A175607 -id:A175607 - cp's OEIS frontend

A175904 Numbers m for which the set of prime divisors of m^2-1 is unique.

2, 3, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 27, 28, 30, 32, 33, 36, 38, 39, 40, 42, 44, 45, 46, 47, 48, 50, 52, 54, 57, 58, 60, 62, 63, 64, 66, 68, 69, 70, 72, 73, 74, 75, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 93, 94, 95, 96, 98, 99

Offset: 1

Artur Jasinski, Oct 12 2010, Oct 21 2010

nonn

Complement of A175903. A proof for the presence of the first 63 terms (for which the largest prime divisor is < 100) follows along the lines of the comment in A175607.

			The unique prime factor sets are {3} (m=2), {2} (m=3), {5,7} (m=6), {3,7} (m=8), {2,5} (m=9) etc.

Cf. A175607, A175901 - A175904, A181447 - A181471.

aa = {}; bb = {}; cc = {}; ff = {}; Do[k = n^2 - 1; kk = FactorInteger[k]; b = {}; Do[AppendTo[b, kk[[m]][[1]]], {m, 1, Length[kk]}]; dd = Position[aa, b]; If[dd == {}, AppendTo[cc, n]; AppendTo[aa, b], AppendTo[ff, n]; AppendTo[bb, cc[[dd[[1]][[1]]]]]], {n, 2, 1000000}]; jj=Table[n,{2,99}]; ss=Union[bb,ff]; Take[Complement[jj,ss],63] (*Artur Jasinski*)

A181452 Numbers k such that 17 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

16, 33, 35, 50, 67, 69, 101, 103, 118, 120, 169, 188, 239, 271, 307, 339, 441, 511, 545, 577, 749, 883, 1121, 1189, 1376, 1429, 1665, 1871, 2024, 2177, 2311, 2449, 2549, 3401, 4115, 4861, 4999, 5201, 9827, 11663, 24751, 28799, 57121, 62425, 74359, 388961, 672281

Offset: 1

Artur Jasinski, Oct 21 2010

Numbers k such that A076605(k) = 17.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(7) = 672281; primepi(17) = 7.

Artur Jasinski, Table of n, a(n) for n = 1..47

Cf. A076605, A175607, A181447-A181451, A181453-A181470, A181568.

[ n: n in [2..350000] | m eq 17 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 19 2011

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..700000] | p mod (n^2-1) eq 0 and (D[#D] eq 17 where D is PrimeDivisors(n^2-1)) ]; // Klaus Brockhaus, Feb 24 2011

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 < 700000, If[GCD[jj, n^2 - 1] == n^2 - 1, k = FactorInteger[n^2 - 1]; kk = Last[k][[1]]; If[kk == 17, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[680000], FactorInteger[#^2-1][[-1, 1]]==17&]

is(n)=n=n^2-1; forprime(p=2, 13, n/=p^valuation(n, p)); n>1 && 17^valuation(n, 17)==n \\ Charles R Greathouse IV, Jul 01 2013

A181453 Numbers k such that 19 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

18, 20, 37, 39, 56, 77, 113, 134, 151, 153, 170, 191, 246, 265, 305, 324, 341, 362, 379, 417, 419, 571, 626, 647, 664, 685, 721, 799, 911, 951, 989, 1025, 1616, 1937, 2431, 2661, 2889, 3041, 3079, 3212, 3457, 3970, 4751, 4863, 5851, 6271, 6499, 8399, 11551, 11857

Offset: 1

Artur Jasinski, Oct 21 2010

Numbers k such that A076605(k) = 19.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(8) = 23718421; primepi(19) = 8.

Artur Jasinski, Table of n, a(n) for n = 1..72

Cf. A076605, A175607, A181447-A181452, A181454-A181470, A181568.

[ n: n in [2..300000] | m eq 19 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 18 2011

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..24000000] | p mod (n^2-1) eq 0 and (D[#D] eq 19 where D is PrimeDivisors(n^2-1)) ]; // Klaus Brockhaus, Feb 24 2011

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 < 24000000, If[GCD[jj, n^2 - 1] == n^2 - 1, k = FactorInteger[n^2 - 1]; kk = Last[k][[1]]; If[kk == 19, AppendTo[rr, n]]]; n++ ]; rr
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==19&]

is(n)=n=n^2-1; forprime(p=2, 17, n/=p^valuation(n, p)); n>1 && 19^valuation(n, 19)==n \\ Charles R Greathouse IV, Jul 01 2013

A181455 Numbers k such that 29 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

28, 57, 59, 86, 115, 144, 146, 175, 231, 233, 289, 349, 376, 407, 436, 463, 494, 521, 579, 639, 666, 755, 811, 987, 1101, 1103, 1217, 1275, 1451, 1565, 1567, 1681, 2029, 2089, 2551, 2872, 2899, 3191, 3249, 3365, 4001, 4003, 4351, 4409, 4523, 4929, 5279

Offset: 1

Artur Jasinski, Oct 21 2010

Numbers k such that A076605(k) = 29.
Sequence is finite. For proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(10) = 354365441; primepi(29) = 10.

Artur Jasinski, Table of n, a(n) for n = 1..126

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

[ n: n in [2..6000] | m eq 29 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 17 2011

Select[Range[5600],FactorInteger[#^2-1][[-1,1]]==29&]  (* Harvey P. Dale, Feb 16 2011 *)

is(n)=n=n^2-1; forprime(p=2, 23, n/=p^valuation(n, p)); n>1 && 29^valuation(n, 29)==n \\ Charles R Greathouse IV, Jul 01 2013

A181457 Numbers k such that 37 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

36, 38, 73, 75, 149, 186, 221, 223, 260, 295, 369, 371, 406, 443, 482, 519, 593, 628, 776, 813, 815, 961, 1000, 1072, 1259, 1331, 1333, 1405, 1407, 1444, 1481, 1701, 1814, 1849, 1886, 1923, 1999, 2071, 2367, 2591, 2663, 2737, 2887, 2959, 3329, 3331, 3403

Offset: 1

Artur Jasinski, Oct 21 2010

Numbers k such that A076605(k) = 37.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(12) = 9447152318; primepi(37) = 12.

Artur Jasinski, Table of n, a(n) for n = 1..208

Cf. A076605, A175607, A181447-A181456, A181458-A181470, A181568.

[ n: n in [2..300000] | m eq 37 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 19 2011

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 37 where D is PrimeDivisors(n^2-1)) ]; // Klaus Brockhaus, Feb 20 2011

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 == 37, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==37&]

is(n)=n=n^2-1; forprime(p=2, 31, n/=p^valuation(n, p)); n>1 && 37^valuation(n, 37)==n \\ Charles R Greathouse IV, Jul 01 2013

A181458 Numbers k such that 41 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

40, 81, 83, 122, 124, 163, 204, 206, 247, 286, 288, 329, 409, 491, 493, 573, 575, 737, 739, 778, 901, 944, 985, 1024, 1065, 1106, 1149, 1231, 1393, 1518, 1559, 1639, 1682, 2049, 2051, 2092, 2295, 2377, 2379, 2623, 2705, 2789, 3035, 3158, 3199, 3361, 3363

Offset: 1

Artur Jasinski, Oct 21 2010

Numbers k such that A076605(k) = 41.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(13) = 127855050751; primepi(41) = 13.

Artur Jasinski, Table of n, a(n) for n = 1..262

Cf. A076605, A175607, A181447-A181457, A181459-A181470, A181568.

[ n: n in [2..300000] | m eq 41 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 19 2011

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 41 where D is PrimeDivisors(n^2-1)) ]; // Klaus Brockhaus, Feb 20 2011

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 == 41, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==41&]

is(n)=n=n^2-1; forprime(p=2, 37, n/=p^valuation(n, p)); n>1 && 41^valuation(n, 41)==n \\ Charles R Greathouse IV, Jul 01 2013

A181459 Numbers k such that 43 is the largest prime factor of k^2 - 1.

Original entry on oeis.org

42, 44, 85, 87, 171, 173, 216, 257, 259, 300, 343, 386, 431, 474, 517, 560, 601, 687, 689, 730, 818, 859, 1074, 1117, 1119, 1289, 1291, 1332, 1420, 1549, 1633, 1721, 1805, 1891, 1977, 1979, 2108, 2321, 2495, 2665, 2667, 2751, 2753, 2794, 2925, 3095, 3484

Offset: 1

Artur Jasinski, Oct 21 2010

Numbers k such that A076605(k) = 43.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(14) = 842277599279; primepi(43) = 14.

Artur Jasinski, Table of n, a(n) for n = 1..343

Cf. A076605, A175607, A181447-A181458, A181460-A181470, A181568.

[ n: n in [2..300000] | m eq 43 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 19 2011

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 43 where D is PrimeDivisors(n^2-1)) ]; // Klaus Brockhaus, Feb 20 2011

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 == 43, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==43&]

is(n)=n=n^2-1; forprime(p=2, 41, n/=p^valuation(n, p)); n>1 && 43^valuation(n, 43)==n \\ Charles R Greathouse IV, Jul 01 2013

A181460 Numbers k such that 47 is the largest prime factor of k^2-1.

Original entry on oeis.org

46, 48, 93, 95, 142, 187, 189, 281, 375, 377, 424, 469, 610, 657, 659, 704, 751, 753, 892, 988, 1033, 1035, 1082, 1174, 1223, 1270, 1364, 1409, 1597, 1599, 1691, 1693, 1926, 1973, 1975, 2022, 2069, 2161, 2255, 2351, 2443, 2584, 2727, 2913, 2915, 3009

Offset: 1

Artur Jasinski, Oct 21 2010

Numbers k such that A076605(k) = 47.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(15) = 2218993446251; primepi(47) = 15.

Artur Jasinski, Table of n, a(n) for n = 1..433

Cf. A076605, A175607, A181447-A181459, A181461-A181470, A181568.

[ n: n in [2..300000] | m eq 47 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 19 2011

Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==47&]

is(n)=n=n^2-1; forprime(p=2, 43, n/=p^valuation(n, p)); n>1 && 47^valuation(n, 47)==n \\ Charles R Greathouse IV, Jul 01 2013

A181461 Numbers k such that 53 is the largest prime factor of k^2-1.

Original entry on oeis.org

52, 54, 105, 107, 160, 211, 319, 370, 476, 529, 531, 584, 637, 741, 743, 847, 849, 900, 902, 953, 1059, 1220, 1273, 1324, 1377, 1379, 1483, 1538, 1644, 1695, 1801, 1803, 2015, 2174, 2278, 2386, 2437, 2543, 2651, 2755, 2861, 2969, 3073, 3181, 3497, 3499

Offset: 1

Artur Jasinski, Oct 21 2010

Numbers k such that A076605(k) = 53.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(16) = 2907159732049; primepi(53) = 16.

Artur Jasinski, Table of n, a(n) for n = 1..507

Cf. A076605, A175607, A181447-A181460, A181462-A181470, A181568.

[ n: n in [2..300000] | m eq 53 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 19 2011

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 53 where D is PrimeDivisors(n^2-1)) ]; // Klaus Brockhaus, Feb 20 2011

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 == 53, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[300000], FactorInteger[#^2-1][[-1, 1]]==53&]

is(n)=n=n^2-1; forprime(p=2, 47, n/=p^valuation(n, p)); n>1 && 53^valuation(n, 53)==n \\ Charles R Greathouse IV, Jul 01 2013

A181462 Numbers k such that 59 is the largest prime factor of k^2-1.

Original entry on oeis.org

58, 117, 119, 176, 235, 237, 296, 353, 471, 530, 532, 589, 591, 650, 766, 827, 945, 1002, 1061, 1063, 1179, 1297, 1299, 1535, 1592, 1594, 1651, 1769, 1828, 1887, 1889, 2066, 2184, 2241, 2243, 2300, 2302, 2479, 2536, 2774, 2951, 3126, 3244, 3305, 3421

Offset: 1

Artur Jasinski, Oct 21 2010

Numbers k such that A076605(k) = 59.
Sequence is finite, for proof see A175607.
Search for terms can be restricted to the range from 2 to A175607(17) = 41257182408961; primepi(59) = 17.

Artur Jasinski, Table of n, a(n) for n = 1..634

Cf. A076605, A175607, A181447-A181461, A181463-A181470, A181568.

[ n: n in [2..300000] | m eq 59 where m is D[#D] where D is PrimeDivisors(n^2-1) ]; // Klaus Brockhaus, Feb 19 2011

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 59 where D is PrimeDivisors(n^2-1)) ]; // Klaus Brockhaus, Feb 20 2011

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 == 59, AppendTo[rr, n]]]; n++ ]; rr (* Artur Jasinski *)
Select[Range[10000],Max[Transpose[FactorInteger[#^2-1]][[1]]]==59&] (* Harvey P. Dale, Nov 13 2010 *)

for(k=2,1e9,vecmax(factor(k^2-1)[,1])==59 & print1(k",")) \\ M. F. Hasler, Nov 13 2010

cp's OEIS Frontend

A175904 Numbers m for which the set of prime divisors of m^2-1 is unique.

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A181452 Numbers k such that 17 is the largest prime factor of k^2 - 1.

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A181453 Numbers k such that 19 is the largest prime factor of k^2 - 1.

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A181455 Numbers k such that 29 is the largest prime factor of k^2 - 1.

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A181457 Numbers k such that 37 is the largest prime factor of k^2 - 1.

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A181458 Numbers k such that 41 is the largest prime factor of k^2 - 1.

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A181459 Numbers k such that 43 is the largest prime factor of k^2 - 1.

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