A247600 -id:A247600 - cp's OEIS frontend

A281197 Least number m such that A247600(m) = n.

9, 7, 6, 998, 5, 6331, 15866, 39860, 100135, 251616, 636787, 1617099, 4124188, 10553076, 27066256, 69709243, 179992154, 465769460, 1208198239, 3140421185, 8179001679, 21338684878, 55762147707, 145935688930, 382465571795, 1003652346718

Offset: 5

Seiichi Manyama, Jan 17 2017

			pi(5*9) = 9 + 5.
pi(6*7) = 7 + 6.
pi(7*6) = 6 + 7.
pi(8*998) = 998 + 8.
pi(9*5) = 5 + 9.
...
pi(15*636787) = 636787 + 15.
pi(16*1617099) = 1617099 + 16.

Cf. A000720, A247600.

pi(n*a(n)) = a(n) + n.

a(19)-a(28) from Chai Wah Wu, May 03 2018

a(29)-a(30) from Chai Wah Wu, May 08 2018

A281196 Number of n such that A247600(n) = 5.

Original entry on oeis.org

9, 10, 11, 12, 13, 14, 19, 20, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 39, 41

Offset: 1

Seiichi Manyama, Jan 17 2017

Conjecture: This sequence has just 22 terms shown.

Sequence is finite and complete. Since pi(x) < 1.25506*x/log(x), it follows that for x > 106, pi(5*x) < x + 5. - Chai Wah Wu, May 01 2018, May 14 2020.

			9 is a term since pi(5*9) = 14 = 9 + 5, and pi(m*9) = 9 + m for no m < 5.

Cf. A000720, A247600, A281197.

pi(5*a(n)) = a(n) + 5.

A247824 Least positive integer m such that m + n divides prime(m) + prime(n).

Original entry on oeis.org

1, 5, 5, 5, 2, 2, 38, 16, 40, 12, 13, 1, 11, 1, 11, 4, 35, 38, 35, 35, 38, 35, 35, 36, 31, 31, 33, 33, 36, 36, 25, 25, 2, 25, 4, 3, 4, 6, 6, 8, 222, 8, 95, 223, 99, 98, 95, 88, 222, 94, 93, 94, 95, 92, 226, 88, 83, 92, 225, 92

Offset: 1

Zhi-Wei Sun, Sep 24 2014

nonn

Conjecture: a(n) exists for any n > 0. Moreover, a(n) < n*(n-1) for all n > 2. - Zhi-Wei Sun, Sep 25 2014
A247869(n) = (prime(a(n)) + prime(n)) / (a(n) + n). - Reinhard Zumkeller, Sep 27 2014
I have verified the conjecture for n up to 10^5, and noted that max{a(n): n=1..10^5} = a(79276) = 3141281384 > 3*10^9. - Zhi-Wei Sun, Oct 08 2014
I would like to offer 500 US dollars as the prize for the first proof of the above conjecture. - Zhi-Wei Sun, Feb 24 2018
Chang Zhang (a student of Nanjing Univ.) has verified the conjecture for n up to 4*10^5. For example, a(337647) = 21342496785. - Zhi-Wei Sun, Jun 22 2020

			a(2) = 5 since 5 + 2 = 7 divides prime(5) + prime(2) = 11 + 3 = 14.
a(10409) = 69804276 since 69804276 + 10409 = 69814685 divides prime(10409) + prime(69804276) = 109481 + 1396184219 = 1396293700 = 20*69814685.
a(35980) = 180302246 since 35980 + 180302246 = 180338226 divides prime(35980) + prime(180302246) = 427727 + 3786675019 = 3787102746 = 21*180338226.
a(79276) = 3141281384 since 79276 + 3141281384 = 3141360660 divides prime(79276) + prime(3141281384) = 1010431 + 75391645409 = 75392655840 = 24*3141360660.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100000
Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685, 2014.
Zhi-Wei Sun, m+n divides prime(m)+prime(n) for some n>0, a message to Number Theory List, Sept. 27, 2014.
Zhi-Wei Sun, A new theorem on the prime-counting function, Ramanujan J. 42(2017), no.1, 59-67.

Cf. A000040, A247600, A247793.

import Data.List (genericIndex)
a247824 n = genericIndex a247824_list (n - 1)
a247824_list = f ips where
   f ((x, p) : xps) = head
     [y | (y, q) <- ips, (p + q) `mod` (x + y) == 0] : f xps
   ips = zip [1..] a000040_list
-- Reinhard Zumkeller, Sep 27 2014

Do[m=1;Label[aa];If[Mod[Prime[m]+Prime[n],m+n]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,60}]
lpi[n_]:=Module[{k=1,p=Prime[n]},While[!Divisible[p+Prime[k],k+n], k++]; k]; Array[lpi,60] (* Harvey P. Dale, Apr 23 2015 *)

a(n) = {m = 1; while ((prime(m) + prime(n)) % (m + n), m++); m;} \\ Michel Marcus, Sep 25 2014

a(n)=my(p=prime(n),m); forprime(q=2,, if((p+q)%(n+m++)==0, return(m))) \\ Charles R Greathouse IV, Sep 25 2014

A247603 Least integer m > 0 with pi(m*n) = sigma(m), where sigma(m) is the sum of all positive divisors of m.

Original entry on oeis.org

1, 2, 23, 61, 8, 22, 16, 12, 202, 386, 30, 36, 174, 10745, 1684, 2804, 1616, 40006, 6764, 996, 5775, 8131355, 19974, 11264, 4446, 27882, 4848, 32466, 162712, 532313373, 2341816, 30864, 14544, 63696, 2880, 390990, 135200, 133992, 1331840, 11621646, 117990

Offset: 2

Zhi-Wei Sun, Sep 21 2014

nonn

Conjecture: a(n) exists for any n > 1.

			a(3) = 2 since pi(3*2) = 3 = sigma(2), and pi(3*1) = 2 > sigma(1) = 1.

Hiroaki Yamanouchi, Table of n, a(n) for n = 2..53
Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685, 2014.

Cf. A000203, A000720, A247600, A247601, A247602, A247604, A247672, A247673.

Do[m=1;Label[aa];If[PrimePi[n*m]==DivisorSigma[1,m],Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];
Label[bb];Continue,{n,2,30}]

a(31)-a(42) from Hiroaki Yamanouchi, Oct 04 2014

A247604 Least integer m > 0 with pi(m*n) = sigma(m+n), where pi(.) and sigma(.) are given by A000720 and A000203.

Original entry on oeis.org

18, 11, 360, 251, 168, 36, 6, 285, 1185, 792, 29, 11, 245078, 5, 1869, 46074, 573, 42863, 11, 5, 8129, 60806, 1443, 452, 15, 39298437, 386891, 1041920, 1290489, 17630, 35569, 10, 8174777, 3152500, 4291325, 57880072, 55991485, 127358, 93462807, 93314912

Offset: 5

Zhi-Wei Sun, Sep 21 2014

nonn

Conjecture: a(n) exists for every n = 5,6,...

			a(5) = 18 since pi(5*18) = 24 = sigma(5+18).

Zhi-Wei Sun and Hiroaki Yamanouchi, Table of n, a(n) for n = 5..52 (terms a(5)-a(40) from Zhi-Wei Sun)
Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685, 2014.

Cf. A000203, A000720, A247600, A247601, A247602, A247603, A247672, A247673.

Do[m=1;Label[aa];If[PrimePi[n*m]==DivisorSigma[1,m+n],Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];
Label[bb];Continue,{n,5,40}]

a(41)-a(44) from Hiroaki Yamanouchi, Oct 04 2014

A247601 Least positive integer m with pi(m*n) = phi(m), where pi(.) is the prime-counting function and phi(.) is Euler's totient function.

Original entry on oeis.org

2, 1, 13, 31, 73, 181, 443, 2249, 238839, 6473, 30001, 40123, 108539, 251707, 637321, 7554079, 4124437, 241895689, 27067097, 69709723, 179992919, 1019958623, 1208198863, 3140421743, 8179002173

Offset: 1

Zhi-Wei Sun, Sep 21 2014

Conjecture: a(n) exists for any n > 0.

This is motivated by Golomb's result that for any n > 1 there is a positive integer m with mn/pi(mn) = n (i.e., pi(mn) = m).

			a(3) = 13 since pi(3*13) = 12 = phi(13).

S. W. Golomb, On the Ratio of N to pi(N), American Mathematical Monthly, 69 (1962), 36-37.
Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685, 2014.

Cf. A000010, A000720, A247600, A247602, A247603, A247604, A247672, A247673.

Do[m=1;Label[aa];If[PrimePi[n*m]==EulerPhi[m],Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];
Label[bb];Continue,{n,1,18}]
Table[m = 1;
While[PrimePi[n*m] != EulerPhi[m], m++]; m, {n,1,12}] (* Robert Price, Sep 08 2019 *)

a(19)-a(25) from Hiroaki Yamanouchi, Oct 04 2014

A247602 Least positive integer m with pi(m*n) = phi(m+n), where pi(.) is the prime-counting function and phi(.) is Euler's totient function.

Original entry on oeis.org

3, 2, 1, 91, 6, 5, 1, 5, 1, 8041, 15870, 39865, 1, 251625, 637064, 1829661, 4124240, 10553093, 1, 69709253, 179992156, 465769749, 1210576800, 3140421235, 13974959892

Offset: 1

Zhi-Wei Sun, Sep 21 2014

Conjecture: a(n) exists for any n > 0.

			a(1) = 3 since pi(1*3) = 2 = phi(1+3).

Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685, 2014.

Cf. A000010, A000720, A247600, A247601, A247603, A247604, A247672, A247673.

Do[m=1;Label[aa];If[PrimePi[n*m]==EulerPhi[m+n],Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];
Label[bb];Continue,{n,1,20}]
Table[m = 1;
While[PrimePi[n*m] != EulerPhi[m + n], m++]; m, {n, 1, 13}] (* Robert Price, Sep 08 2019 *)

a(n) = {my(m = 1); while (primepi(m*n) != eulerphi(m+n), m++); m;} \\ Michel Marcus, Sep 22 2014

a(21)-a(25) from Hiroaki Yamanouchi, Oct 04 2014

A247673 Least integer m > 0 with pi(m*n) = sigma(m) + sigma(n), where pi(.) and sigma(.) are given by A000720 and A000203 respectively.

Original entry on oeis.org

23, 47, 359, 25, 11, 33, 9, 17, 182, 11, 15, 304, 12, 160, 6105, 444, 22676, 408, 5, 60, 8, 17888, 9, 125526, 1616818, 334976, 22584, 19548, 10, 286780, 21540, 6698792, 640720, 2466378, 75999272, 646104, 573678, 801525615, 1116040868, 3565308, 127408112

Offset: 5

Zhi-Wei Sun, Sep 22 2014

nonn

Conjecture: a(n) exists for every n = 5, 6, ... .

			a(5) = 23 since pi(5*23) = 30 = sigma(5) + sigma(23).

Zhi-Wei Sun and Hiroaki Yamanouchi, Table of n, a(n) for n = 5..53 (terms a(5)-a(41) from Zhi-Wei Sun)
Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685, 2014.

Cf. A000203, A000720, A247600, A247601, A247602, A247603, A247604, A247672.

Do[m=1; Label[aa]; If[PrimePi[m*n]==DivisorSigma[1,m]+DivisorSigma[1,n], Print[n, " ", m]; Goto[bb]]; m=m+1; Goto[aa]; Label[bb]; Continue, {n, 5, 41}]

a(42)-a(45) from Hiroaki Yamanouchi, Oct 04 2014

A247672 Least integer m > 0 with pi(m*n) = phi(m) + phi(n), where pi(.) is the prime-counting function and phi(.) is Euler's totient function.

Original entry on oeis.org

6, 2, 2, 23, 3, 1, 3, 1033, 2, 6449, 15887, 1, 100169, 268393, 636917, 2113589, 70324093, 1, 27852457, 78848479, 2, 468329417, 4, 1, 10220118551

Offset: 1

Zhi-Wei Sun, Sep 22 2014

Conjecture: a(n) exists for every n > 0.

			a(1) = 6 since pi(6) = 3 = phi(1) + phi(6), and pi(1*m) = phi(1) + phi(m) for no m < 6.

Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685 [math.NT], 2014-2017.

Cf. A000010, A000720, A247600, A247601, A247602, A247603, A247604, A247673.

Table[m = 1; While[PrimePi[n*m] != EulerPhi[m] + EulerPhi[n], m++]; m, {n, 1, 12}] (* Robert Price, Sep 08 2019 *)

use ntheory ":all"; for my $n (1..16) { my $m=1; $m++ until (prime_count($m*$n) == euler_phi($m) + euler_phi($n)); say "$n $m"; } # Dana Jacobsen, Mar 07 2023

a(19)-a(25) from Hiroaki Yamanouchi, Oct 04 2014

A247793 Least integer m > 0 such that pi(m*n) divides prime(m) + prime(n), where pi(x) denotes the number of primes not exceeding x.

Original entry on oeis.org

2, 1, 75, 10, 18, 1, 75, 41, 58, 2, 94, 107, 14, 13, 2, 14, 14, 1, 84, 527, 124, 715, 13, 4, 1, 4, 276, 310, 2, 4, 11216, 3074, 3470, 14, 2, 15, 5, 947, 538839, 2, 8, 2, 1592, 4, 8, 16813, 2293, 1, 2755, 3007, 3272, 32203, 5357440, 6, 17, 17, 374252, 9, 17, 6905

Offset: 1

Zhi-Wei Sun, Sep 23 2014

nonn

Conjecture: a(n) exists for any n > 0.

			a(4) = 10 since pi(4*10) = 12 divides prime(4) + prime(10) = 7 + 29 = 36.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100
Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685, 2014.

Cf. A000040, A000720, A247600, A247601, A247602, A247603, A247604, A247672, A247673.

a247793_list = 2 : f (zip [2..] $ tail a000040_list) where
   f ((x, p) : xps) = m : f xps where
     m = head [y | y <- [1..], (p + a000040 y) `mod` a000720 (x * y) == 0]
-- Reinhard Zumkeller, Sep 24 2014

Do[m=1;Label[aa];If[m*n>1&&Mod[Prime[m]+Prime[n],PrimePi[m*n]]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,60}]

cp's OEIS Frontend

A281197 Least number m such that A247600(m) = n.

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A281196 Number of n such that A247600(n) = 5.

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A247824 Least positive integer m such that m + n divides prime(m) + prime(n).

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A247603 Least integer m > 0 with pi(m*n) = sigma(m), where sigma(m) is the sum of all positive divisors of m.

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A247604 Least integer m > 0 with pi(m*n) = sigma(m+n), where pi(.) and sigma(.) are given by A000720 and A000203.

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A247601 Least positive integer m with pi(m*n) = phi(m), where pi(.) is the prime-counting function and phi(.) is Euler's totient function.

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A247602 Least positive integer m with pi(m*n) = phi(m+n), where pi(.) is the prime-counting function and phi(.) is Euler's totient function.

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A247673 Least integer m > 0 with pi(m*n) = sigma(m) + sigma(n), where pi(.) and sigma(.) are given by A000720 and A000203 respectively.

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