A128470 -id:A128470 - cp's OEIS frontend

A271114 Expansion of (1+x)*(2+x)/(1-x)^2.

2, 7, 13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, 91, 97, 103, 109, 115, 121, 127, 133, 139, 145, 151, 157, 163, 169, 175, 181, 187, 193, 199, 205, 211, 217, 223, 229, 235, 241, 247, 253, 259, 265, 271, 277, 283, 289, 295, 301, 307, 313, 319, 325

Offset: 0

Colin Barker, Mar 31 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A016921, A270700.

Cf. A016969, A082369, A100764, A128470, A132231, A156376.

Join[{2}, LinearRecurrence[{2, -1}, {7, 13}, 100]] (* G. C. Greubel, Mar 31 2016 *)

PARI
```
Vec((1+x)*(2+x)/(1-x)^2 + O(x^70))
```

G.f.: (1+x)*(2+x)/(1-x)^2.
a(n) = A270700(n)/6.
a(n) = 6*n+1 = A016921(n) for n>0.
a(n) = 2*a(n-1)-a(n-2) for n>2.
E.g.f.: 1 + (1+6*x)*exp(x). - G. C. Greubel, Mar 31 2016
From Bruno Berselli and G. C. Greubel, Mar 31 2016: (Start)
a(5*m+1) = 30*m + 7 = A132231(m+1).
a(5*m+2) = 30*m + 13 = A082369(m+1).
a(5*m+3) = 30*m + 19 = A156376(m).
a(5*m+4) = 30*m + 25 = 5*A016969(m).
a(5*m+5) = 30*m + 31 = A128470(m+1). (End)
a(n) = A100764(n+3) for n >= 1. - Georg Fischer, Oct 30 2018

A051646 Primes of the form 30*p + 1 where p is also prime.

Original entry on oeis.org

61, 151, 211, 331, 571, 691, 1231, 1291, 1831, 2011, 2131, 2371, 2671, 3271, 3391, 3931, 4111, 5011, 5431, 5791, 6691, 6871, 6991, 8311, 8431, 9391, 9511, 9931, 10111, 10771, 11491, 13171, 13291, 13711, 13831, 14011, 14731, 15091, 15271, 16231, 16411, 17791, 17971

Offset: 1

Labos Elemer

Analogous to A005385, safe primes. Can be called 30-safe primes.

			61 is in the sequence because both 2 and 30*2 + 1 = 61 are primes.

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

Cf. A005384, A005385, A007693, A051645, A128470.

Select[30 * Prime[Range[120]] + 1, PrimeQ] (* Amiram Eldar, Feb 24 2025 *)

isok(k) = isprime(k) && k % 30 == 1 && isprime((k-1)/30); \\ Amiram Eldar, Feb 24 2025

a(n) = A128470(A051645(n)) = 30 * A051645(n) + 1. - Amiram Eldar, Feb 24 2025

A375646 Products of prime 8-tuples (p, p+2, p+6, p+8, p+12, p+18, p+20, p+26) where p = A022011(n).

Original entry on oeis.org

35336848261, 3806030828359338048562146944333316580734237696172777384261, 187864049264599549789422153639990042809117925774238076689261, 5904346651213195922822968174735596082317962321492475630960754261, 1225364761514380727859407545185568342040426059188911173818624004261

Offset: 1

Michael De Vlieger, Aug 24 2024

nonn

Primes p in A022011 belong to 11 (mod 210), thus a(n) is congruent to the product of residues {11, 13, 17, 19, 23, 29, 31, 37} (mod 210), i.e., 1 (mod 210).

Gaps between primes are {2, 4, 2, 4, 6, 2, 6}.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A022011, A128470, A375647, A375648.

Map[Times @@ NextPrime[#, Range[0, 7]] &, Import["https://oeis.org/A022011/b022011.txt", "Data"][[;; 12, -1]] ]

A375647 Products of prime 8-tuples (p, p+2, p+6, p+12, p+14, p+20, p+24, p+26) where p = A022012(n).

Original entry on oeis.org

435656388001, 7667061486004435747476001, 26887071293271756518203932603297162186001, 1967190066500349361284627627321478140655499961186001, 34207121652717644163491129612663352350226660003697376196001, 131790860746164880099394335252801389818740796081899944471402001

Offset: 1

Michael De Vlieger, Aug 24 2024

nonn

Primes p in A022012 belong to either 17 or 167 (mod 210).
Therefore a(n) is either congruent to the product of residues {17, 19, 23, 29, 31, 37, 41, 43} (mod 210), or {167, 169, 173, 179, 181, 187, 191, 193} (mod 210), so a(n) is congruent to 121 (mod 210).
Gaps between prime factors have a symmetric arrangement {2, 4, 6, 2, 6, 4, 2}.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A022012, A128470, A375646, A375648.

Map[Times @@ NextPrime[#, Range[0, 7]] &, Import["https://oeis.org/A022012/b022012.txt", "Data"][[;; 12, -1]]]

A375648 Products of prime 8-tuples (p, p+6, p+8, p+14, p+18, p+20, p+24, p+26) where p = A022013(n).

Original entry on oeis.org

3868985835982814590518552822749329543261, 43207320984601757696213691690377119115644261, 287530494211069388143263747303929618940138523261, 2991325021830996455943969680355510324042937309261, 3433715221252595293789329211184553889095776281330363261, 523198428668721638888114210837839571392856841008842698982189261

Offset: 1

Michael De Vlieger, Aug 24 2024

nonn

Primes p in A022013 belong to 173 (mod 210). Thus a(n) is congruent to the product of residues {173, 179, 181, 187, 191, 193, 197, 199} (mod 210), i.e., 1 (mod 210).

Gaps between primes are {6, 2, 6, 4, 2, 4, 2}.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A022012, A128470, A375646, A375647.

Map[Times @@ NextPrime[#, Range[0, 7]] &, Import["https://oeis.org/A022013/b022013.txt", "Data"][[;; 12, -1]]]

A330670 Squares of primes congruent to 1 (mod 30).

Original entry on oeis.org

121, 361, 841, 961, 1681, 3481, 3721, 5041, 6241, 7921, 10201, 11881, 17161, 19321, 22201, 22801, 32041, 32761, 36841, 39601, 44521, 52441, 57121, 58081, 63001, 72361, 73441, 78961, 96721, 109561, 121801, 128881, 143641, 151321, 166801, 167281, 175561

Offset: 1

Harry E. Neel, Dec 24 2019

nonn

Sequence lists squares of prime numbers that end in 1 or 9.

			Prime 11*11=121 and 19*19=361; 121 and 361 are terms of this sequence.
Prime 13*13=169 and 17*17=289; 169 and 289 are not terms of this sequence.

Intersection of A001248 and A128470.

Select[Range[360], PrimeQ[#] && Mod[#^2, 30] == 1 &]^2 (* Amiram Eldar, Dec 28 2019 *)

isok(m) = issquare(m) && isprime(sqrtint(m)) && ((m % 30) == 1); \\ Michel Marcus, Dec 26 2019

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A271114 Expansion of (1+x)*(2+x)/(1-x)^2.

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A051646 Primes of the form 30*p + 1 where p is also prime.

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A375646 Products of prime 8-tuples (p, p+2, p+6, p+8, p+12, p+18, p+20, p+26) where p = A022011(n).

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A375647 Products of prime 8-tuples (p, p+2, p+6, p+12, p+14, p+20, p+24, p+26) where p = A022012(n).

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A375648 Products of prime 8-tuples (p, p+6, p+8, p+14, p+18, p+20, p+24, p+26) where p = A022013(n).

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A330670 Squares of primes congruent to 1 (mod 30).

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