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.

User: Tim Johannes Ohrtmann

Tim Johannes Ohrtmann's wiki page.

Tim Johannes Ohrtmann has authored 162 sequences. Here are the ten most recent ones:

A349030 Lucas-Carmichael numbers with 11 prime factors.

Original entry on oeis.org

20576473996736735, 42380075646230399, 75943207554554879, 83668951228080959, 96195222056687039, 116436396482735615, 132525862783734959, 134052021887096159, 162544912900261199, 175900784368936319, 186326804496197519, 190523141606006495, 196467189590024639
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Nov 06 2021

Keywords

Examples

			20576473996736735 = 5*7*11*17*23*31*47*53*71*107*233 and 6, 8, 12, 18, 24, 32, 48, 54, 72, 108, and 234 all divide 20576473996736736.

Links

Crossrefs

Intersection of A006972 and A069272.
Cf. A216928 (least Lucas-Carmichael number with n prime factors).
Cf. A216925, A216926, A216927, A217002, A217003, A217091, A349028, A349029 (Lucas-Carmichael numbers with 3-10 prime factors).

Programs

  • PARI
    is(n)={omega(n)==11&&is_A006972(n)}

A349029 Lucas-Carmichael numbers with 10 prime factors.

Original entry on oeis.org

989565001538399, 1250312791224959, 1419432982021439, 1518134614712639, 2240225337903839, 2493922560242399, 2708548708646879, 2786001880066559, 2807577905060159, 2808521396058455, 3157015238986895, 3210972445532159, 3221015190555239, 3407706183722399, 3614740529402519
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Nov 06 2021

Keywords

Examples

			989565001538399 = 11*13*17*19*29*31*41*47*83*149 and 12, 14, 18, 20, 30, 32, 42, 48, 84, and 150 all divide 989565001538400.

Links

Crossrefs

Intersection of A006972 and A046314.
Cf. A216928 (least Lucas-Carmichael number with n prime factors).
Cf. A216925, A216926, A216927, A217002, A217003, A217091, A349028, A349030 (Lucas-Carmichael numbers with 3-9 and 11 prime factors).

Programs

  • PARI
    is(n)={omega(n)==10&&is_A006972(n)}

A349028 Lucas-Carmichael numbers with 9 prime factors.

Original entry on oeis.org

14563696180319, 16569718534655, 20203946790335, 22034564147519, 23315834862719, 23889526894079, 27074874805055, 28932092649215, 31534433588735, 34236981827279, 34249223161439, 45373136257295, 45593377151399, 50103079391519, 50415330959279, 50683388926247
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Nov 06 2021

Keywords

Examples

			14563696180319 = 11*13*17*23*29*41*47*59*79 and 12, 14, 18, 24, 30, 42, 48, 60, and 80 all divide 14563696180320.

Links

Crossrefs

Intersection of A006972 and A046312.
Cf. A216928 (least Lucas-Carmichael number with n prime factors).
Cf. A216925, A216926, A216927, A217002, A217003, A217091, A349029, A349030 (Lucas-Carmichael numbers with 3-8, 10 and 11 prime factors).

Programs

  • PARI
    is(n)={omega(n)==9&&is_A006972(n)}

A338525 Numbers k such that (11^k + 6^k)/17 is prime.

Original entry on oeis.org

5, 7, 107, 383, 17359, 21929, 26393
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Nov 01 2020

Keywords

Comments

All terms are prime.
The corresponding primes are 9931, 1162771, ...

Links

Crossrefs

Cf. A057177, A125957, A128070, A224501, A128340.

Programs

  • Magma
    [n: n in [1..10000] |IsPrime((11^n + 6^n)/17)]
  • Mathematica
    Select[Range[1, 10000], PrimeQ[(11^# + 6^#)/17] &]
  • PARI
    for(n=1, 10000, if(isprime((11^n + 6^n)/17), print1(n, ", ")))

A338443 Carmichael numbers with 11 prime factors.

Original entry on oeis.org

60977817398996785, 105083995864811041, 107473646345582881, 132819104923908481, 145671955835893201, 161802381510126721, 165167398073764801, 206063729626916161, 263076030916096321, 292433912163313921, 292561243007134465, 337365329710615921, 388219799621120545
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Oct 28 2020

Keywords

Examples

			60977817398996785 = 5*7*17*19*23*37*53*73*79*89*233 and 4, 6, 16, 18, 22, 36, 52, 72, 78, 88, 232 all divide 60977817398996784.

Links

Crossrefs

Cf. A002997 (Carmichael numbers).
Cf. A006931 (Least Carmichael number with n prime factors).
Cf. A299710 (Number of terms less than 10^n).
Cf. A087788, A074379, A112428, A112429, A112430, A112431, A112432, A338442 (Carmichael numbers with 3-10 prime factors).

Programs

  • PARI
    is(n)={omega(n)==11&&is_A002997(n)}

Formula

Equals A002997 intersect A069272.

A338442 Carmichael numbers with 10 prime factors.

Original entry on oeis.org

1436697831295441, 1493812621027441, 2094319836529921, 2349991949342401, 2842648863161185, 2859959706040801, 3455134500424321, 3871703982953521, 4177950872896801, 4289150794129201, 4937378437571041, 5071419883911745, 5778659093725441, 6665161459969441, 6682056104892961
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Oct 28 2020

Keywords

Examples

			1436697831295441 = 11*13*19*29*31*37*41*43*71*127 and 10, 12, 18, 28, 30, 36, 40, 42, 70, 126 all divide 1436697831295440.

Links

Crossrefs

Cf. A002997 (Carmichael numbers).
Cf. A006931 (Least Carmichael number with n prime factors).
Cf. A299710 (Number of terms less than 10^n).
Cf. A087788, A074379, A112428, A112429, A112430, A112431, A112432, A338443 (Carmichael numbers with 3-9 and 11 prime factors).

Programs

  • PARI
    is(n)={omega(n)==10&&is_A002997(n)}

Formula

Equals A002997 intersect A046314.

A338412 Numbers k such that k * 20^k + 1 is prime.

Original entry on oeis.org

3, 6207, 8076, 22356, 151456
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Oct 25 2020

Keywords

Comments

a(6) > 219976.

Links

Crossrefs

Numbers k such that k * b^k + 1 is prime: A006093 (b=1), A005849 (b=2), A006552 (b=3), A007646 (b=4), A242176 (b=6), A242177 (b=7), A242178 (b=8), A265013 (b=9), A007647(b=10), A242196(b=12), A242197 (b=14), A242198 (b=15), A242199 (b=16), A007648 (b=18), this sequence (b=20).

Programs

  • Magma
    [n: n in [1..10000] |IsPrime(n*20^n+1)]
  • Mathematica
    Select[Range[1, 10000], PrimeQ[n*20^n+1] &]
  • PARI
    for(n=1, 10000, if(isprime(n*20^n+1), print1(n, ", ")))

A328660 Numbers k such that (10^k + 7^k)/17 is prime.

Original entry on oeis.org

3, 5, 11, 19, 1259, 1399, 2539, 2843, 5857, 10589
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Oct 24 2019

Keywords

Comments

All terms are odd primes. Proof: a(n) cannot be even, because (10^(2*k) + 7^(2*k))/17 is not an integer. If odd number k = x*y, then (10^x + 7^x) and (10^y + 7^y) are nontrivial factors of (10^(x*y) + 7^(x*y)). In conclusion, a(n) must be odd and prime. - Daniel Suteu, Jan 22 2020
The corresponding primes are 79, 6871, 5998666279, 588905817363845479, ...
a(11) > 60000. - Michael S. Branicky, Jul 11 2024

Crossrefs

Cf. A001562, A128069, A217095.

Programs

  • Magma
    [p: p in PrimesUpTo(1000) | IsPrime((10^p+7^p) div 17)]; // Modified by Jinyuan Wang, Jan 22 2020
  • Mathematica
    Select[Table[Prime[n], {n, 500}], PrimeQ[(10^#+7^#)/17] &] (* Modified by Jinyuan Wang, Jan 22 2020 *)
  • PARI
    forprime(k=3, 10000, if(isprime((10^k+7^k)/17), print1(k, ", ")))

A301510 Smallest positive number b such that ((b+1)^prime(n) + b^prime(n))/(2*b + 1) is prime, or 0 if no such b exists.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 3, 16, 1, 11, 6, 37, 1, 9, 120, 9, 1, 2, 67, 16, 1, 26, 103, 12, 60, 1, 239, 4, 40, 2, 44, 174, 33, 1, 3, 260, 114, 1, 161, 70, 1, 3, 2, 3, 50, 45, 472, 228, 183, 66, 37, 7, 122, 235, 68, 102, 294, 8, 13, 1, 40, 62, 143, 1, 61, 7
Offset: 2

Views

Author

Tim Johannes Ohrtmann, Mar 22 2018

Keywords

Comments

Conjecture: a(n) > 0 for every n > 1.
Records: 1, 4, 16, 37, 120, 239, 260, 472, 917, 1539, 6633, 7050, 12818, ..., which occur at n = 2, 10, 13, 17, 20, 32, 41, 52, 72, 128, 171, 290, 309, ... - Robert G. Wilson v, Jun 16 2018

Examples

			a(10) = 4 because (5^29 + 4^29)/9 = 2149818248341 is prime and (2^29 + 1^29)/3, (3^29 + 2^29)/5 and (4^29 + 3^29)/7 are all composite.

Links

Crossrefs

Numbers n such that ((b+1)^n + b^n)/(2*b + 1) is prime for b = 1 to 18: A000978, A057469, A128066, A128335, A128336, A187805, A181141, A187819, A217095, A185239, A213216, A225097, A224984, A221637, A227170, A228573, A227171, A225818.
Cf. A058013, A103794, A222119, A247244, A250201.

Programs

  • Mathematica
    Table[p = Prime[n]; k = 1; While[q = ((b+1)^n+b^n)/(2*b+1); ! PrimeQ[q], k++]; k, {n, 200}]
f[n_] := Block[{b = 1, p = Prime@ n}, While[! PrimeQ[((b +1)^p + b^p)/(2b +1)], b++]; b]; Array[f, 70, 2] (* Robert G. Wilson v, Jun 13 2018 *)
  • PARI
    for(n=2, 200, b=0; until(isprime((((b+1)^prime(n)+b^prime(n))/(2*b+1))), b++); print1(b,", ")) \\ corrected by Eric Chen, Jun 06 2018

Formula

a(n) = A250201(2*prime(n)) - 1 for n >= 2. - Eric Chen, Jun 06 2018

A301369 Numbers k such that (9^k + 7^k)/16 is prime.

Original entry on oeis.org

3, 107, 197, 2843, 3571, 4451, 31517, 44819
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Mar 19 2018

Keywords

Comments

All terms are prime.
The corresponding primes are 67, 79401467172644850007356716446663549450843749853576087044440771380676673442288169290888310265443988907, ...

Crossrefs

Cf. A057175, A125956, A211409, A128339, A187819.

Programs

  • Magma
    [n: n in [1..10000] |IsPrime((9^n+7^n)/16)]
  • Maple
    select(n->isprime((9^n+7^n)/16),[seq(n,n=1..10000,2)]); # Muniru A Asiru, Mar 27 2018
  • Mathematica
    Select[Range[1, 10000], PrimeQ[(9^n+7^n)/16] &]
  • PARI
    forprime(n=3, 10000, if(isprime((9^n+7^n)/16), print1(n, ", ")))

Extensions

a(7) from Michael S. Branicky, Apr 29 2023
a(8) from Michael S. Branicky, Jun 22 2024

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