Rafael Parra Machio - cp's OEIS frontend

A225893 Numbers of the form p^2 * (p^2 + 1), where p = A224718.

2450, 2827442, 3420650, 131091050, 607597850, 1387525250, 3262865762, 3969189002, 4362536450, 7370136650, 8882968250, 38513866250, 43618113650, 96254752250, 110842051970, 123657370850, 135755034050, 170940489050, 304758650450, 328385729450, 472300879322

Offset: 1

Rafael Parra Machio, May 20 2013

nonn

The sequence A224718 of Zak Seidov, about prime numbers p such that p^2+1 is not squarefree.

Cf. A224718.

p = Select[Prime[Range[250]], !SquareFreeQ[#^2+1]&]; p^2 * (p^2+1)

A225892 Numbers of the form p^2 * (p^2 + 1) where p is in A225856.

Original entry on oeis.org

20, 90, 650, 14762, 28730, 83810, 130682, 280370, 708122, 924482, 1875530, 4881890, 7893290, 12120842, 13849562, 20155610, 25416722, 28403570, 38956322, 47465210, 62750162, 88538690, 104070602, 112561490, 141170042, 163060130, 260160770, 294517082, 352294130

Offset: 1

Rafael Parra Machio, May 20 2013

nonn

			a(2) = 3^2(3^2+1) = 3^4+3^2 = 90.
a(5) = 13^2(13^2+1) = 13^4+13^2 = 28730.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A071253, A225856 (primes p such that p^2+1 is squarefree).

p = Select[Prime[Range[60]], SquareFreeQ[#^2+1]&]; p^2 * (p^2+1)
#^2 (#^2+1)&/@Select[Prime[Range[50]],SquareFreeQ[#^2+1]&] (* Harvey P. Dale, Jun 17 2022 *)

a(n) = A071253(A225856(n)). - Amiram Eldar, Feb 23 2021

A225856 Primes p such that p^2 + 1 is squarefree.

Original entry on oeis.org

2, 3, 5, 11, 13, 17, 19, 23, 29, 31, 37, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 109, 113, 127, 131, 137, 139, 149, 151, 163, 167, 173, 179, 181, 191, 197, 199, 211, 223, 227, 229, 233, 241, 263, 269, 271, 277, 281, 283, 311, 313, 317, 331, 337

Offset: 1

Rafael Parra Machio, May 18 2013

nonn

Primes of the sequence A224718 generating squarefree.

			23 is a term since 23^2+1 = 530 = 2*5*53, is squarefree.
43 is not a term since 43^2+1 = 1850 = 2*5^2*7, is not squarefree.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A000040 and A049533.

Cf. A224718.

Select[Prime[Range[100]], SquareFreeQ[#^2+1]&]

A192228 Primes of the form (n+1)^6+(n+2)^6+(n+3)^6-666.

Original entry on oeis.org

425783, 263145359, 744158711, 1805712959, 32484102023, 103206118583, 271979814143, 324434645039, 454854785303, 626321908703, 6944429711711, 21648847849679, 23586002145119

Offset: 1

Rafael Parra Machio, Jun 26 2011

nonn

Sum of three consecutive numbers with exponent 6, the difference with 666 generate prime number of the form 3n^6 +36n^5 +210n^4 +720n^3 +1470n^2 +1656n +128.

			425783 = 6^6+7^6+8^6-666 and 744158711 = 24^6+25^6+26^6-666 are in the sequence.

lst={};Do[If[PrimeQ[p=(n+1)^6+(n+2)^6+(n+3)^6-666],AppendTo[lst,p]],{n,200}];lst
Select[Total/@(Partition[Range[200],3,1]^6)-666,PrimeQ] (* Harvey P. Dale, Dec 14 2011 *)

for(n=1,1e3,if(isprime(k=(n+1)^6+(n+2)^6+(n+3)^6-666),print1(k", "))) \\ Charles R Greathouse IV, Jul 01 2011

A191683 Smallest representative squarefree composite n with prescribed number of prime factors and prescribed, prime arithmetic average of these factors.

Original entry on oeis.org

21, 33, 57, 69, 93, 105, 129, 177, 195, 213, 217, 237, 249, 265, 309, 393, 417, 445, 465, 483, 489, 565, 573, 597, 633, 645, 669, 753, 813, 865, 915, 933, 973, 987, 993, 1057, 1077, 1137, 1149, 1185, 1257, 1285, 1329, 1365, 1389, 1393, 1417, 1437, 1465, 1477, 1497, 1545, 1569, 1689, 1743, 1765, 1857, 1893, 1897, 1945, 1977

Offset: 1

Rafael Parra Machio, Jun 11 2011

nonn

A187073 contains numbers n = q_1*q_2*q_3*... *q_k with k distinct prime factors q subject to the condition that the arithmetic average (q_1+q_2+...+q_k)/k is some prime p.

This sequence here is a subsequence of A187073 and lists only the smallest n associated with the two parameters k and p. If a larger/later number in A187073 represents the same prime p with the same number k, it is not copied into this sequence here.

			195 and 231 are representatives of the prime average p=7 with k=3 primes in A187073. The smaller 195 is, but the larger 231 is not in this sequence here.
57 and 85 are representatives of p=11 with k=2 primes in A187073. Only the smaller 57 is in here.
93, 145 and 253 are representatives of p=17 with k=2 primes in A187073. Only the smallest representative 93 is in this sequence here.

Carlos Sánchez y Rita Roldán, Goldbach: Una Conjetura Indómita, Nivola, 2009, p. 105

Antonio Roldán, Numeros y hoja de calculo: primos por tadas partes
Antonio Roldán, Numeros y hoja de calculo: números Arolmar
Rafael Parra Machío, Números Arolmar (PDF)

Cf. A187073

A191865 Primes of the form (n-1)^6 + n^5 + (n+1)^4.

Original entry on oeis.org

17, 563, 67559, 758677727, 5639788283, 12519315713, 228317617103, 2215267259747, 2458514680949, 5331791014853, 9754511753219, 11469661520567, 60568409162663, 64329745367417, 148696534573127, 164890314104507, 1843608625927967, 2182930574787737, 5990875533026939

Offset: 1

Rafael Parra Machio, Jun 18 2011

nonn

Sum of three consecutive numbers using exponents 6, 5, and 4 to generate prime numbers from n^6 - 5n^5 + 16n^4 - 16n^3 + 21n^2 - 2n + 2 = (n-1)^6 + n^5 + (n+1)^4.

			2^6 + 3^5 + 4^4 = 563 and 6^6 + 7^5 + 8^4 = 67559 are primes in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000
Rafael Parra Machío, PROPIEDADES DEL 2011: Un paseo a través de los números primos

R:= NULL: count:= 0:
for n from 1 by 2 while count < 100 do
   v:= (n-1)^6+n^5+(n+1)^4;
   if isprime(v) then count:= count+1; R:= R,v; fi
od:
R; # Robert Israel, Jan 05 2021

lst={};Do[If[PrimeQ[p=(n-1)^6+n^5+(n+1)^4], AppendTo[lst, p]],{n,200}];lst
lst={};Do[If[PrimeQ[p=n^6-5n^5+16n^4-16n^3+21n^2-2n+2], AppendTo[lst, p]],{n,200}];lst

forstep(n=1,1e3,2,if(isprime(k=(n-1)^6+n^5+(n+1)^4),print1(k", "))) \\ Charles R Greathouse IV, Jun 19 2011

A191589 Primes of the form 3n^4+12n^2+2, n > 0.

Original entry on oeis.org

17, 353, 7793, 45377, 588737, 1603073, 2131937, 2782097, 23705153, 27488177, 36393857, 142457633, 156688577, 288296417, 423617057, 780627473, 830968337, 938914433, 1254730193, 5724613457, 9150064577, 13500386657, 15247220033

Offset: 1

Rafael Parra Machio, Jun 07 2011

nonn

Prime sums of three consecutive fourth powers, since 3*n^4+12*n^2+2 = (n-1)^4+n^4+(n+1)^4.

Primes in A160827.

			2^4+3^4+4^4 = 353 is prime and therefore in the sequence.

Cf. A160827.

[ p: n in [0..300] | IsPrime(p) where p is n^4+(n+1)^4+(n+2)^4 ];

lst={};Do[If[PrimeQ[p=(n+1)^4+n^4+ (n-1)^4], AppendTo[lst,p]],{n, 100}];lst
lst={};Do[If[PrimeQ[p=3*n^4+12*n^2+2], AppendTo[lst, p]],{n,100}];lst

A188269 Prime numbers of the form k^4 + k^3 + 4k^2 + 7k + 5 = k^4 + (k+1)^3 + (k+2)^2.

Original entry on oeis.org

5, 59, 348077, 10023053, 30414227, 55367063, 72452489, 85856933, 109346759, 182679473, 254112143, 305966369, 433051637, 727914497, 2029672529, 4178961167, 6528621257, 8346080159, 12783893813, 17220494579, 17993776223, 19618171127, 23673478589, 29448235247, 43333033853

Offset: 1

Rafael Parra Machio, Jun 09 2011

nonn

Bunyakovsky's conjecture implies that this sequence is infinite. - Charles R Greathouse IV, Jun 09 2011

All the terms in the sequence are congruent to 2 mod 3. - K. D. Bajpai, Apr 11 2014

			5 is prime and appears in the sequence because 0^4 + 1^3 + 2^2 = 5.
59 is prime and appears in the sequence because 2^4 + 3^3 + 4^2 = 59.
348077 = 24^4 + (24+1)^3 + (24+2)^2 = 24^4 + 25^3 + 26^2.
10023053 = 56^4 + (56+1)^3 + (56+2)^2 = 56^4 + 57^3 + 58^2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A088548, A088550, A156018.

select(isprime, [n^4+(n+1)^3+(n+2)^2$n=0..1000])[]; # K. D. Bajpai, Apr 11 2014

lst={};Do[If[PrimeQ[p=n^4+n^3+4*n^2+7*n+5], AppendTo[lst, p]],{n,200}];lst
Select[Table[n^4+n^3+4n^2+7n+5,{n,500}],PrimeQ] (* Harvey P. Dale, Jun 19 2011 *)

for(n=1,1e3,if(isprime(k=n^4+n^3+4*n^2+7*n+5),print1(k", "))) \\ Charles R Greathouse IV, Jun 09 2011

Duplicate Mathematica program deleted by Harvey P. Dale, Jun 19 2011

Missing term 5 inserted by Alois P. Heinz, Sep 21 2024

A190780 a(n) = 2(n^8 + 224n^4 + 256)^2.

Original entry on oeis.org

131072, 462722, 33554432, 1246103042, 30324948992, 563669272322, 7763186941952, 79452617800322, 626224351281152, 3963462651845762, 20906139893891072, 94733225757031682, 377800938372595712, 1351791004705013762, 4406854039510188032, 13253329257388072322

Offset: 0

Rafael Parra Machio, May 19 2011

Each term equals the sum of three eighth powers and also twice a perfect square: a(n)= 2*(n^8+14n^4*2^4+2^8)^2.

More generally, a(n,k) = 2*(n^8+14*n^4*k^4+k^8)^2 = x^8+y^8+z^8, where x=n^2-k^2; y=n^2+k^2; z=2*n*k.

			462722 = 3^8+5^8+4^8 = 2*481^2.
563669272322 = 21^8+29^8+20^8 = 2*481^2.
Triplets (x,y,z) for k=2: {-3,5,4}, {0,8,8}, {5,13,12}, {12,20,16}, {21,29,20}, {32,40,24}, {45,53,28}, {60, 68,32}, {77,85,36},
{96,104,40}, see A028347 for x, A087475 for y, A008586 for z.

Robert Carmichael, Diophantine Analysis, Ed. 1915 by Mathematical Monographs, pages 96

Rafael Parra Machío, Ecuaciones Diofánticas, Tema XI, page 19.
Rafael Parra Machío, Teoría de Números, Web Site.
Index entries for linear recurrences with constant coefficients, signature (17,-136,680,-2380,6188, -12376,19448,-24310,24310,-19448,12376, -6188,2380,-680,136,-17,1).

Table[2(m^8+14m^4n^4+n^8)^2,{m,1,10}]/. n -> 2
Table[(m^2-n^2)^8+(m^2+n^2)^8+(2*m*n), {m, 1, 10}]/. n -> 2
Table[{(m^2-2^2), (m^2+2^2), (2*m*2)}, {m, 1, 5}], (* triples x, y, z *)
Table[2(n^8+224n^4+256)^2,{n,0,20}] (* Harvey P. Dale, Jun 19 2011 *)

a(n)=2*(n^4+4*n^3+8*n^2-16*n+16)^2*(n^4-4*n^3+8*n^2+16*n+16)^2 ; \\ Charles R Greathouse IV, May 19 2011

a(n) = 2*(n^8+14*n^4*2^4+2^8)^2.

G.f.: ( -131072 +1765502*x -43513950*x^2 -649478930*x^3 -13701900430*x^4 -195088344234*x^5 -1536270678326*x^6 -6277763482330*x^7 -12900117572550*x^8 -12896931212230*x^9 -6280312570586*x^10 -1534648531254*x^11 -195899417770*x^12 -13389949070*x^13 -738607890*x^14 -25688158*x^15 -462722*x^16 ) / (x-1)^17. - R. J. Mathar, Jun 04 2011

More terms from Harvey P. Dale, Jun 29 2011

A190176 a(n) = n^4 + 2^4 + (n+2)^4.

Original entry on oeis.org

32, 98, 288, 722, 1568, 3042, 5408, 8978, 14112, 21218, 30752, 43218, 59168, 79202, 103968, 134162, 170528, 213858, 264992, 324818, 394272, 474338, 566048, 670482, 788768, 922082, 1071648, 1238738, 1424672, 1630818, 1858592

Offset: 0

Rafael Parra Machio, May 19 2011

Each term equals the sum of three fourth powers and also twice a perfect square: n^4 + 2^4 + (n+2)^4 = 2*(n^2 + 2*n + 2^2)^2.

More generally, n^4 + k^4 + (n+k)^4 = 2*(n^2 + n*k + k^2)^2; in this case, k=2.

			a(3) = 722 = 3^4 +2^4+(3+2)^4 = 2(3^2+3*2+2^2)^2 = 2*19^2.
a(13) = 79202 = 13^4+2^4+(13 + 2)^4 = 2(13^2+13*2+2^2)^2 = 2*199^2.

Robert Carmichael, Diophantine Analysis, Ed. 1915 by Mathematical Monographs, pages 66-67.

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Rafael Parra Machío, dofanticas.pdf, pages 14-15
Rafael Parra Machío, Educaciones iofanticas.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

[n^4+2^4+(n+2)^4: n in [0..35]]; // Vincenzo Librandi, Jun 09 2011

Table[n^4+2^4+(n+2)^4,{n,0,20}]
CoefficientList[Series[(32 - 62*x + 118*x^2 - 58*x^3 + 18*x^4)/(1-x)^5, {x,0,50}], x] (* G. C. Greubel, Dec 28 2017 *)
LinearRecurrence[{5,-10,10,-5,1},{32,98,288,722,1568},50] (* Harvey P. Dale, May 26 2023 *)

a(n)=2*(n^2+2*n+4)^2 \\ Charles R Greathouse IV, Jun 08 2011

x='x+O('x^30); Vec((32 - 62*x + 118*x^2 - 58*x^3 + 18*x^4)/(1-x)^5 ) \\ G. C. Greubel, Dec 28 2017

G.f.: (32 - 62*x + 118*x^2 - 58*x^3 + 18*x^4)/(1-x)^5.

cp's OEIS Frontend

User: Rafael Parra Machio

A225893 Numbers of the form p^2 * (p^2 + 1), where p = A224718.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A225892 Numbers of the form p^2 * (p^2 + 1) where p is in A225856.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A225856 Primes p such that p^2 + 1 is squarefree.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A192228 Primes of the form (n+1)^6+(n+2)^6+(n+3)^6-666.

Views

Author

Keywords

Comments

Examples

Programs

Mathematica

PARI

A191683 Smallest representative squarefree composite n with prescribed number of prime factors and prescribed, prime arithmetic average of these factors.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A191865 Primes of the form (n-1)^6 + n^5 + (n+1)^4.

Views

Author

Keywords

Comments

Examples

Links

Programs

Maple

Mathematica

PARI

A191589 Primes of the form 3*n^4+12*n^2+2, n > 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

A188269 Prime numbers of the form k^4 + k^3 + 4*k^2 + 7*k + 5 = k^4 + (k+1)^3 + (k+2)^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A191589 Primes of the form 3n^4+12n^2+2, n > 0.

A188269 Prime numbers of the form k^4 + k^3 + 4k^2 + 7k + 5 = k^4 + (k+1)^3 + (k+2)^2.

A190780 a(n) = 2(n^8 + 224n^4 + 256)^2.