A247672 -id:A247672 - cp's OEIS frontend

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

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

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}]

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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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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.

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