A132232 -id:A132232 - 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

A132235 Primes congruent to 23 (mod 30).

Original entry on oeis.org

23, 53, 83, 113, 173, 233, 263, 293, 353, 383, 443, 503, 563, 593, 653, 683, 743, 773, 863, 953, 983, 1013, 1103, 1163, 1193, 1223, 1283, 1373, 1433, 1493, 1523, 1553, 1583, 1613, 1733, 1823, 1913, 1973, 2003, 2063, 2153, 2213, 2243, 2273, 2333, 2393

Offset: 1

Omar E. Pol, Aug 15 2007

Primes (excluding 3) ending in 3 with (SOD-1)/3 non-integer where SOD is sum of digits. - Ki Punches
The sequence is infinite by Dirichlet's theorem. - Arkadiusz Wesolowski, Apr 02 2014
Terms are non-twin primes A007510. - Omar E. Pol, Jul 25 2019

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
C. K. Caldwell, The Prime Pages.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Omar E. Pol, Prime numbers in a sieve with 30 columns

Cf. A000040, A039949.

Cf. A132230, A132231, A132232, A132233, A132234, A132236.

[p: p in PrimesUpTo(3000) | p mod 30 eq 23 ]; // Vincenzo Librandi, Aug 14 2012

Select[Prime[Range[1000]],MemberQ[{23},Mod[#,30]]&] (* Vincenzo Librandi, Aug 14 2012 *)
Select[Range[23,2400,30],PrimeQ] (* Harvey P. Dale, Jan 27 2020 *)

is(n)=isprime(n) && n%30==23 \\ Charles R Greathouse IV, Jul 01 2016

a(n) = A158791(n)*30 + 23. - Ray Chandler, Apr 07 2009

Intersection of A030431 and A007528. - Ray Chandler, Apr 07 2009

Extended by Ray Chandler, Apr 07 2009

A158614 Numbers n such that 30*n + 11 is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 9, 10, 13, 14, 15, 16, 17, 21, 23, 25, 27, 29, 30, 31, 32, 34, 35, 36, 38, 39, 43, 45, 48, 49, 50, 52, 53, 57, 60, 62, 63, 64, 69, 70, 71, 78, 79, 80, 81, 84, 86, 87, 90, 91, 93, 95, 100, 101, 106, 107, 108, 112, 115, 116, 119, 122, 123, 125, 127, 128

Offset: 1

Ki Punches, Mar 22 2009, Mar 29 2009

nonn

Encoded primes with LSD 1 and (SOD-1)/3 non-integer, (LSD, least significant digit; SOD, sum of digits). Divide any such number by 30, if the whole number portion of the quotient is in the sequence, the number is prime.

			Example: 3191, with LSD 1 and (SOD-1)/3 = 4.33 (non-integer); Then 3191/30=106.367, or 106 which is in the sequence, thus 3191 is prime.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A111175, A158573, A158648, A158746, A158791, A158806, A158850.

Select[Range[0,130],PrimeQ[30#+11]&] (* Harvey P. Dale, Jul 26 2011 *)

is(n)=isprime(30*n+11) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = (A132232(n) - 11)/30 = floor(A132232(n)/30). - Ray Chandler, Apr 07 2009

Edited by Ray Chandler, Apr 07 2009

A132247 Twin primes congruent to {1, 11, 13, 17, 19, 29} mod 30.

Original entry on oeis.org

11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73, 101, 103, 107, 109, 137, 139, 149, 151, 179, 181, 191, 193, 197, 199, 227, 229, 239, 241, 269, 271, 281, 283, 311, 313, 347, 349, 419, 421, 431, 433, 461, 463, 521, 523, 569, 571, 599, 601, 617, 619, 641, 643

Offset: 1

Omar E. Pol, Aug 18 2007

Twin primes that are greater than 7. - Omar E. Pol, Oct 31 2013

C. K. Caldwell, The Prime Pages
C. K. Caldwell, Twin Primes
Omar E. Pol, Prime numbers in a sieve with 30 columns
Omar E. Pol, Sobre el patrón de los números primos

Cf. A001097, A039949, A132230, A132232-A132234, A132236, A132238-A132246, A132248-A132259.

a(n) = A001097(n+3). - Michel Marcus, Nov 03 2013

A282321 Lesser of twin primes congruent to 11 (mod 30).

Original entry on oeis.org

11, 41, 71, 101, 191, 281, 311, 431, 461, 521, 641, 821, 881, 1031, 1061, 1091, 1151, 1301, 1451, 1481, 1721, 1871, 1931, 2081, 2111, 2141, 2381, 2591, 2711, 2801, 3251, 3371, 3461, 3581, 3671, 3821, 3851, 4001, 4091, 4241, 4271, 4421, 4481, 4721, 4931, 5021

Offset: 1

Martin Renner, Feb 11 2017

nonn

The union of [this sequence and A282322] is A132241.
The union of [{3, 5}, this sequence, A282323 and A060229] is the lesser of twin primes sequence A001359.
The union of [{3, 5, 7}, A282321 to A282326] is the twin primes sequence A001097.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Subset of A001097, A001359, A132232, A132241 and A132247.

Cf. A006512, A232880, A232881, A232882, A282322, A282323, A060229, A060229, A282326.

[p: p in PrimesUpTo(5000) | IsPrime(p+2) and p mod 30 eq 11 ]; // Vincenzo Librandi, Feb 12 2017

a:={}:
for i from 1 to 1229 do
  if isprime(ithprime(i)+2) and ithprime(i) mod 30 = 11 then
    a:={op(a),ithprime(i)}:
  fi:
od:
a;

list(lim)=my(v=List(), p=2); forprime(q=3, lim+2, if(q-p==2 && q%30==13, listput(v, p)); p=q); Vec(v) \\ Charles R Greathouse IV, Feb 14 2017

A140378 Lengths of runs of consecutive primes and nonprimes in A007775.

Original entry on oeis.org

1, 12, 1, 6, 1, 3, 1, 6, 2, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 4, 2, 1, 2, 6, 1, 1, 1, 1, 1, 6, 2, 1, 2, 4, 3, 2, 2, 4, 1, 1, 1, 1, 1, 3, 1, 2, 2, 1, 1, 2, 1, 2, 1, 3, 1, 4, 2, 1, 1, 2, 2, 3, 2, 2, 4, 2, 2, 1, 1, 4, 2, 1, 1, 4, 1, 3, 2, 1, 1, 3, 1, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 2, 2

Offset: 1

Juri-Stepan Gerasimov, Jun 13 2008

nonn

Primes can be classified according to their remainder modulo 30: remainder 1 (A136066), 7 (A132231), 11 (A132232), 13 (A132233), 17 (A039949), 19 (A132234), 23 (A132235), or 29 (A132236). In the sequence A007775 of all numbers (prime or nonprime) in any of these remainder classes, we look for runs of numbers that are successively prime or nonprime and place the lengths of these runs in this sequence.

			Groups of runs in A007775 are (1), (7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47), (49), (53, 59, 61, 67, 71, 73), (77), (79, 83,...), which is 1 nonprime followed by 12 primes followed by 1 nonprime followed by 6 primes etc.

Cf. A136066, A132231, A132232, A132233, A039919, A132234, A132235, A132236.

A007775 := proc(n) option remember ; local a; if n = 1 then 1; else for a from A007775(n-1)+1 do if a mod 2 <>0 and a mod 3 <>0 and a mod 5 <> 0 then RETURN(a) ; fi ; od: fi ; end: A := proc() local al,isp,n; al := 0: isp := false ; n := 1: while n< 300 do a := A007775(n) ; if isprime(a) <> isp then printf("%d,",al) ; al := 1; isp := not isp ; else al := al+1 ; fi ; n := n+1: od: end: A() ; # R. J. Mathar, Jun 16 2008

Edited by R. J. Mathar, Jun 16 2008

A339463 Primes p such that (p-1)/gpf(p-1) = 2^q * 5^r with q, r >= 1, where gpf(m) is the greatest prime factor of m, A006530.

Original entry on oeis.org

71, 101, 131, 191, 251, 281, 311, 401, 431, 461, 521, 701, 761, 821, 881, 941, 971, 1031, 1061, 1091, 1151, 1181, 1301, 1361, 1451, 1481, 1511, 1571, 1601, 1721, 1811, 1901, 1931, 2081, 2111, 2141, 2351, 2411, 2441, 2621, 2711, 2741, 2801, 3041, 3251, 3371

Offset: 1

Bernard Schott, Dec 13 2020

nonn

These primes that are all congruent to 11 (mod 30) form a subsequence of A132232. The first terms of A132232 that are not terms here are 11, 41, 491, ... (see examples)

			41 is prime, 40/5 = 8 = 2^3, hence 41 is not a term.
101 is prime, 100/5 = 20 = 2^2 * 5, hence 101 is a term.
491 is prime, 490/7 = 70 = 2 * 5 * 7, hence 491 is not a term.
521 is prime, 520/13 = 40 = 2^3 * 5, hence 521 is a term.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B46, p. 154.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006093 (prime(n)-1), A006530, A052126, A339464.
Cf. A074781 (ratio=2^k), A339465 (ratio=2^q*3^r).
Subsequence of A132232 and of A339466.

alias(pf = NumberTheory:-PrimeFactors): gpf := n -> max(pf(n)):
is_a := n -> isprime(n) and pf((n-1)/gpf(n-1)) = {2, 5}:
select(is_a, [$5..3371]); # Peter Luschny, Dec 13 2020

q[n_] := Divisible[n, 10] && ((PrimeQ[(r = n/2^IntegerExponent[n, 2]/5^(e = IntegerExponent[n, 5]))] && r > 5) || (r == 1 && e > 1)); Select[Range[3500], PrimeQ[#] && q[# - 1] &] (* Amiram Eldar, Dec 13 2020 *)

A351597 Primes p such that the 6 consecutive primes starting with p are congruent to 11, 13, 17, 19, 23, 29 (modulo 30) in this order.

Original entry on oeis.org

11, 1481, 27701, 165701, 317921, 326141, 397751, 558791, 585911, 661091, 716411, 739391, 959831, 1015361, 1022501, 1068701, 1156031, 1161401, 1246361, 1265861, 1461401, 1514321, 1917731, 1940711, 2183921, 2188871, 2296871, 2725781, 2896931, 3058871, 3075341

Offset: 1

Zak Seidov, May 02 2022

nonn

All terms are congruent to 11 (modulo 30) by definition.

			The six consecutive primes starting with 1481: 1481, 1483, 1487, 1489, 1493, 1499 are congruent to 11, 13, 17, 19, 23, 29 (modulo 30) in this order.

Subsequence of A132232.

Select[Partition[Prime[Range[222000]], 6, 1], Mod[#, 30] == {11, 13, 17, 19, 23, 29} &][[;; , 1]] (* Amiram Eldar, May 03 2022 *)

A140387 Binary encoding of the location of primes in integer sets r+30*n with remainder r=1,7,11,..,29.

Original entry on oeis.org

1, 32, 16, 129, 73, 36, 194, 6, 42, 176, 225, 12, 21, 89, 18, 97, 25, 243, 44, 44, 196, 34, 166, 90, 149, 152, 109, 66, 135, 225, 89, 169, 169, 28, 82, 210, 33, 213, 179, 170, 38, 92, 15, 96, 252, 171, 94, 7, 209, 2, 187, 22, 153, 9, 236, 197, 71, 179, 212, 197, 186

Offset: 1

Juri-Stepan Gerasimov, Jun 10 2008

Classify all integers 30n+r with r= 1, 7, 11, 13, 17, 19, 23 or 29 as nonprime or prime and assign bit positions 0=LSB, 1, 2, 3, .., 7=MSB to the 8 remainders in the same order. Raise the bit if 30n+r is nonprime, erase it if 30n+r is prime.

The sequence interprets this as a number in base 2 and shows the decimal representation.

			For n=1, the 8 numbers 31 (r=1), 37 (r=7), 41 (r=11), 43 (r=17), 47 (r=17), 49 (r=19), 53 (r=23) and 59 (r=29) are prime, prime, prime, prime, prime, nonprime, prime, prime, prime, which is rendered into the binary 000100000 = 2^5=32=a(1).

Cf. A132236, A132235, A132234, A039919, A132233, A132232, A136066, A140378.

Cf. A105052 (analog in base 10, prime = bit 1, remainder 1 = MSB), A140891 (analog in base 14, prime = bit 0, remainder 1 = LSB).

Edited by R. J. Mathar, Jun 17 2008

A229947 Primes congruent to {1, 11, 13, 17, 19, 29} mod 30.

Original entry on oeis.org

11, 13, 17, 19, 29, 31, 41, 43, 47, 59, 61, 71, 73, 79, 89, 101, 103, 107, 109, 131, 137, 139, 149, 151, 163, 167, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 239, 241, 251, 257, 269, 271, 281, 283, 311, 313, 317, 331, 347, 349, 359, 373, 379, 389

Offset: 1

Omar E. Pol, Oct 27 2013

For twin primes congruent to {1, 11, 13, 17, 19, 29} mod 30 see A132247.

Complement of A132237, primes congruent to 7 or 23 (mod 30), in the set of primes > 5. - M. F. Hasler, Nov 02 2013

Omar E. Pol, Prime numbers in a sieve with 30 columns

Cf. A000040, A001097, A039949, A132230, A132232-A132234, A132236, A132238-A132259, A230462.

[p: p in PrimesUpTo(500) | p mod 30 in {1,11,13,17, 19,29} ]; // Vincenzo Librandi, Apr 05 2015

Select[Flatten[Table[30n + {1, 11, 13, 17, 19, 29}, {n, 0, 11}]], PrimeQ] (* Alonso del Arte, Nov 01 2013 *)
Select[Prime@Range[100], MemberQ[{1, 11, 13, 17, 19, 29}, Mod[#, 30]] &] (* Vincenzo Librandi, Apr 05 2015 *)

is(n)=isprime(n) && setsearch([1,11,13,17,19,29], n%30) \\ Charles R Greathouse IV, Mar 08 2015

a(n) ~ 4/3 n log n. - Charles R Greathouse IV, Mar 08 2015

cp's OEIS Frontend

A030430 Primes of the form 10*n+1.

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A132235 Primes congruent to 23 (mod 30).

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A158614 Numbers n such that 30*n + 11 is prime.

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A132247 Twin primes congruent to {1, 11, 13, 17, 19, 29} mod 30.

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A282321 Lesser of twin primes congruent to 11 (mod 30).

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A140378 Lengths of runs of consecutive primes and nonprimes in A007775.

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A339463 Primes p such that (p-1)/gpf(p-1) = 2^q * 5^r with q, r >= 1, where gpf(m) is the greatest prime factor of m, A006530.

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