A017305 -id:A017305 - cp's OEIS frontend

A355309 Carmichael numbers ending in 3.

52633, 63973, 334153, 670033, 997633, 2508013, 2628073, 5968873, 6733693, 13696033, 15829633, 15888313, 18900973, 26280073, 27336673, 46483633, 53711113, 65241793, 67653433, 75765313, 124630273, 133344793, 158864833, 182356993, 227752993, 242641153, 292244833, 426821473, 577240273, 580565233, 600892993

Offset: 1

Omar E. Pol, Jul 25 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Carmichael numbers

Intersection of A002997 and A017305.

Cf. A352970, A354609, A355305, A355307.

Select[10*Range[0, 10^7] + 3, CompositeQ[#] && Divisible[# - 1, CarmichaelLambda[#]] &] (* Amiram Eldar, Jul 25 2022 *)

from itertools import count, islice
from sympy import factorint
def A355309_gen(): # generator of terms
    for n in count(3,10):
        f = factorint(n)
        if len(f) == sum(f.values()) > 1 and not any((n-1) % (p-1) for p in f):
            yield n
A355309_list = list(islice(A355309_gen(),5)) # Chai Wah Wu, Jul 26 2022

A017198 a(n) = (9*n + 3)^2.

Original entry on oeis.org

9, 144, 441, 900, 1521, 2304, 3249, 4356, 5625, 7056, 8649, 10404, 12321, 14400, 16641, 19044, 21609, 24336, 27225, 30276, 33489, 36864, 40401, 44100, 47961, 51984, 56169, 60516, 65025, 69696

Offset: 0

N. J. A. Sloane

a(n) = A000290(A017197(n)) = A156677(n+2) + A017305(n). - Reinhard Zumkeller, Jul 13 2010

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

[(9*n+3)^2: n in [0..35]]; // Vincenzo Librandi, Jul 23 2011

(9*Range[0, 30] + 3)^2 (* or *) LinearRecurrence[{3,-3,1},{9,144,441},30] (* Harvey P. Dale, Dec 25 2015 *)

a(n)=(9*n+3)^2 \\ Charles R Greathouse IV, Jun 17 2017

a(0)=9, a(1)=144, a(2)=441, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Dec 25 2015

G.f.: (-9 - 117*x - 36*x^2) / (x-1)^3. - R. J. Mathar, Jul 14 2016

A136191 Primes p such that 2p-3 and 2p+3 are both prime (A092110), with last decimal being 3.

Original entry on oeis.org

13, 43, 53, 113, 193, 223, 283, 563, 613, 643, 743, 773, 1033, 1193, 1453, 1483, 1543, 1583, 1663, 1733, 2143, 2393, 2503, 2843, 3163, 3413, 3433, 3793, 3823, 4133, 4463, 4483, 4523, 4603, 4673, 4813, 5443, 5743, 5953, 6073, 6133, 6163, 6553, 6733, 6863

Offset: 1

Carlos Alves, Dec 20 2007

Except for p=5, the decimals in A092110 end in 3 or 7.

Theorem: If in the triple (2n-3,n,2n+3) all numbers are primes then n=5 or the decimal representation of n ends in 3 or 7. Proof: Consider Q=(2n-3)n(2n+3), by hypothesis factorized into primes. If n is prime, n=10k+r with r=1,3,7 or 9. We want to exclude r=1 and r=9. Case n=10k+1. Then Q=5(-1+6k+240k^2+800k^3) and 5 is a factor; thus 2n-3=5 or n=5 or 2n+1=5 : this means n=4 (not prime); or n=5 (included); or n=2 (impossible, because 2n-3=1). Case n=10k+9. Then Q=5(567+1926k+2160k^2+800k^3) and 5 is a factor; the arguments, for the previous case, also hold.

Intersection of A092110 and A017305.

Cf. A136192.

Select[Prime[Range[1000]],AllTrue[{2#-3,2#+3},PrimeQ]&&IntegerDigits[#][[-1]]==3&] (* James C. McMahon, Apr 30 2025 *)

isok(n)  = (n % 10 == 3) && isprime(n) && isprime(2*n-3) && isprime(2*n+3); \\ Michel Marcus, Sep 02 2013

A154408 Primes p such that (p^2 + 1)/10 is also prime.

Original entry on oeis.org

7, 13, 17, 23, 37, 53, 67, 97, 103, 113, 127, 137, 163, 167, 197, 223, 227, 263, 277, 283, 347, 367, 373, 383, 397, 433, 503, 547, 587, 617, 653, 673, 677, 683, 773, 797, 823, 877, 883, 937, 947, 953, 997, 1063, 1103, 1117, 1163, 1187, 1213, 1367, 1423, 1447

Offset: 1

Vincenzo Librandi, Jan 09 2009

			37 is in the sequence because both 37 and (37^2 + 1)/10 = 137 are primes. [_Emeric Deutsch_, Jan 21 2009]

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

Cf. A017305.

[p: p in PrimesInInterval(7, 2500) | IsPrime((p^2 + 1) div 10)]; // Vincenzo Librandi, Oct 15 2012

a := proc (n) if isprime(n) = true and type((1/10)*n^2+1/10, integer) = true and isprime((1/10)*n^2+1/10) = true then n else end if end proc: seq(a(n), n = 2 .. 1700); # Emeric Deutsch, Jan 21 2009

Select[Prime[Range[200]], PrimeQ[(#^2 + 1)/10] &] (* Vincenzo Librandi, Oct 15 2012 *)

Corrected and extended by Emeric Deutsch, Jan 21 2009

A017307 a(n) = (10*n + 3)^3.

Original entry on oeis.org

27, 2197, 12167, 35937, 79507, 148877, 250047, 389017, 571787, 804357, 1092727, 1442897, 1860867, 2352637, 2924207, 3581577, 4330747, 5177717, 6128487, 7189057, 8365427, 9663597, 11089567, 12649337, 14348907, 16194277, 18191447, 20346417, 22665187, 25153757, 27818127

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 (4,-6,4,-1).

Cf. A000578, A017305.

[(10*n+3)^3: n in [0..35]]; // Vincenzo Librandi, Jul 31 2011

(10 Range[0,30]+3)^3 (* or *) LinearRecurrence[{4,-6,4,-1},{27,2197,12167,35937},30] (* Harvey P. Dale, Jul 05 2021 *)

a(n) = (10*n + 3)^3; \\ Michel Marcus, Apr 14 2022

G.f.: (27 + 2089*x + 3541*x^2 + 343*x^3)/(x-1)^4. - R. J. Mathar, Mar 20 2018

a(n) = A000578(A017305(n)). - Michel Marcus, Apr 14 2022

A152161 a(n) = 100n^2 + 100n + 21.

Original entry on oeis.org

21, 221, 621, 1221, 2021, 3021, 4221, 5621, 7221, 9021, 11021, 13221, 15621, 18221, 21021, 24021, 27221, 30621, 34221, 38021, 42021, 46221, 50621, 55221, 60021, 65021, 70221, 75621, 81221, 87021, 93021, 99221, 105621, 112221, 119021, 126021, 133221, 140621, 148221

Offset: 0

Paul Curtz, Nov 27 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. A001622, A017305, A017353, A061037.

[(10*n+3)*(10*n+7): n in [0..40]]; // Vincenzo Librandi, Jul 28 2011

Table[100*n^2 + 100*n + 21, {n,0,50}] (* G. C. Greubel, Sep 20 2018 *)
LinearRecurrence[{3,-3,1},{21,221,621},40] (* Harvey P. Dale, Dec 06 2018 *)

a(n)=100*n*(n+1)+21 \\ Charles R Greathouse IV, Jul 28 2011

a(n) = A017305(n)*A017353(n) = A061037(10*n+3).
From Amiram Eldar, Feb 20 2023: (Start)
Sum_{n>=0} 1/a(n) = sqrt(5-2*sqrt(5))*Pi/40.
Sum_{n>=0} (-1)^n/a(n) = (sqrt(10-2*sqrt(5))*log(cot(Pi/20)) + sqrt(10+2*sqrt(5))*log(tan(3*Pi/20)))/40.
Product_{n>=0} (1 - 1/a(n)) = 2*cos(sqrt(5)*Pi/10)/phi, where phi is the golden ratio (A001622).
Product_{n>=0} (1 + 1/a(n)) = 2*cos(sqrt(3)*Pi/10)/phi. (End)
From Elmo R. Oliveira, Nov 27 2024: (Start)
G.f.: (21 + 158*x + 21*x^2)/(1-x)^3.
E.g.f.: (21 + 200*x + 100*x^2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

A163676 Triangle T(n,m) = 4mn + 2m + 2n - 1 read by rows.

Original entry on oeis.org

7, 13, 23, 19, 33, 47, 25, 43, 61, 79, 31, 53, 75, 97, 119, 37, 63, 89, 115, 141, 167, 43, 73, 103, 133, 163, 193, 223, 49, 83, 117, 151, 185, 219, 253, 287, 55, 93, 131, 169, 207, 245, 283, 321, 359, 61, 103, 145, 187, 229, 271, 313, 355, 397, 439, 67, 113, 159

Offset: 1

Vincenzo Librandi, Aug 03 2009

2 + T(n,m) = (2*n+1)*(2*m+1) are composite numbers. - clarified by R. J. Mathar, Oct 16 2009

First column: A016921, second column: A017305, third column: A126980. - Vincenzo Librandi, Nov 21 2012

			Triangle begins:
   7;
  13,  23;
  19,  33,  47;
  25,  43,  61,  79;
  31,  53,  75,  97, 119;
  37,  63,  89, 115, 141, 167;
  43,  73, 103, 133, 163, 193, 223;
  49,  83, 117, 151, 185, 219, 253, 287;
  55,  93, 131, 169, 207, 245, 283, 321, 359;
  61, 103, 145, 187, 229, 271, 313, 355, 397, 439;

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A016921, A017305, A126980, A155151, A155156.

[4*n*k + 2*n + 2*k - 1: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 21 2012

t[n_,k_]:=4 n*k + 2n + 2k - 1; Table[t[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 21 2012 *)

for(n=1,10, for(k=1,n, print1(4*n*k + 2*n + 2*k - 1, ", "))) \\ G. C. Greubel, Aug 02 2017

T(n,m) = A155151(n,m) - 3 = A155156(n,m) - 1. - R. J. Mathar, Oct 16 2009

A168437 a(n) = 3 + 10*floor(n/2).

Original entry on oeis.org

3, 13, 13, 23, 23, 33, 33, 43, 43, 53, 53, 63, 63, 73, 73, 83, 83, 93, 93, 103, 103, 113, 113, 123, 123, 133, 133, 143, 143, 153, 153, 163, 163, 173, 173, 183, 183, 193, 193, 203, 203, 213, 213, 223, 223, 233, 233, 243, 243, 253, 253, 263, 263, 273, 273, 283

Offset: 1

Vincenzo Librandi, Nov 25 2009

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

Bisections of A168437 are A017305 and (A017305 MINUS {3}). - Rick L. Shepherd, Jun 17 2010

[3+10*Floor(n/2): n in [1..70]]; // Vincenzo Librandi, Sep 19 2013

Table[3 + 10 Floor[n/2], {n, 70}] (* or *) CoefficientList[Series[(3 + 10 x - 3 x^2)/((1 + x) (x - 1)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Sep 19 2013 *)
LinearRecurrence[{1,1,-1},{3,13,13},70] (* Harvey P. Dale, May 26 2021 *)

a(n)=n\2*10+3 \\ Charles R Greathouse IV, Jan 11 2012

a(n) = 10*n - a(n-1) - 4, with n>1, a(1) = 3.
a(n) = 10*floor(n/2) + 3 = A168641(n) + 3. - Rick L. Shepherd, Jun 17 2010
G.f.: x*(3 + 10*x - 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
a(n) = (1 + 5*(-1)^n + 10*n)/2. - Bruno Berselli, Sep 19 2013
E.g.f.: (1/2)*(5 - 6*exp(x) + (10*x + 1)*exp(2*x))*exp(-x). - G. C. Greubel, Jul 22 2016

Edited by Rick L. Shepherd, Jun 17 2010

Definition rewritten, using Shepherd's formula, by Vincenzo Librandi, Sep 19 2013

A385623 Array read by ascending antidiagonals: A(n,k) is the number obtained by concatenation of n with k in that order, with k >= 0.

Original entry on oeis.org

0, 10, 1, 20, 11, 2, 30, 21, 12, 3, 40, 31, 22, 13, 4, 50, 41, 32, 23, 14, 5, 60, 51, 42, 33, 24, 15, 6, 70, 61, 52, 43, 34, 25, 16, 7, 80, 71, 62, 53, 44, 35, 26, 17, 8, 90, 81, 72, 63, 54, 45, 36, 27, 18, 9, 100, 91, 82, 73, 64, 55, 46, 37, 28, 19, 10, 110, 101, 92, 83, 74, 65, 56, 47, 38, 29, 110, 11

Offset: 0

Stefano Spezia, Jul 05 2025

			Array begins as:
   0,  1,  2,  3,  4,  5,  6,  7, ...
  10, 11, 12, 13, 14, 15, 16, 17, ...
  20, 21, 22, 23, 24, 25, 26, 27, ...
  30, 31, 32, 33, 34, 35, 36, 37, ...
  40, 41, 42, 43, 44, 45, 46, 47, ...
  50, 51, 52, 53, 54, 55, 56, 57, ...
  60, 61, 62, 63, 64, 65, 66, 67, ...
  ...

Stefano Spezia, Table of n, a(n) for n = 0..11475 (first 151 antidiagonals of the array, flattened)

Columns for k=0..9 give: A008592, A017281, A017293, A017305, A017317, A017329, A017341, A017353, A017365, A017377.

Cf. A001477 (1st row), A020338 (main diagonal), A055642, A385624 (antidiagonal sums).

A[n_,k_]:=FromDigits[Join[IntegerDigits[n],IntegerDigits[k]]]; Table[A[n,k],{n,0,6},{k,0,7}] (* or *)
A[n_,k_]:=If[k==0,10n,n*10^(Floor[Log10[k]]+1)+k]; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten

T(n, k) = fromdigits(concat(digits(n), digits(k))); \\ Michel Marcus, Jul 06 2025

A(n,0) = 10*n and A(n,k) = n*10^(floor(log_10(k)) + 1) + k for k > 0.

A017312 a(n) = (10*n + 3)^8.

Original entry on oeis.org

6561, 815730721, 78310985281, 1406408618241, 11688200277601, 62259690411361, 248155780267521, 806460091894081, 2252292232139041, 5595818096650401, 12667700813876161, 26584441929064321, 52389094428262881

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 (9, -36, 84, -126, 126, -84, 36, -9, 1).

Cf. A001016, A017305.

[(10*n+3)^8: n in [0..20]]; // Vincenzo Librandi, Jul 31 2011

A017312:=n->(10*n+3)^8; seq(A017312(n), n=0..20); # Wesley Ivan Hurt, Jan 23 2014

Table[(10*n + 3)^8, {n, 0, 20}] (* Wesley Ivan Hurt, Jan 23 2014 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{6561,815730721,78310985281,1406408618241,11688200277601,62259690411361,248155780267521,806460091894081,2252292232139041},20] (* Harvey P. Dale, Oct 16 2023 *)

a(n) = A017305(n)^8 = A001016(A017305(n)). - Wesley Ivan Hurt, Jan 23 2014

cp's OEIS Frontend

A355309 Carmichael numbers ending in 3.

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A017198 a(n) = (9*n + 3)^2.

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A136191 Primes p such that 2p-3 and 2p+3 are both prime (A092110), with last decimal being 3.

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A154408 Primes p such that (p^2 + 1)/10 is also prime.

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A017307 a(n) = (10*n + 3)^3.

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A152161 a(n) = 100*n^2 + 100*n + 21.

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A163676 Triangle T(n,m) = 4mn + 2m + 2n - 1 read by rows.

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A168437 a(n) = 3 + 10*floor(n/2).

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A152161 a(n) = 100n^2 + 100n + 21.