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.

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A181463 Numbers k such that 61 is the largest prime factor of k^2-1.

Original entry on oeis.org

60, 62, 121, 123, 184, 243, 245, 365, 367, 426, 428, 487, 489, 550, 609, 611, 794, 1036, 1099, 1160, 1219, 1221, 1343, 1463, 1585, 1646, 1709, 1768, 1770, 1951, 2014, 2073, 2256, 2319, 2439, 2441, 2500, 2561, 2624, 2807, 2927, 3173, 3537, 3539, 3659, 3781
Offset: 1

Views

Author

Artur Jasinski, Oct 21 2010

Keywords

Comments

Numbers k such that A076605(k) = 61.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(18) = 63774701665793; primepi(61) = 18.

Links

Crossrefs

Cf. A076605, A175607, A181447-A181462, A181464-A181470, A181568.

Programs

  • Magma
    [ n: n in [2..300000] | m eq 61 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 19 2011
  • Mathematica
    jj = 2^36*3^23*5^15*7^13*11^10*13^9*17^8*19^8*23^8*29^7*31^7*37^7*41^6 *43^6*47^6*53^6*59^6*61^6*67^6*71^5*73^5*79^5*83^5*89^5*97^5; rr = {}; n = 2; While[n < 14000000, If[GCD[jj, n^2 - 1] == n^2 - 1, k = FactorInteger[n^2 - 1]; kk = Last[k][[1]]; If[kk == 61, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==61&]
  • PARI
    is(n)=n=n^2-1; forprime(p=2, 59, n/=p^valuation(n, p)); n>1 && 61^valuation(n, 61)==n \\ Charles R Greathouse IV, Jul 01 2013

A181464 Numbers k such that 67 is the largest prime factor of k^2-1.

Original entry on oeis.org

66, 68, 133, 135, 202, 267, 269, 334, 401, 604, 671, 736, 805, 937, 939, 1004, 1006, 1205, 1272, 1274, 1341, 1473, 1475, 1540, 1609, 1676, 1741, 2078, 2143, 2145, 2210, 2279, 2545, 2547, 2746, 2813, 2815, 2882, 2949, 3081, 3349, 3550, 3552, 3751, 3887
Offset: 1

Views

Author

Artur Jasinski, Oct 21 2010

Keywords

Comments

Numbers k such that A076605(k) = 67.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(19) = 25640240468751; primepi(67) = 19.

Links

Crossrefs

Cf. A076605, A175607, A181447-A181463, A181465-A181470, A181568.

Programs

  • Magma
    [ n: n in [2..300000] | m eq 67 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 19 2011
  • Mathematica
    jj = 2^36*3^23*5^15*7^13*11^10*13^9*17^8*19^8*23^8*29^7*31^7*37^7*41^6 *43^6*47^6*53^6*59^6*61^6*67^6*71^5*73^5*79^5*83^5*89^5*97^5; rr = {}; n = 2; While[n < 14000000, If[GCD[jj, n^2 - 1] == n^2 - 1, k = FactorInteger[n^2 - 1]; kk = Last[k][[1]]; If[kk == 67, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==67&]
  • PARI
    is(n)=n=n^2-1; forprime(p=2, 61, n/=p^valuation(n, p)); n>1 && 67^valuation(n, 67)==n \\ Charles R Greathouse IV, Jul 01 2013

A181465 Numbers k such that 71 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

70, 141, 143, 214, 283, 285, 356, 425, 496, 569, 638, 709, 780, 782, 851, 853, 924, 993, 1135, 1208, 1277, 1279, 1561, 1563, 1703, 1847, 2058, 2129, 2131, 2344, 2413, 2626, 2699, 2839, 2841, 3054, 3265, 3267, 3336, 3338, 3409, 3478, 3480, 3551, 3620, 3691
Offset: 1

Views

Author

Artur Jasinski, Oct 21 2010

Keywords

Comments

Numbers k such that A076605(k) = 71.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(20) = 238178082107393; primepi(71) = 20.

Links

Crossrefs

Row 20 of A223701.
Cf. A076605, A175607, A181447-A181464, A181466-A181470, A181568.

Programs

  • Magma
    [ n: n in [2..300000] | m eq 71 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 19 2011
  • Mathematica
    jj = 2^36*3^23*5^15*7^13*11^10*13^9*17^8*19^8*23^8*29^7*31^7*37^7*41^6 *43^6*47^6*53^6*59^6*61^6*67^6*71^5*73^5*79^5*83^5*89^5*97^5; rr = {}; n = 2; While[n < 14000000, If[GCD[jj, n^2 - 1] == n^2 - 1, k = FactorInteger[n^2 - 1]; kk = Last[k][[1]]; If[kk == 71, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==71&]
  • PARI
    is(n)=n=n^2-1; forprime(p=2, 67, n/=p^valuation(n, p)); n>1 && 71^valuation(n, 71)==n \\ Charles R Greathouse IV, Jul 01 2013

A181466 Numbers k such that 73 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

72, 74, 145, 147, 218, 220, 291, 293, 364, 439, 512, 729, 731, 804, 875, 1021, 1023, 1167, 1169, 1240, 1313, 1315, 1459, 1461, 1607, 1678, 1680, 1751, 1826, 1899, 2045, 2116, 2262, 2481, 2483, 2554, 2702, 2773, 2848, 3067, 3284, 3359, 3576, 3649, 3722
Offset: 1

Views

Author

Artur Jasinski, Oct 21 2010

Keywords

Comments

Numbers k such that A076605(k) = 73.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(21) = 4573663454608289; primepi(73) = 21.

Links

Crossrefs

Row 21 of A223701
Cf. A076605, A175607, A181447-A181465, A181467-A181470, A181568.

Programs

  • Magma
    [ n: n in [2..300000] | m eq 73 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 21 2011
  • Magma
    p:=(97*89*83*79*73*71)^5 *(67*61*59*53*47*43*41)^6 *(37*31*29)^7 *(23*19*17)^8 *13^9 *11^10 *7^13 *5^15 *3^23 *2^36; [ n: n in [2..50000000] | p mod (n^2-1) eq 0 and (D[#D] eq 73 where D is PrimeDivisors(n^2-1)) ]; // Klaus Brockhaus, Feb 21 2011
  • Mathematica
    jj = 2^36*3^23*5^15*7^13*11^10*13^9*17^8*19^8*23^8*29^7*31^7*37^7*41^6 *43^6*47^6*53^6*59^6*61^6*67^6*71^5*73^5*79^5*83^5*89^5*97^5; rr = {}; n = 2; While[n < 3222617400, If[GCD[jj, n^2 - 1] == n^2 - 1, k = FactorInteger[n^2 - 1]; kk = Last[k][[1]]; If[kk == 73, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==73&]
  • PARI
    is(n)=n=n^2-1; forprime(p=2, 71, n/=p^valuation(n, p)); n>1 && 73^valuation(n, 73)==n \\ Charles R Greathouse IV, Jul 01 2013

A181467 Numbers k such that 79 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

78, 80, 157, 159, 236, 317, 473, 475, 552, 554, 631, 712, 791, 868, 870, 947, 949, 1026, 1028, 1105, 1184, 1421, 1737, 1739, 1816, 1897, 2053, 2134, 2211, 2213, 2369, 2450, 2529, 2685, 2687, 2843, 2924, 3001, 3161, 3477, 3554, 3870, 3949, 3951, 4186, 4188
Offset: 1

Views

Author

Artur Jasinski, Oct 21 2010

Keywords

Comments

Numbers k such that A076605(k) = 79.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(22) = 19182937474703818751; primepi(79) = 22.

Links

Crossrefs

Row 22 of A223701.
Cf. A076605, A175607, A181447-A181466, A181468-A181470, A181568.

Programs

  • Magma
    [ n: n in [2..300000] | m eq 79 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 21 2011
  • Magma
    p:=(97*89*83*79*73*71)^5 *(67*61*59*53*47*43*41)^6 *(37*31*29)^7 *(23*19*17)^8 *13^9 *11^10 *7^13 *5^15 *3^23 *2^36; [ n: n in [2..50000000] | p mod (n^2-1) eq 0 and (D[#D] eq 79 where D is PrimeDivisors(n^2-1)) ]; // Klaus Brockhaus, Feb 21 2011
  • Mathematica
    jj = 2^36*3^23*5^15*7^13*11^10*13^9*17^8*19^8*23^8*29^7*31^7*37^7*41^6 *43^6*47^6*53^6*59^6*61^6*67^6*71^5*73^5*79^5*83^5*89^5*97^5; rr = {}; n = 2; While[n < 3222617400, If[GCD[jj, n^2 - 1] == n^2 - 1, k = FactorInteger[n^2 - 1]; kk = Last[k][[1]]; If[kk == 79, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==79&]
  • PARI
    is(n)=n=n^2-1; forprime(p=2, 73, n/=p^valuation(n, p)); n>1 && 79^valuation(n, 79)==n \\ Charles R Greathouse IV, Jul 01 2013

A181468 Numbers k such that 83 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

82, 84, 165, 167, 248, 331, 414, 497, 499, 582, 665, 829, 831, 914, 995, 1080, 1161, 1246, 1327, 1329, 1495, 1576, 1825, 1910, 2076, 2157, 2159, 2323, 2406, 2408, 2738, 2821, 2906, 2989, 3070, 3238, 3319, 3485, 3568, 3651, 3653, 3817, 4149, 4234, 4481
Offset: 1

Views

Author

Artur Jasinski, Oct 21 2010

Keywords

Comments

Numbers k such that A076605(k) = 83.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(23) = 34903240221563713 = a(2082); pi(83) = 23.

Links

Crossrefs

Row 23 of A223701.
Cf. A000720, A076605, A175607, A181447-A181467, A181469-A181470, A181568.

Programs

  • Magma
    [ n: n in [2..300000] | m eq 83 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 21 2011
  • Magma
    p:=(97*89*83*79*73*71)^5 *(67*61*59*53*47*43*41)^6 *(37*31*29)^7 *(23*19*17)^8 *13^9 *11^10 *7^13 *5^15 *3^23 *2^36; [ n: n in [2..50000000] | p mod (n^2-1) eq 0 and (D[#D] eq 83 where D is PrimeDivisors(n^2-1)) ]; // Klaus Brockhaus, Feb 21 2011
  • Mathematica
    jj = 2^36*3^23*5^15*7^13*11^10*13^9*17^8*19^8*23^8*29^7*31^7*37^7*41^6 *43^6*47^6*53^6*59^6*61^6*67^6*71^5*73^5*79^5*83^5*89^5*97^5; rr = {}; n = 2; While[n < 3222617400, If[GCD[jj, n^2 - 1] == n^2 - 1, k = FactorInteger[n^2 - 1]; kk = Last[k][[1]]; If[kk == 83, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==83&]
  • PARI
    is(n)=n=n^2-1; forprime(p=2, 79, n/=p^valuation(n, p)); n>1 && 83^valuation(n, 83)==n \\ Charles R Greathouse IV, Jul 01 2013

A181469 Numbers n such that 89 is the largest prime factor of n^2-1.

Original entry on oeis.org

88, 90, 177, 179, 266, 355, 533, 535, 622, 624, 711, 713, 800, 802, 889, 1067, 1156, 1158, 1247, 1334, 1423, 1425, 1601, 1781, 1959, 2224, 2313, 2402, 2404, 2491, 2493, 2582, 2669, 2849, 3025, 3381, 3383, 3739, 3826, 4093, 4095, 4184, 4271, 4451, 4716
Offset: 1

Views

Author

Artur Jasinski, Oct 21 2010

Keywords

Comments

Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(24) = 332110803172167361; primepi(89) = 24.

Links

  • Lucas A. Brown, Table of n, a(n) for n = 1..2371 (terms 1..432 and 434..2371 from _Artur Jasinski_, term 433 from _Lucas A. Brown_, Oct 01 2022. This is the complete list of terms assuming an abc conjecture c < rad(abc)^2).
  • From _David A. Corneth_, Oct 02 2022: (Start)
  • To verify full I listed all 89-smooth numbers k that are a multiple of 89 below (inclusive) A175607(24) + 2. I then checked if k+2 is 89-smooth. If so, k+1 is a term. Then similarily I checked if k-2 is 89-smooth. If so, k-1 is a term.
  • Doing so found the 2371 terms from the b-file. All candidates have been checked completing the proof of full. (End)

Crossrefs

Cf. A175607, A181447-A181468, A181470, A181568.

Programs

  • Magma
    [ n: n in [2..300000] | m eq 89 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 21 2011
  • Magma
    p:=(97*89*83*79*73*71)^5 *(67*61*59*53*47*43*41)^6 *(37*31*29)^7 *(23*19*17)^8 *13^9 *11^10 *7^13 *5^15 *3^23 *2^36; [ n: n in [2..50000000] | p mod (n^2-1) eq 0 and (D[#D] eq 89 where D is PrimeDivisors(n^2-1)) ]; // Klaus Brockhaus, Feb 21 2011
  • Mathematica
    jj = 2^36*3^23*5^15*7^13*11^10*13^9*17^8*19^8*23^8*29^7*31^7*37^7*41^6 *43^6*47^6*53^6*59^6*61^6*67^6*71^5*73^5*79^5*83^5*89^5*97^5; rr = {}; n = 2; While[n < 3222617400, If[GCD[jj, n^2 - 1] == n^2 - 1, k = FactorInteger[n^2 - 1]; kk = Last[k][[1]]; If[kk == 89, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==89&]
  • PARI
    is(n)=n=n^2-1; forprime(p=2, 83, n/=p^valuation(n, p)); n>1 && 89^valuation(n, 89)==n \\ Charles R Greathouse IV, Jul 01 2013

A181448 Numbers k such that 5 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

4, 9, 11, 19, 26, 31, 49, 161
Offset: 1

Views

Author

Artur Jasinski, Oct 21 2010

Keywords

Comments

Numbers k such that A076605(k) = 3.
Sequence is finite and complete, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(3) = 161; primepi(5) = 3.

Crossrefs

Row 3 of A223701.
Cf. A076605, A175607, A181447, A181449-A181470, A181568.

Programs

  • Magma
    [ n: n in [2..200] | m eq 5 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 17 2011
  • Mathematica
    Select[Range[200], FactorInteger[#^2-1][[-1, 1]]==5&]
  • PARI
    is(n)=n=n^2-1; n>>=valuation(n,2); n/=3^valuation(n,3); n>1 && 5^valuation(n, 5)==n \\ Charles R Greathouse IV, Jul 01 2013

A181449 Numbers k such that 7 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

6, 8, 13, 15, 29, 41, 55, 71, 97, 99, 127, 244, 251, 449, 4801, 8749
Offset: 1

Views

Author

Artur Jasinski, Oct 21 2010

Keywords

Comments

Numbers k such that A076605(k) = 7.
Sequence is finite and complete, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(4) = 8749; primepi(7) = 4.

Crossrefs

Row 4 of A223701.
Cf. A076605, A175607, A181447, A181448, A181450-A181470, A181568.

Programs

  • Magma
    [ n: n in [2..9000] | m eq 7 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 17 2011
  • Mathematica
    Select[Range[9000], FactorInteger[#^2-1][[-1, 1]]==7&]
  • PARI
    is(n)=n=n^2-1; forprime(p=2, 5, n/=p^valuation(n, p)); n>1 && 7^valuation(n, 7)==n \\ Charles R Greathouse IV, Jul 01 2013

A181450 Numbers k such that 11 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

10, 21, 23, 34, 43, 65, 76, 89, 109, 111, 197, 199, 241, 351, 485, 769, 881, 1079, 6049, 19601
Offset: 1

Views

Author

Artur Jasinski, Oct 21 2010

Keywords

Comments

Numbers k such that A076605(k) = 11.
Sequence is finite and complete, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(5) = 19601; primepi(11) = 5.

Crossrefs

Row 5 of A223701.
Cf. A076605, A175607, A181447-A181449, A181451-A181470, A181568.

Programs

  • Magma
    [ n: n in [2..20000] | m eq 11 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 18 2011
  • Mathematica
    Select[Range[20000], FactorInteger[#^2-1][[-1, 1]]==11&]
  • PARI
    is(n)=n=n^2-1; forprime(p=2, 7, n/=p^valuation(n, p)); n>1 && 11^valuation(n, 11)==n \\ Charles R Greathouse IV, Jul 01 2013
Previous Showing 21-30 of 42 results. Next

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