41, 42, 43, 52, 45, 58, 53, 48, 49, 50, 51, 44, 47, 54, 65, 56, 57, 46, 61, 60, 59, 74, 63, 64, 55, 78, 71, 68, 87, 70, 79, 72, 67, 62, 75, 76, 91, 66, 89, 80, 81, 82, 101

Offset: 1

Antti Karttunen, Feb 27 2020

Antti Karttunen, Table of n, a(n) for n = 1..26927

Cf. A000720, A332806, A108549 (fixed points), A332818, A332819.

Inverse permutation is A108548, from which this differs for the first time at n=67, where a(67) = 71, while A108548(67) = 73.

up_to = 10000;
A332806list(up_to) = { my(v=vector(2), xs=Map(), lista=List([]), p,q,u); v[2] = 3; v[1] = 5; mapput(xs,1,1); mapput(xs,2,2); mapput(xs,3,3);  for(n=4,up_to, p = v[2-(n%2)]; q = nextprime(1+p); while(q%4 != p%4, q=nextprime(1+q)); v[2-(n%2)] = q; mapput(xs,primepi(q),n)); for(i=1, oo, if(!mapisdefined(xs, i, &u), return(Vec(lista)), listput(lista, prime(u)))); };
v332806 = A332806list(up_to);
A332806(n) = v332806[n];
A332808(n) = { my(f=factor(n)); f[,1] = apply(A332806,apply(primepi,f[,1])); factorback(f); };

A365400 a(n) = 64 + A000720(n) - A365339(n).

Original entry on oeis.org

63, 63, 63, 62, 62, 62, 62, 62, 61, 61, 61, 61, 61, 61, 60, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 59, 58, 58, 58, 58, 58, 58, 58, 58, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 57, 56

Offset: 1

Peter Luschny, Sep 06 2023

nonn

It is conjectured that A365339(n) = PrimePi(n) + 64 for all n >= 31957 (Pollack et al.). Assuming this conjecture a(n) = 0 for n > 31956.

a is not monotonically decreasing.

Paul Pollack, Carl Pomerance, and Enrique Treviño, Sets of monotonicity for Euler's totient function, preprint. See M(n).
Paul Pollack, Carl Pomerance, and Enrique Treviño, Sets of monotonicity for Euler's totient function, Ramanujan J. 30 (2013), no. 3, pp. 379-398.
Terence Tao, Monotone non-decreasing sequences of the Euler totient function, arXiv:2309.02325 [math.NT], 2023.

Cf. A000720, A365339, A365474.

# Computes the first N terms of the sequence.
using Nemo
function A365400List(N)
    phi = Int64[i for i in 1:N + 1]
    for i in 2:N + 1
        if phi[i] == i
            for j in i:i:N + 1
                phi[j] -= div(phi[j], i)
    end end end
    lst = zeros(Int64, N)
    dyn = zeros(Int64, N)
    pi = 64
    for n in 1:N
        p = phi[n]
        nxt = dyn[p] + 1
        while p <= N && dyn[p] < nxt
            dyn[p] = nxt
            p += 1
        end
        pi += is_prime(n) ? 1 : 0
        lst[n] = pi - dyn[n]
    end
    return lst
end
println(A365400List(32000))

from bisect import bisect
from sympy import totient, primepi
def A365400(n):
    plist, qlist, c = tuple(totient(i) for i in range(1,n+1)), [0]*(n+1), 0
    for i in range(n):
        qlist[a:=bisect(qlist,plist[i],lo=1,hi=c+1,key=lambda x:plist[x])]=i
        c = max(c,a)
    return 64+primepi(n)-c # Chai Wah Wu, Sep 06 2023

A212210 Triangle read by rows: T(n,k) = pi(n) + pi(k) - pi(n+k), n >= 1, 1 <= k <= n, where pi() = A000720().

Original entry on oeis.org

-1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, 1, 2, -1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 2, 1, 2, 0, 1, 1, 1, 1, 1, 2, 2, 0, 0, 1, 0, 1, 1, 2, 1, 1, -1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 2, 1, 2, 1, 1, 1, 2, -1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 3, 3

Offset: 1

N. J. A. Sloane, May 04 2012

It is conjectured that pi(x)+pi(y) >= pi(x+y) for 1 < y <= x.

A006093 gives row numbers of rows containing at least one negative term. [Reinhard Zumkeller, May 05 2012]

			Triangle begins:
  -1
  -1 0
   0 0 1
  -1 0 0 0
   0 0 1 1 2
  -1 0 1 1 1 1
   0 1 2 1 2 1 2
   0 1 1 1 1 1 2 2
   0 0 1 0 1 1 2 1 1
  -1 0 0 0 1 1 1 1 0 0
  ...

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VII.5, p. 235.

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
P. Erdos and J. L. Selfridge, Complete prime subsets of consecutive integers. Proceedings of the Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1971), pp. 1--14. Dept. Comput. Sci., Univ. Manitoba, Winnipeg, Man., 1971. MR0337828 (49 #2597).

Cf. A000720, A212211, A212212, A212213, A060208, A047885, A047886.

Left diagonal is -A010051.

import Data.List (inits, tails)
a212210 n k = a212210_tabl !! (n-1) !! (k-1)
a212210_row n = a212210_tabl !! (n-1)
a212210_tabl = f $ tail $ zip (inits pis) (tails pis) where
   f ((xs,ys) : zss) = (zipWith (-) (map (+ last xs) (xs)) ys) : f zss
   pis = a000720_list
-- Reinhard Zumkeller, May 04 2012

t[n_, k_] := PrimePi[n] + PrimePi[k] - PrimePi[n + k]; Table[t[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 17 2012 *)

A214935 Index of the primes of A205827, A000720(A205827(n)).

Original entry on oeis.org

1, 2, 4, 9, 30, 189, 217, 2225, 3385, 14357, 30802, 31545, 104071, 149689, 1094421, 1319945, 10655462, 23163298, 112228683, 182837804, 203615628, 486570087, 1094330259, 11992433550, 17883926781, 50070452577, 52302956123, 72178455400

Offset: 1

John W. Nicholson, Oct 28 2012

nonn

A000040(a(n)) = A205827(n).

With pi(x) being the prime counting function, A000720(x), for n from 1 to 3, a(n) = pi(A111870(n)) = A241542(n), for n from 5 to 28, a(n) = pi(A111870(n-1)) = A241542(n-1). - John W. Nicholson, May 10 2014

			a(4) = 9, A000040(9) = 23, and A205827(4) = 23.

John W. Nicholson, Table of n, a(n) for n = 1..38

Cf. A205827.

record=0;i=0;A214935=vectorsmall(38);for(n=1,75,current=(A000101[n]/A002386[n]*1.)^A005669[n];if(current>record,record=current;i++;A214935[i]=A005669[n]))

a(n) = pi(A205827(n)) = A000720(A205827(n)).

a(13)-a(28) from Donovan Johnson, Oct 28 2012

a(29)-a(38) from John W. Nicholson, Dec 01 2013

A237612 Least positive integer k such that A000720(k*n) is a square, or 0 if such a number k does not exist.

Original entry on oeis.org

1, 1, 3, 2, 2, 4, 1, 1, 1, 1, 5, 2, 2, 2, 28, 34, 9, 3, 3, 5, 20, 7, 1, 1, 1, 1, 1, 1, 2, 14, 5, 17, 3, 16, 12, 23, 18, 4, 4, 30, 46, 10, 50, 23, 36, 18, 40, 14, 2, 2, 3, 3, 1, 1, 1, 1, 1, 1, 32, 7, 11, 68, 19, 79, 29, 267, 10, 8, 12, 6

Offset: 1

Zhi-Wei Sun, Feb 10 2014

nonn

According to the conjecture in A237598, we should always have 0 < a(n) < prime(n).

			a(3) = 3 since A000720(3*3) = 4 is a square, but neither A000720(1*3) = 2  nor A000720(2*3) = 3 is a square.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000290, A000720, A237578, A237597, A237598, A237614.

sq[n_]:=IntegerQ[Sqrt[PrimePi[n]]]
Do[Do[If[sq[k*n],Print[n," ",k];Goto[aa]],{k,1,Prime[n]-1}];
Print[n," ",0];Label[aa];Continue,{n,1,100}]

A354202 Fully multiplicative with a(p^e) = A354200(A000720(p))^e.

Original entry on oeis.org

1, 5, 7, 25, 13, 35, 11, 125, 49, 65, 19, 175, 17, 55, 91, 625, 29, 245, 23, 325, 77, 95, 31, 875, 169, 85, 343, 275, 37, 455, 43, 3125, 133, 145, 143, 1225, 41, 115, 119, 1625, 53, 385, 47, 475, 637, 155, 59, 4375, 121, 845, 203, 425, 61, 1715, 247, 1375, 161, 185, 67, 2275, 73, 215, 539, 15625, 221, 665, 71, 725

Offset: 1

Antti Karttunen, May 23 2022

Permutation of A007310. Preserves the prime signature.

Index entries for sequences computed from indices in prime factorization

Cf. A007310 (terms sorted into ascending order), A354200, A354203 (left inverse), A354204 (Möbius transform), A354205 (inverse Möbius transform).

Cf. also A003961, A108548, A267099, A332818, A348746, A354091 for similar constructions.

A354200(n) = if(1==n,5,my(p=prime(n), m=p%4); forprime(q=1+p,,if(m==(q%4),return(q))));
A354202(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A354200(primepi(f[k,1]))); factorback(f); };

A038626 Smallest positive integer m such that m = pi(nm) = A000720(nm).

Original entry on oeis.org

1, 9, 24, 66, 168, 437, 1051, 2614, 6454, 15927, 40071, 100346, 251706, 637197, 1617172, 4124436, 10553399, 27066969, 69709679, 179992838, 465769802, 1208198523, 3140421715, 8179002095, 21338685402, 55762149023, 145935689357, 382465573481, 1003652347080, 2636913002890, 6935812012540

Offset: 2

Jud McCranie

nonn

Golomb shows that solutions exist for each n>1.
For all known terms, we have 2.4*a(n) < a(n+1) < 2.7*a(n) + 7. A038627(n) gives number of natural solutions of the equation m = pi(n*m). - Farideh Firoozbakht, Jan 09 2005
a(n) grows as exp(n)/n. Thus, a(n+1)/a(n) tends to e=exp(1) as n grows. - Max Alekseyev, Oct 15 2017

			pi(3059) = 437 and 3059/437 = 7, so a(7)=437.

Giovanni Resta, Table of n, a(n) for n = 2..50
S. W. Golomb, On the Ratio of N to pi(N), American Mathematical Monthly, 69 (1962), 36-37.
Eric Weisstein's World of Mathematics, Prime Counting Function.

Cf. A038623, A038624, A038625, A038627, A102281, A087237, A293529.

a(n) = limit of f^(k)(1) as k grows, where f(x)=A000720(n*x). Also, a(n) = f^(A293529(n))(1). - Max Alekseyev, Oct 11 2017

a(n) = A038625(n) / n. - Max Alekseyev, Oct 13 2023

a(24) from Farideh Firoozbakht, Jan 09 2005
Edited by N. J. A. Sloane at the suggestion of Chris K. Caldwell, Apr 08 2008
a(25)-a(32) from Max Alekseyev, Jul 18 2011, Oct 14 2017
a(33)-a(50) obtained from the values of A038625 computed by Jan Büthe. - Giovanni Resta, Aug 31 2018

A212213 Array read by antidiagonals: pi(n) + pi(k) - pi(n+k), where pi() = A000720.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 2, 1, 2, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 2, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0

Offset: 2

N. J. A. Sloane, May 04 2012

It is conjectured that pi(x) + pi(y) >= pi(x+y) for 1 < y <= x.

			Array begins:
  0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, ...
  0, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, ...
  0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, ...
  0, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, ...
  0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, ...
  1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, ...
  ...

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VII.5, p. 235.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
P. Erdős and J. L. Selfridge, Complete prime subsets of consecutive integers. Proceedings of the Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1971), pp. 1-14. Dept. Comput. Sci., Univ. Manitoba, Winnipeg, Man., 1971.

Cf. A000720, A047885, A047886, A060208, A212210-A212213.

t[n_, k_] := PrimePi[n] + PrimePi[k] - PrimePi[n + k]; Table[t[n - k + 2, k], {n, 0, 15}, {k, 2, n}] // Flatten (* Jean-François Alcover, Dec 31 2012 *)

cp's OEIS Frontend

A237284 Number of ordered ways to write 2*n = p + q with p, q and A000720(p) all prime.

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A291440 a(n) = pi(n^2) - pi(n)^2, where pi(n) = A000720(n).

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A332808 Fully multiplicative with a(p) = A332806(A000720(p)).

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A365400 a(n) = 64 + A000720(n) - A365339(n).

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Julia

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A212210 Triangle read by rows: T(n,k) = pi(n) + pi(k) - pi(n+k), n >= 1, 1 <= k <= n, where pi() = A000720().

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A214935 Index of the primes of A205827, A000720(A205827(n)).

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A237612 Least positive integer k such that A000720(k*n) is a square, or 0 if such a number k does not exist.

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A354202 Fully multiplicative with a(p^e) = A354200(A000720(p))^e.

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A038626 Smallest positive integer m such that m = pi(nm) = A000720(nm).