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.

Showing 1-7 of 7 results.

A247600 Least positive integer m with pi(m*n) = m + n, where pi(x) denotes the number of primes not exceeding x.

Original entry on oeis.org

9, 7, 6, 998, 5, 5, 5, 5, 5, 5, 636787, 1617099, 4124188, 10553076, 5, 5, 179992154, 465769460, 1208198239, 3140421185, 5, 5, 5, 145935688930, 5, 5, 5, 5, 5, 5, 5, 5, 5
Offset: 5

Views

Author

Zhi-Wei Sun, Sep 21 2014

Keywords

Comments

The author proved that a(n) exists for every n >= 5.
a(39) = a(41) = 5. - Chai Wah Wu, Jun 06 2024

Examples

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

Links

Crossrefs

Cf. A247601, A247602, A247603, A247604, A000720, A281197.

Programs

  • Mathematica
    Do[m=1;Label[aa];If[PrimePi[n*m]==m+n,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,5,21}]
Table[m = 1; While[PrimePi[m*n] != m + n, m++]; m, {n, 5, 14}] (* Robert Price, Mar 20 2019 *)

Extensions

a(22)-a(37) from Chai Wah Wu, May 03 2018

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

Views

Author

Zhi-Wei Sun, Sep 21 2014

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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}]

Extensions

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

Views

Author

Zhi-Wei Sun, Sep 21 2014

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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}]

Extensions

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

Views

Author

Zhi-Wei Sun, Sep 21 2014

Keywords

Comments

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).

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

a(19)-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

Views

Author

Zhi-Wei Sun, Sep 22 2014

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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}]

Extensions

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

Views

Author

Zhi-Wei Sun, Sep 22 2014

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

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

Extensions

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

Views

Author

Zhi-Wei Sun, Sep 23 2014

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Haskell
    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
  • Mathematica
    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}]
Showing 1-7 of 7 results.

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