A017353 -id:A017353 - cp's OEIS frontend

A017361 a(n) = (10*n + 7)^9.

40353607, 118587876497, 7625597484987, 129961739795077, 1119130473102767, 6351461955384057, 27206534396294947, 95151694449171437, 285544154243029527, 760231058654565217, 1838459212420154507, 4108400332687853397

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (10, -45, 120, -210, 252, -210, 120, -45, 10, -1).

Cf. A017353 (10n+7), A001017 (n^9).

[(10*n+7)^9: n in [0..20]]; // Vincenzo Librandi, Aug 30 2011

(10*Range[0,30]+7)^9 (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{40353607,118587876497,7625597484987,129961739795077,1119130473102767,6351461955384057,27206534396294947,95151694449171437,285544154243029527,760231058654565217},30] (* Harvey P. Dale, Dec 28 2011 *)

vector(20, n, n--; (10*n+7)^9) \\ G. C. Greubel, Nov 10 2018

a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10); a(0)=40353607, a(1)=118587876497, a(2)=7625597484987, a(3)=129961739795077, a(4)=1119130473102767, a(5)=6351461955384057, a(6)=27206534396294947, a(7)=95151694449171437, a(8)=285544154243029527, a(9)=760231058654565217. - Harvey P. Dale, Dec 28 2011

A017362 a(n) = (10*n + 7)^10.

Original entry on oeis.org

282475249, 2015993900449, 205891132094649, 4808584372417849, 52599132235830049, 362033331456891249, 1822837804551761449, 7326680472586200649, 24842341419143568849, 73742412689492826049

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Cf. A017353 (10n+7), A008454 (n^10).

[(10*n+7)^10: n in [0..20]]; // Vincenzo Librandi, Aug 30 2011

(10*Range[0,10]+7)^10 (* Harvey P. Dale, Nov 12 2015 *)

vector(20, n, n--; (10*n+7)^10) \\ G. C. Greubel, Nov 10 2018

A032588 Lucky numbers ending with digit 7.

Original entry on oeis.org

7, 37, 67, 87, 127, 237, 267, 297, 307, 327, 357, 367, 427, 477, 487, 517, 537, 577, 717, 727, 777, 787, 867, 897, 927, 937, 957, 997, 1057, 1087, 1107, 1117, 1147, 1167, 1197, 1357, 1387, 1417, 1497, 1567, 1587, 1597, 1737, 1767, 1777, 1797, 1827, 1857

Offset: 1

Patrick De Geest, Apr 15 1998

Also, lucky numbers (A000959) which are congruent to 2 mod 5 (because only numbers ending in 2 could make a difference, but these are removed in the 1st step of the lucky sieve). - R. J. Mathar, Apr 29 2008

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

Intersection of A000959 and A017353.

Edited by N. J. A. Sloane, May 15 2008 at the suggestion of R. J. Mathar

A053180 Numbers ending in 7 which are not prime.

Original entry on oeis.org

27, 57, 77, 87, 117, 147, 177, 187, 207, 217, 237, 247, 267, 287, 297, 327, 357, 377, 387, 407, 417, 427, 437, 447, 477, 497, 507, 517, 527, 537, 567, 597, 627, 637, 657, 667, 687, 697, 707, 717, 737, 747, 767, 777, 807, 817, 837, 847, 867, 897, 917, 927

Offset: 1

Enoch Haga, Feb 29 2000

G. C. Greubel, Table of n, a(n) for n = 1..10000

Intersection of A002808 (composites) and A017353 (ending with 7). - Michel Marcus, Sep 04 2018

Select[Range[7,1000,10],!PrimeQ[#]&] (* Harvey P. Dale, Nov 03 2013 *)

A152579 a(n) = (10n+3)(10*n+17).

Original entry on oeis.org

51, 351, 851, 1551, 2451, 3551, 4851, 6351, 8051, 9951, 12051, 14351, 16851, 19551, 22451, 25551, 28851, 32351, 36051, 39951, 44051, 48351, 52851, 57551, 62451, 67551, 72851, 78351, 84051, 89951, 96051, 102351, 108851, 115551, 122451, 129551, 136851, 144351, 152051, 159951

Offset: 0

Paul Curtz, Dec 08 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005563, A017270, A017305, A017353, A056220, A144396, A152161.

[(10*n+3)*(10*n+17): n in [0..40]]; // Vincenzo Librandi, Jul 28 2011

a(n)=100*n*(n+2)+51 \\ Charles R Greathouse IV, Jul 28 2011

a(n) = 100*n*(n+2) + 51 = 100*A005563(n) + 51 = 100*(n+1)^2 - 49 = A017270(n+1) - 49.
a(n) = 2*a(n-2) - a(n-2) + 200.
a(n) = 50*A056220(n+1) + 1.
a(n+1) - a(n) = 200*n + 300 = 100*A144396(n+1).
G.f.: (-51 - 198*x + 49*x^2)/(x-1)^3. - R. J. Mathar, Jul 01 2011
From Elmo R. Oliveira, Oct 27 2024: (Start)
E.g.f.: exp(x)*(51 + 300*x + 100*x^2).
a(n) = A017305(n)*A017353(n+1).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A154412 Primes of the form 10n^2+14n+5, n >= 0.

Original entry on oeis.org

5, 29, 73, 137, 449, 593, 757, 941, 1613, 1877, 2161, 2789, 3881, 5153, 6101, 7129, 7673, 8237, 8821, 12041, 13469, 15761, 17389, 18233, 26729, 27773, 28837, 29921, 34457, 38069, 39313, 40577, 45833, 60373, 63521, 66749, 71741, 75169, 76913

Offset: 1

Vincenzo Librandi, Jan 09 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A017353, A154408.

[a: n in [0..100] | IsPrime(a) where a is 10*n^2+14*n+5]; // Vincenzo Librandi, Jul 23 2012

Select[Table[10n^2+14n+5,{n,0,200}],PrimeQ] (* Vincenzo Librandi, Jul 23 2012 *)

{for(n=0, 100, if(isprime(k=10*n^2+14*n+5), print1(k, ", ")))}; \\ Vincenzo Librandi, Jul 23 2012

Corrected by Don Reble, Jun 16 2010

A168458 a(n) = 7 + 10*floor((n-1)/2).

Original entry on oeis.org

7, 7, 17, 17, 27, 27, 37, 37, 47, 47, 57, 57, 67, 67, 77, 77, 87, 87, 97, 97, 107, 107, 117, 117, 127, 127, 137, 137, 147, 147, 157, 157, 167, 167, 177, 177, 187, 187, 197, 197, 207, 207, 217, 217, 227, 227, 237, 237, 247, 247, 257, 257, 267, 267, 277, 277, 287

Offset: 1

Vincenzo Librandi, Nov 26 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A017353.

[7+10*Floor((n-1)/2): n in [1..70]]; // Vincenzo Librandi Sep 19 2013

A168458:=n->7 + 10*floor((n-1)/2); seq(A168458(k), k=1..100); # Wesley Ivan Hurt, Nov 08 2013

Table[7 + 10 Floor[(n - 1)/2], {n, 70}] (* or *) CoefficientList[Series[(7 + 3 x^2)/((1 + x) (x - 1)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Sep 19 2013 *)
LinearRecurrence[{1,1,-1},{7,7,17},60] (* Harvey P. Dale, Apr 12 2018 *)

a(n) = 10*n - a(n-1) - 6, with n>1, a(1)=7.
G.f.: x*(7 + 3*x^2)/((1+x)*(x-1)^2). - Vincenzo Librandi, Sep 19 2013
a(n) = a(n-1) +a(n-2) -a(n-3). - Vincenzo Librandi, Sep 19 2013
From G. C. Greubel, Jul 23 2016: (Start)
a(n) = (10*n - 5*(-1)^n - 1)/2.
E.g.f.: (1/2)*(-5 + 6*exp(x) + (10*x - 1)*exp(2*x))*exp(-x). (End)

New definition by Vincenzo Librandi, Sep 19 2013

A268867 Number of integers of the form m^2+1 between two consecutive pairs of primes of the same form.

Original entry on oeis.org

0, 7, 7, 27, 67, 77, 177, 77, 167, 7, 67, 377, 47, 27, 67, 27, 37, 57, 187, 47, 57, 7, 277, 87, 27, 7, 307, 47, 77, 127, 167, 87, 207, 167, 227, 217, 17, 247, 127, 17, 187, 237, 7, 117, 47, 7, 157, 57, 37, 197, 217, 87, 17, 137, 147, 287, 67, 547, 37, 187, 787

Offset: 1

Michel Lagneau, Feb 15 2016

nonn

Or number of integers of the form m^2+1 between two consecutive twin k^2+1 primes.
a(n)==7 mod 10 for n>1.
The primes of the sequence are 7, 17, 37, … (A030432), subsequence of A017353.
Conjecture 1: the sequence is infinite.
Conjecture 2: for n>1, the sequence sorted in ascending order with distinct values generates the set B = {b(k)} = {7, 17, 27, 37, …} = {7 + 10k}, k = 0,1,2,…  and {a(n)}/qZ = B/qZ with q = 2^m, m = 1, 2,… is a multiplicative group.
This has been verified for n up to 10^8.
A remarkable simple way to perceive properties of invariability in this sequence (or other particular sequences) is the use of finite groups of integers modulo q. This study can provide interesting interpretations for some regularities which describe properties in other mathematical spaces.
However, this concept is based on a conjecture if it is impossible to prove the infinity character of a sequence. This requires, for the calculations, a large number of elements in the sequence.
The groups of integers modulo 2^m are:
q = 2 => B/2Z = {1}, the trivial group.
q = 4 => B/4Z = {1,3}, the cyclic group of order 2 with two elements.
q = 8 => B/8Z = {1,3,5,7}, the group of order 4 with generating set {3,7} => the Klein four-group of order 2. The square of each element of B/8Z is 1. The group is not cyclic.
q = 16 => B/16Z = {1,3,5,7,9,11,13,15}, the group of order 8 with generating set {3,15}. The powers of 3 {1,3,9,11} are a subgroup of order 4, as are the powers of 5, {1,5,9,13}. The group B/16Z is not cyclic.
For higher powers q = 2^k, k>2,  B/(2^k)Z = {1,3,5,…,2^k-1}, with generating set {3, 2^k-1}. The group B/(2^k)Z is not cyclic.
The order of the group is given by Euler’s totient function (A000010): this is the product of the orders of the cyclic groups in the direct product (see the links).

			a(1)=0 because there exists 0 number of the form m^2+1 between the two consecutive pairs of primes(2^2+1, 4^2+1) and (4^2+1, 6^2+1);
a(2) = 7 because there exists 7 numbers of the form m^2+1 between the two consecutive pairs of primes(4^2+1, 6^2+1) and (14^2+1, 16^2+1): 50, 65, 82, 101, 122, 145 and 170.

Michel Lagneau, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Finite Group
Wikipedia, Finite group
Wikipedia, Multiplicative group of integers modulo n

Cf. A005574, A030432, A017353, A096012, A206328.

nn:=10000:T:=array(1..200):kk:=0:
for n from 4 by 2 to nn do:
   p1:=n^2+1:p2:=(n+2)^2+1:
    if isprime(p1) and isprime(p2)
     then
     kk:=kk+1:T[kk]:=n:
     else
    fi:
  od:
    for m from 1 to kk-1 do:
      q:=T[m+1]-T[m]-3:printf(`%d, `,q):
    od:

lst={};Do[If[PrimeQ[n^2+1],AppendTo[lst,n]],{n,1,10000}];Module[{tr=Transpose[Select[Partition[lst,2,1],#[[2]]-#[[1]]==2&]],fir,las},fir=Rest[tr[[1]]];las=Most[tr[[2]]];Flatten[Abs[Differences/@Thread[{fir,las}]]]-1/.{-1->0}]

A160912 [1, 3, 5, 7, ...] convolved with [1, 4, 0, 0, 0, ...].

Original entry on oeis.org

1, 7, 17, 27, 37, 47, 57, 67, 77, 87, 97, 107, 117, 127, 137, 147, 157, 167, 177, 187, 197, 207, 217, 227, 237, 247, 257, 267, 277, 287, 297, 307, 317, 327, 337, 347, 357, 367, 377, 387, 397, 407, 417, 427, 437, 447

Offset: 1

Gary W. Adamson, May 30 2009

nonn

			a(3) = 17 = (5, 3, 1) * (1, 4, 0) = (5 + 12 + 0).

Index entries for linear recurrences with constant coefficients, signature (2, -1).

Essentially a duplicate of A017353.

Table[ListConvolve[Range[1,2n-1,2],PadRight[{1,4},n,0]],{n,50}]//Flatten (* or *) Join[{1},Range[7,450,10]] (* or *) LinearRecurrence[{2,-1},{1,7,17},50] (* Harvey P. Dale, Apr 28 2018 *)

Odd integers convolved with [1, 4, 0, 0, 0, ...].

More terms from N. J. A. Sloane, May 31 2009

A270968 Reduced 5x+1 function R applied to the odd integers: a(n) = R(2n-1), where R(k) = (5k+1)/2^r, with r as large as possible.

Original entry on oeis.org

3, 1, 13, 9, 23, 7, 33, 19, 43, 3, 53, 29, 63, 17, 73, 39, 83, 11, 93, 49, 103, 27, 113, 59, 123, 1, 133, 69, 143, 37, 153, 79, 163, 21, 173, 89, 183, 47, 193, 99, 203, 13, 213, 109, 223, 57, 233, 119, 243, 31, 253, 129, 263, 67, 273, 139, 283, 9, 293, 149, 303

Offset: 1

Michel Lagneau, Mar 27 2016

The odd-indexed terms a(2i+1) = 10i+3 = A017305(i), i>=0;
a(4i+4) = 10i+9 = A017377(i), i>=0;
a(8i+6) = 10i+7 = A017353(i), i>=0;
a(16i+2) = 10i+1 = A017281(i), i>=0.
Note that a(n) = a(16n-6) = a(6n-2)/3. No multiple of 5 is in this sequence.
a(n) = R(2n-1) < 2n-1 for n = 2, 6, 10, ..., 2+4i,...

			a(4)=9 because (2*4-1) = 7  -> (5*7+1)/2^2 = 9.

Cf. A075677, A017281, A017305, A017353, A017377, A133423, A133424, A174795, A133419, A133420, A232711, A259207, A267969.

nextOddK[n_] := Module[{m=5n+1}, While[EvenQ[m], m=m/2]; m]; (* assumes odd n *) Table[nextOddK[n], {n, 1, 200, 2}]

a(n) = my(m = 2*n-1, c = 5*m+1); c/2^valuation(c, 2); \\ Michel Marcus, Mar 27 2016

a(n) = A000265(A017341(n-1)). - Michel Marcus, Mar 27 2016

cp's OEIS Frontend

A017361 a(n) = (10*n + 7)^9.

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A017362 a(n) = (10*n + 7)^10.

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Magma

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A032588 Lucky numbers ending with digit 7.

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A053180 Numbers ending in 7 which are not prime.

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A152579 a(n) = (10*n+3)*(10*n+17).

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A154412 Primes of the form 10n^2+14n+5, n >= 0.

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Magma

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A168458 a(n) = 7 + 10*floor((n-1)/2).

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A268867 Number of integers of the form m^2+1 between two consecutive pairs of primes of the same form.

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A152579 a(n) = (10n+3)(10*n+17).