A293624 -id:A293624 - 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

A293625 Generators of Fermat pseudoprimes to base 2 that are square pyramidal numbers: numbers k such that 12k+1, 18k+1 and 36*k+1 are all primes.

Original entry on oeis.org

1, 15, 45, 56, 71, 85, 121, 141, 155, 176, 185, 206, 255, 275, 301, 346, 350, 380, 401, 470, 506, 511, 540, 680, 710, 745, 786, 801, 871, 946, 1025, 1156, 1200, 1211, 1326, 1380, 1395, 1421, 1480, 1505, 1515, 1590, 1676, 1696, 1710, 1830, 1941, 2066, 2171

Offset: 1

Amiram Eldar, Oct 13 2017

nonn

Rotkiewicz proved that if n is in the sequence then P((2^(2(18n+1))-1)/3) is a square pyramidal Fermat pseudoprime to base 2, where P(k) = k*(k+1)*(2k+1)/6 (A000330).

The generated numbers are terms in A293624. The first term is 256409721410526509996425240557391, the next 2 terms are about 3.683...*10^487 and 8.007...*10^1462.

			1 is in the sequence since 12*1+1 = 13, 18*1+1 = 19 and 36*1+1 = 37 are all primes. P((2^(2(18*1+1))-1)/3) = P(91625968981) = 256409721410526509996425240557391 is a Fermat pseudoprime to base 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andrzej Rotkiewicz, On pyramidal numbers of order 4, Elemente der Mathematik, Vol. 28 (1973), pp. 14-16.

Cf. A000330, A001567, A293624.

Select[Range[1, 1000], PrimeQ[12#+1] && PrimeQ[18#+1] && PrimeQ[36#+1] &]

A321866 Indices of tetrahedral numbers that are Fermat pseudoprimes to base 2.

Original entry on oeis.org

3457, 16705, 21169, 28297, 30577, 45481, 114601, 123121, 127297, 140977, 156601, 159337, 312841, 393121, 418177, 437977, 443017, 453601, 509737, 518017, 521137, 539401, 545161, 545617, 657841, 679297, 704161, 717817, 762121, 775057, 832801, 904801, 996601

Offset: 1

Amiram Eldar, Nov 20 2018

nonn

Numbers n such that n(n+1)(n+2)/6 is a Fermat pseudoprimes to base 2.
The corresponding tetrahedral Fermat pseudoprimes are 6891657409, 777080801185, 1581289265305, 3776730328549, 4765143438329, 15680770945781, 250856489370101, 311068284648121, 343806031110049, ...
Sierpinski asked for the existence of these numbers in 1965.

			3457 is in the sequence since A000292(3457) = 6891657409 is a Fermat pseudoprime to base 2.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Andrzej Rotkiewicz, On some problems of W. Sierpinski, Acta Arithmetica, Vol. 21 (1972), pp. 251-259.

Cf. A000292, A001567, A293622, A293624.

fermatQ[n_,k_] := CompositeQ[n] && PowerMod[k,n-1,n]==1; p[n_] := n(n+1)(n+2)/6; seq={}; Do[p1=p[n]; If[fermatQ[p1,2], AppendTo[seq,n]], {n,1,1000000,2}]; seq

isok(n) = my(t = n*(n+1)*(n+2)/6); (t != 1) && (Mod(2, t)^t == 2); \\ Michel Marcus, Nov 20 2018

cp's OEIS Frontend

A321868 Fermat pseudoprimes to base 2 that are octagonal.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A321870 Fermat pseudoprimes to base 2 that are decagonal.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A322123 Fermat pseudoprimes to base 2 that are tetradecagonal.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A322160 Fermat pseudoprimes to base 2 that are octadecagonal.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A322130 Fermat pseudoprimes to base 2 that are hexagonal.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A293625 Generators of Fermat pseudoprimes to base 2 that are square pyramidal numbers: numbers k such that 12*k+1, 18*k+1 and 36*k+1 are all primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A321866 Indices of tetrahedral numbers that are Fermat pseudoprimes to base 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A293625 Generators of Fermat pseudoprimes to base 2 that are square pyramidal numbers: numbers k such that 12k+1, 18k+1 and 36*k+1 are all primes.