A017305 -id:A017305 - cp's OEIS frontend

A017313 a(n) = (10*n + 3)^9.

19683, 10604499373, 1801152661463, 46411484401953, 502592611936843, 3299763591802133, 15633814156853823, 58871586708267913, 186940255267540403, 520411082988487293, 1304773183829244583, 3004041937984268273

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. A001017 (n^9), A017305 (10n+3).

[(10*n+3)^9: n in [0..15]]; // Vincenzo Librandi, Jul 31 2011

(10*Range[0,20]+3)^9 (* or *) LinearRecurrence[ {10,-45,120,-210,252,-210,120,-45,10,-1},{19683,10604499373,1801152661463,46411484401953,502592611936843,3299763591802133,15633814156853823,58871586708267913,186940255267540403,520411082988487293},30] (* Harvey P. Dale, Sep 14 2013 *)

for n in range(0, 15): print((10*n + 3)**9, end=", ") # Stefano Spezia, Oct 20 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)=19683, a(1)=10604499373, a(2)=1801152661463, a(3)=46411484401953, a(4)=502592611936843, a(5)=3299763591802133, a(6)=15633814156853823, a(7)=58871586708267913, a(8)=186940255267540403, a(9)=520411082988487293. - Harvey P. Dale, Sep 14 2013

A032586 Lucky numbers ending with digit 3.

Original entry on oeis.org

3, 13, 33, 43, 63, 73, 93, 133, 163, 193, 223, 273, 283, 303, 393, 433, 463, 483, 553, 583, 613, 643, 673, 693, 723, 823, 873, 883, 903, 933, 993, 1023, 1053, 1093, 1123, 1183, 1203, 1233, 1263, 1303, 1323, 1473, 1503, 1533, 1543, 1563, 1663, 1693, 1723

Offset: 1

Patrick De Geest, Apr 15 1998

Also, lucky numbers (A000959) which are congruent to 3 mod 5 (because only numbers ending in 8 could make a difference, but these are removed in the first 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 A017305.

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

A135580 Largest prime visible as a substring of 10n+3.

Original entry on oeis.org

3, 13, 23, 3, 43, 53, 3, 73, 83, 3, 103, 113, 23, 13, 43, 53, 163, 173, 83, 193, 3, 13, 223, 233, 43, 53, 263, 73, 283, 293, 3, 313, 23, 3, 43, 353, 3, 373, 383, 3, 3, 41, 23, 433, 443, 53, 463, 73, 83, 3, 503, 13, 523, 53, 43, 53, 563, 73, 83, 593, 3, 613, 23, 3, 643, 653, 3

Offset: 0

Zak Seidov, Feb 24 2008

			a(0) = 3 because 10 * 0 + 3 = 3 (prime),
a(1) = 13 because 10 * 1 + 3 = 13 (prime),
a(3) = 3 because 10 * 3 + 3 = 33 (composite). Substrings of 33 are 0, 3, 33 and the largest  prime of these is 3.
a(78) = 83 because 10 * 78 + 3 = 783 (composite). The largest prime substring is 83.

Cf. A017305, A030431, A047814.

a[n_]:=Max[Select[FromDigits/@Subsequences[IntegerDigits[10n+3]],PrimeQ]] (* James C. McMahon, Apr 16 2025 *)

a(n) = A047814(A017305(n)). - Michel Marcus, Apr 16 2025

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)

A154409 Primes of the form 10n^2+6n+1.

Original entry on oeis.org

17, 53, 109, 281, 397, 1061, 1277, 2341, 2657, 4973, 6917, 8009, 9181, 10433, 13177, 13913, 14669, 18749, 20521, 25301, 26317, 28409, 32833, 42641, 45293, 46649, 56701, 58217, 59753, 67733, 69389, 76213, 77969, 83357, 85193, 87049, 90821

Offset: 1

Vincenzo Librandi, Jan 09 2009

Subsequence of A207337. Primes of the form n^2+(3n+1)^2. - Bruno Berselli, Jul 17 2012

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

Cf. A017305, A154408.

[a: n in [0..200] | IsPrime(a) where a is 10*n^2+6*n+1]; // Vincenzo Librandi, Jul 17 2012

Select[Table[10n^2+6n+1,{n,0,700}],PrimeQ] (* Vincenzo Librandi, Jul 17 2012 *)

Misleading formula removed - R. J. Mathar, Oct 18 2010

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

A384735 Numbers that are prime or end in a prime number (of any length).

Original entry on oeis.org

2, 3, 5, 7, 11, 12, 13, 15, 17, 19, 22, 23, 25, 27, 29, 31, 32, 33, 35, 37, 41, 42, 43, 45, 47, 52, 53, 55, 57, 59, 61, 62, 63, 65, 67, 71, 72, 73, 75, 77, 79, 82, 83, 85, 87, 89, 92, 93, 95, 97, 101, 102, 103, 105, 107, 109, 111, 112, 113, 115, 117

Offset: 1

Mohd Anwar Jamal Faiz, Jun 08 2025

If k is a term, so is m*10^A055642(k) + k for all m > 0. - Michael S. Branicky, Jun 10 2025

			2, 3, 5, 7 and 11 are terms since they are prime.
15 is a term since it ends in the prime 5.
111 is a term since it ends in the prime 11.

Mohd Anwar Jamal Faiz, Primender Sequence: A Novel Mathematical Construct for Testing Symbolic Inference and AI Reasoning, arXiv:2506.10585 [cs.AI], 2025. See pp. 4, 9.

Cf. A000040, A033664, A055642.

Contains A017293, A017305, A017329, A017353.

q:= n-> isprime(n) or (k-> k>1 and q(n mod 10^(k-1)))(length(n)):
select(q, [$1..150])[];  # Alois P. Heinz, Jun 08 2025

q[n_] := AnyTrue[Range[1, IntegerLength[n]-1], PrimeQ[Mod[n, 10^#]] &]; Select[Range[120], PrimeQ[#] || q[#] &] (* Amiram Eldar, Jun 10 2025 *)

isok(x) = my(y=x, nb=0); while(y>1, y/=10; nb++; if (isprime(x%(10^nb)), return(1))); \\ Michel Marcus, Jun 10 2025

from sympy import isprime
def ok(n):
    s = str(n)
    return any(isprime(int(s[i:])) for i in range(len(s)))
print([k for k in range(118) if ok(k)])

cp's OEIS Frontend

A017313 a(n) = (10*n + 3)^9.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Python

Formula

A032586 Lucky numbers ending with digit 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A135580 Largest prime visible as a substring of 10n+3.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

PARI

Formula

A154409 Primes of the form 10n^2+6n+1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Extensions

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A384735 Numbers that are prime or end in a prime number (of any length).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

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