A045453 -id:A045453 - cp's OEIS frontend

A030430 Primes of the form 10*n+1.

11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, 271, 281, 311, 331, 401, 421, 431, 461, 491, 521, 541, 571, 601, 631, 641, 661, 691, 701, 751, 761, 811, 821, 881, 911, 941, 971, 991, 1021, 1031, 1051, 1061, 1091, 1151, 1171, 1181, 1201, 1231, 1291

Offset: 1

Warut Roonguthai

Also primes of form 5*n+1 or equivalently 5*n+6.
Primes p such that the arithmetic mean of divisors of p^4 is an integer: A000203(p^4)/A000005(p^4) = C. - Ctibor O. Zizka, Sep 15 2008
Being a subset of A141158, this is also a subset of the primes of form x^2-5*y^2. - Tito Piezas III, Dec 28 2008
5 is quadratic residue of primes of this form. - Vincenzo Librandi, Jun 25 2014
Primes p such that 5 divides sigma(p^4), cf. A274397. - M. F. Hasler, Jul 10 2016

Michael B. Porter, Table of n, a(n) for n = 1..100000
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 519.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.

Cf. A024912, A045453, A049511, A081759, A017281, A010051, A004615 (multiplicative closure).
Cf. A001583 (subsequence).
Union of A132230 and A132232. - Ray Chandler, Apr 07 2009

a030430 n = a030430_list !! (n-1)
a030430_list = filter ((== 1) . a010051) a017281_list
-- Reinhard Zumkeller, Apr 16 2012

Select[Prime@Range[210], Mod[ #, 10] == 1 &] (* Ray Chandler, Dec 06 2006 *)
Select[Range[11,1291,10],PrimeQ] (*Zak Seidov, Aug 14 2011*)

is(n)=n%10==1 && isprime(n) \\ Charles R Greathouse IV, Sep 06 2012

lista(nn) = forprime(p=11, nn, if(p%10==1, print1(p, ", "))) \\ Iain Fox, Dec 30 2017

a(n) = 10*A024912(n)+1 = 5*A081759(n)+6.
A104146(floor(a(n)/10)) = 1.
Union of A132230 and A132232. - Ray Chandler, Apr 07 2009
a(n) ~ 4n log n. - Charles R Greathouse IV, Sep 06 2012
Intersection of A000040 and A017281. - Iain Fox, Dec 30 2017

A066500 Numbers k such that 5 divides phi(k).

Original entry on oeis.org

11, 22, 25, 31, 33, 41, 44, 50, 55, 61, 62, 66, 71, 75, 77, 82, 88, 93, 99, 100, 101, 110, 121, 122, 123, 124, 125, 131, 132, 142, 143, 150, 151, 154, 155, 164, 165, 175, 176, 181, 183, 186, 187, 191, 198, 200, 202, 205, 209, 211, 213, 217, 220, 225, 231, 241

Offset: 1

Benoit Cloitre, Jan 04 2002

nonn

Related to the equation x^5 == 1 (mod k): sequence gives values of k such there are solutions 1 < x < k of x^5 == 1 (mod k).
If k is a term of this sequence, then G =  is a non-abelian group of order 5k, where 1 < r < n and r^5 == 1 (mod k). For example, G can be the subgroup of GL(2, Z_k) generated by x = {{1, 1}, {0, 1}} and y = {{r, 0}, {0, 1}}. - Jianing Song, Sep 17 2019
The asymptotic density of this sequence is 1 (Dressler, 1975). - Amiram Eldar, May 23 2022

			x^5 == 1 (mod 11) has solutions 1 < x < 11, namely {3,4,5,9}.

Harry J. Smith, Table of n, a(n) for n = 1..1000
Robert E. Dressler, A property of the phi and sigma_j functions, Compositio Mathematica, Vol. 31, No. 2 (1975), pp. 115-118.

Cf. A000010, A045453, A066498, A066499, A066501, A066502.

Column k=3 of A277915.

Select[Range[250], Divisible[EulerPhi[#], 5] &] (* Amiram Eldar, May 23 2022 *)

isok(k) = { eulerphi(k)%5 == 0 } \\ Harry J. Smith, Feb 18 2010

a(n) are the numbers generated by 5^2 = 25 and all primes congruent to 1 mod 5 (A045453). Hence sequence gives all k such that k == 0 (mod A045453(n)) for some n > 1 or k == 0 (mod 25).

Simpler definition from Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 25 2003

Extended by Ray Chandler, Nov 06 2003

A108594 Numbers k such that 10*k + 101 is prime.

Original entry on oeis.org

0, 3, 5, 8, 9, 11, 14, 15, 17, 18, 21, 23, 30, 32, 33, 36, 39, 42, 44, 47, 50, 53, 54, 56, 59, 60, 65, 66, 71, 72, 78, 81, 84, 87, 89, 92, 93, 95, 96, 99, 105, 107, 108, 110, 113, 119, 120, 122, 126, 128, 135, 137, 138, 141, 143, 147, 150, 152, 162, 164, 170, 171, 173

Offset: 1

Parthasarathy Nambi, Jul 05 2005

			If n=33 then 10*n + 101 = 431 (prime).

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

Cf. A030430, A045453, A108595.

[n: n in [0..300] | IsPrime(10*n+101)]; // G. C. Greubel, Oct 21 2023

a:=proc(n) if isprime(10*n+101)=true then n else fi end: seq(a(n),n=0..200); # Emeric Deutsch, Jul 13 2005

Select[Range[0,300], PrimeQ[10*# +101] &] (* G. C. Greubel, Oct 21 2023 *)

is(n)=isprime(10*n+101) \\ Charles R Greathouse IV, Jun 13 2017

[n for n in (0..300) if is_prime(10*n+101)] # G. C. Greubel, Oct 21 2023

More terms from Emeric Deutsch, Jul 13 2005

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A030430 Primes of the form 10*n+1.

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A066500 Numbers k such that 5 divides phi(k).

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A108594 Numbers k such that 10*k + 101 is prime.

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