A293623 -id:A293623 - cp's OEIS frontend

A321868 Fermat pseudoprimes to base 2 that are octagonal.

341, 645, 2465, 2821, 4033, 5461, 8321, 15841, 25761, 31621, 68101, 83333, 162401, 219781, 282133, 348161, 530881, 587861, 653333, 710533, 722261, 997633, 1053761, 1082401, 1193221, 1246785, 1333333, 1357441, 1398101, 1489665, 1584133, 1690501, 1735841

Offset: 1

Amiram Eldar, Nov 20 2018

nonn

Rotkiewicz proved that under Schinzel's Hypothesis H this sequence is infinite.
Intersection of A001567 and A000567.
The corresponding indices of the octagonal numbers are 11, 15, 29, 31, 37, 43, 53, 73, 93, 103, 151, 167, 233, 271, 307, 341, 421, 443, 467, 487, 491, 577, 593, 601, 631, 645, 667, 673, 683, 705, 727, 751, 761, 901, 911, 919, 991, ...
First differs from A216170 at n = 505.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andrzej Rotkiewicz, On some problems of W. Sierpinski, Acta Arithmetica, Vol. 21 (1972), pp. 251-259.
Wikipedia, Schinzel's Hypothesis H.

Cf. A000567, A001567, A216170, A293623, A293624, A321869.

oct[n_]:=n(3n-2); Select[oct[Range[1, 1000]], PowerMod[2, (# - 1), #]==1 &]

isok(n) = (n>1) && ispolygonal(n, 8) && !isprime(n) && (Mod(2, n)^n==2); \\ Daniel Suteu, Nov 29 2018

A321870 Fermat pseudoprimes to base 2 that are decagonal.

Original entry on oeis.org

1105, 1387, 2047, 3277, 6601, 13747, 16705, 19951, 31417, 74665, 83665, 88357, 90751, 275887, 390937, 514447, 604117, 642001, 741751, 748657, 769567, 916327, 1092547, 1293337, 1302451, 1433407, 1520905, 1530787, 1809697, 1907851, 2008597, 2205967, 2387797

Offset: 1

Amiram Eldar, Nov 20 2018

nonn

Rotkiewicz proved that under Schinzel's Hypothesis H this sequence is infinite.
Intersection of A001567 and A001107.
The corresponding indices of the decagonal numbers are 17, 19, 23, 29, 41, 59, 65, 71, 89, 137, 145, 149, 151, 263, 313, 359, 389, 401, 431, 433, 439, 479, 523, 569, 571, 599, 617, 619, 673, 691, 709, 743, 773, 829, 863, 883, 911, 919, 941, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andrzej Rotkiewicz, On some problems of W. Sierpinski, Acta Arithmetica, Vol. 21 (1972), pp. 251-259.
Wikipedia, Schinzel's Hypothesis H.

Cf. A001107, A001567, A293623, A293624, A321871.

dec[n_] := n(4n-3); Select[dec[Range[1, 1000]], PowerMod[2, (# - 1), #]==1 &]

isok(n) = (n>1) && ispolygonal(n, 10) && !isprime(n) && (Mod(2, n)^n==2); \\ Daniel Suteu, Nov 29 2018

A322123 Fermat pseudoprimes to base 2 that are tetradecagonal.

Original entry on oeis.org

31609, 60701, 458989, 513629, 679729, 729061, 745889, 1207361, 1994689, 2746589, 4361389, 4974971, 5173601, 5444489, 6749021, 9056501, 12659989, 13295281, 15525241, 15757741, 16070429, 16705021, 20770621, 21400481, 23822329, 23966011, 27492581, 34003061

Offset: 1

Amiram Eldar, Nov 27 2018

nonn

Rotkiewicz proved that under Schinzel's Hypothesis H this sequence is infinite.
Intersection of A001567 and A051866.
The corresponding indices of the tetradecagonal numbers are 73, 101, 277, 293, 337, 349, 353, 449, 577, 677, 853, 911, 929, 953, 1061, 1229, 1453, 1489, 1609, 1621, 1637, 1669, 1861, 1889, 1993, 1999, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Andrzej Rotkiewicz, On some problems of W. Sierpinski, Acta Arithmetica, Vol. 21 (1972), pp. 251-259.
Wikipedia, Schinzel's Hypothesis H.

Cf. A001567, A051866, A293623, A293624, A322124.

tetradec[n_] := n(6n-5); Select[tetradec[Range[1, 1000]], PowerMod[2, (# - 1), #]==1 &]
Select[PolygonalNumber[14,Range[2400]],PowerMod[2,#-1,#]==1&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 11 2018 *)

isok(n) = ispolygonal(n, 14) && (Mod(2, n)^n==2) && !isprime(n) && (n>1); \\ Michel Marcus, Nov 28 2018

A322160 Fermat pseudoprimes to base 2 that are octadecagonal.

Original entry on oeis.org

8481, 14491, 29341, 62745, 196093, 396271, 526593, 2184571, 2513841, 5256091, 7017193, 8137585, 13448593, 15247621, 16053193, 16879501, 18740971, 20494401, 29878381, 33704101, 35703361, 36724591, 41607721, 42709591, 69741001, 70593931, 80927821, 82976181

Offset: 1

Amiram Eldar, Nov 29 2018

nonn

Rotkiewicz proved that under Schinzel's Hypothesis H this sequence is infinite.
Intersection of A001567 and A051870.
The corresponding indices of the octadecagonal numbers are 33, 43, 61, 89, 157, 223, 257, 523, 561, 811, 937, 1009, 1297, 1381, 1417, 1453, 1531, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andrzej Rotkiewicz, On some problems of W. Sierpinski, Acta Arithmetica, Vol. 21 (1972), pp. 251-259.
Wikipedia, Schinzel's Hypothesis H.

Cf. A001567, A051870, A293623, A293624, A322161.

octadec[n_]:=n(8n-7); Select[octadec[Range[1, 1000]], PowerMod[2, (# - 1), #]==1 &]

isok(n) = (n>1) && ispolygonal(n, 18) && !isprime(n) && (Mod(2, n)^n==2); \\ Michel Marcus, Nov 29 2018

A322130 Fermat pseudoprimes to base 2 that are hexagonal.

Original entry on oeis.org

561, 2701, 4371, 8911, 10585, 18721, 33153, 49141, 93961, 104653, 115921, 157641, 226801, 289941, 314821, 334153, 534061, 665281, 721801, 831405, 873181, 915981, 1004653, 1373653, 1537381, 1755001, 1815465, 1987021, 2035153, 2233441, 2284453, 3059101, 3363121

Offset: 1

Amiram Eldar, Nov 27 2018

nonn

Rotkiewicz proved that under Schinzel's Hypothesis H this sequence is infinite. His proof is the same as that of triangular pseudoprimes, since all the triangular numbers that he generates are also hexagonal (see comment in A320599).
Intersection of A001567 and A000384.
Subsequence of A293622.
The corresponding indices of the hexagonal numbers are 17, 37, 47, 67, 73, 97, 129, 157, 217, 229, 241, 281, 337, 381, 397, 409, 517, 577, 601, 645, 661, 677, 709, 829, 877, 937, 953, 997, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andrzej Rotkiewicz, On some problems of W. Sierpinski, Acta Arithmetica, Vol. 21 (1972), pp. 251-259.
Wikipedia, Schinzel's Hypothesis H.

Cf. A000384, A001567, A293622, A293623, A293624, A320599.

hex[n_] := n(2n-1); Select[hex[Range[1, 1000]], PowerMod[2, (# - 1), #]==1 &]

isok(n) = ispolygonal(n, 6) && (Mod(2, n)^n==2) && !isprime(n) && (n>1); \\ Michel Marcus, Nov 28 2018

A371759 a(n) is the smallest n-gonal number that is a Fermat pseudoprime to base 2 (A001567), or -1 if no such number exists.

Original entry on oeis.org

561, 1194649, 7957, 561, 23377, 341, 129889, 1105, 35333, 561, 204001, 31609, 2940337, 1105, 493697, 8481, 13981, 1905, 88561, 41665, 10680265, 1729, 107185, 264773, 449065, 6601, 2165801, 23001, 1141141, 13981, 272251, 4369, 17590957, 15841, 137149, 2821, 561

Offset: 3

Amiram Eldar, Apr 05 2024

nonn

The corresponding indices of the n-gonal numbers are 33, 1093, 73, 17, 97, ... (A371760).

			a(4) = A001220(1)^2 = 1093^2 = 1194649. The only known square base-2 pseudoprimes are the squares of the Wieferich primes (A001220).

Amiram Eldar, Table of n, a(n) for n = 3..10000
Eric Weisstein's World of Mathematics, Polygonal Number.
Eric Weisstein's World of Mathematics, Poulet Number.
Wikipedia, Polygonal number.
Wikipedia, Pseudoprime.
Index entries for sequences related to pseudoprimes.

Cf. A001220, A001567, A293622, A293623, A322130, A321868, A321870, A322123, A322160, A371760.

p[k_, n_] := ((n-2)*k^2 - (n-4)*k)/2; pspQ[n_] := CompositeQ[n] && PowerMod[2, n - 1, n] == 1; a[n_] := Module[{k = 2}, While[! pspQ[p[k, n]], k++]; p[k, n]]; Array[a, 50, 3]

p(k, n) = ((n-2)*k^2 - (n-4)*k)/2;
ispsp(n) = !isprime(n) && Mod(2, n)^(n-1) == 1;
a(n) = {my(k = 2); while(!ispsp(p(k, n)), k++); p(k, n);}

a(n) = ((n-2)*k^2 - (n-4)*k)/2, where k = A371760(n).

A371760 a(n) is the smallest number k such that the k-th n-gonal number is a Fermat pseudoprime to base 2 (A001567), or -1 if no such number exists.

Original entry on oeis.org

33, 1093, 73, 17, 97, 11, 193, 17, 89, 11, 193, 73, 673, 13, 257, 33, 41, 15, 97, 65, 1009, 13, 97, 149, 190, 23, 401, 41, 281, 31, 133, 17, 1033, 31, 89, 13, 6, 59, 241, 157, 1217, 91, 145, 37, 937, 29, 1289, 73, 97, 41, 617, 19, 137, 151, 34, 103, 8641, 47, 82

Offset: 3

Amiram Eldar, Apr 05 2024

nonn

The corresponding pseudoprimes are in A371759.

Amiram Eldar, Table of n, a(n) for n = 3..10000
Eric Weisstein's World of Mathematics, Polygonal Number.
Eric Weisstein's World of Mathematics, Poulet Number.
Wikipedia, Polygonal number.
Wikipedia, Pseudoprime.
Index entries for sequences related to pseudoprimes.

Cf. A001567, A293622, A293623, A322130, A321868, A321870, A322123, A322160, A371759.

p[k_, n_] := ((n - 2)*k^2 - (n - 4)*k)/2; pspQ[n_] := CompositeQ[n] && PowerMod[2, n - 1, n] == 1; a[n_] := Module[{k = 2}, While[! pspQ[p[k, n]], k++]; k]; Array[a, 100, 3]

p(k, n) = ((n-2)*k^2 - (n-4)*k)/2;
ispsp(n) = !isprime(n) && Mod(2, n)^(n-1) == 1;
a(n) = {my(k = 2); while(!ispsp(p(k, n)), k++); k;}

cp's OEIS Frontend

A321868 Fermat pseudoprimes to base 2 that are octagonal.

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A321870 Fermat pseudoprimes to base 2 that are decagonal.

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A322123 Fermat pseudoprimes to base 2 that are tetradecagonal.

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A322160 Fermat pseudoprimes to base 2 that are octadecagonal.

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A322130 Fermat pseudoprimes to base 2 that are hexagonal.

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A371759 a(n) is the smallest n-gonal number that is a Fermat pseudoprime to base 2 (A001567), or -1 if no such number exists.

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A371760 a(n) is the smallest number k such that the k-th n-gonal number is a Fermat pseudoprime to base 2 (A001567), or -1 if no such number exists.

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