A078403 -id:A078403 - cp's OEIS frontend

A078400 Iterated sum-of-digits of A078403(n).

2, 3, 5, 7, 2, 5, 2, 5, 7, 2, 5, 7, 7, 2, 7, 2, 5, 5, 2, 5, 7, 5, 2, 2, 7, 2, 5, 7, 5, 2, 7, 2, 5, 5, 7, 2, 7, 5, 7, 2, 7, 5, 2, 5, 5, 7, 7, 2, 7, 2, 2, 5, 5, 7, 5, 2, 2, 5, 7, 5, 7, 2, 5, 2, 7, 2, 7, 7, 7, 5, 5, 5, 2, 2, 7, 2, 5, 7, 2, 2, 5, 2, 5, 2, 7, 5, 2, 5, 7, 5, 7, 7, 7, 2, 5, 2, 7, 2, 2, 5, 7, 2, 5, 7, 7

Offset: 1

Cino Hilliard, Dec 24 2002

Cino Hilliard, Proof of the Digital Root Theorem, posted on Yahoo group B2LCC, Feb 17 2003

Cf. A038194, A078403.

drp(n) = { forprime(x=2,n, r = x%9; if(isprime(r), print1(r" "); ); ); }

More terms from Joshua Zucker, Jul 24 2006

A167134 Primes congruent to {2, 3, 5, 7} mod 11.

Original entry on oeis.org

2, 3, 5, 7, 13, 29, 47, 71, 73, 79, 101, 113, 137, 139, 157, 167, 179, 181, 211, 223, 227, 233, 269, 271, 277, 293, 311, 313, 337, 359, 379, 401, 409, 421, 431, 443, 467, 487, 491, 509, 541, 557, 563, 577, 599, 601, 607, 619, 641, 643, 673, 709, 733, 739, 751

Offset: 1

Klaus Brockhaus, Oct 28 2009

Primes p such that p mod 11 is prime.
Primes of the form 11*n+r where n >= 0 and r is in {2, 3, 5, 7}.
2 and primes congruent to {3, 5, 7, 13} mod 22. - Chai Wah Wu, Apr 29 2025

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

Cf. A003627, A045326, A003631, A045309, A045314, A042987, A078403, A042993, A167134, A167135, A167119: primes p such that p mod k is prime, for k = 3..13 resp.

[ p: p in PrimesUpTo(760) | p mod 11 in {2, 3, 5, 7} ];
[ p: p in PrimesUpTo(760) | exists(t){ n: n in [0..p div 11] | exists(u){ r: r in {2, 3, 5,7} | p eq (11*n+r) } } ];

Select[Prime[Range[600]],MemberQ[{2, 3, 5, 7},Mod[#,11]]&] (* Vincenzo Librandi, Aug 05 2012 *)

A167135 Primes congruent to {2, 3, 5, 7, 11} mod 12.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 19, 23, 29, 31, 41, 43, 47, 53, 59, 67, 71, 79, 83, 89, 101, 103, 107, 113, 127, 131, 137, 139, 149, 151, 163, 167, 173, 179, 191, 197, 199, 211, 223, 227, 233, 239, 251, 257, 263, 269, 271, 281, 283, 293, 307, 311, 317, 331, 347, 353, 359

Offset: 1

Klaus Brockhaus, Oct 28 2009

Primes p such that p mod 12 is prime.
Primes of the form 12*n+r where n >= 0 and r is in {2, 3, 5, 7, 11}.
Except for the prime 2, these are the primes that are encountered in the set of numbers {x, f(f(x))} where x is of the form 4k+3 with k>=0, and where f(x) is the 3x+1-problem function, and f(f(x)) the second iteration value. Indeed this sequence is the set union of 2 and A002145 (4k+3 primes) and A007528 (6k+5 primes), since f(f(4k+3))=6k+5. Equivalently one does not get any prime from A068228 (the complement of the present sequence). - Michel Marcus and Bill McEachen, May 07 2016

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

Subsequences: A002145, A007528. Complement: A068228.

Cf. A003627, A045326, A003631, A045309, A045314, A042987, A078403, A042993, A167134, A167135, A167119: primes p such that p mod k is prime, for k = 3..13 resp.

[ p: p in PrimesUpTo(760) | p mod 12 in {2, 3, 5, 7, 11} ];

[ p: p in PrimesUpTo(760) | exists(t){ n: n in [0..p div 12] | exists(u){ r: r in {2, 3, 5,7, 11} | p eq (12*n+r) } } ];

isA167135  := n -> isprime(n) and not modp(n, 12) != 1:
select(isA167135, [$1..360]); # Peter Luschny, Mar 28 2018

Select[Prime[Range[400]],MemberQ[{2,3, 5, 7, 11},Mod[#,12]]&] (* Vincenzo Librandi, Aug 05 2012 *)
Select[Prime[Range[72]], Mod[#, 12] != 1 &] (* Peter Luschny, Mar 28 2018 *)

A167119 Primes congruent to 2, 3, 5, 7 or 11 (mod 13).

Original entry on oeis.org

2, 3, 5, 7, 11, 29, 31, 37, 41, 59, 67, 83, 89, 107, 109, 137, 163, 167, 193, 197, 211, 223, 239, 241, 263, 271, 293, 317, 349, 353, 367, 379, 397, 401, 419, 421, 431, 449, 457, 479, 499, 509, 523, 557, 577, 587, 601, 613, 631, 653, 661, 683, 691, 709, 733, 739, 743, 757

Offset: 1

Vladimir Joseph Stephan Orlovsky, Oct 27 2009

Primes which have a remainder mod 13 that is prime.

Union of A141858, A100202, A102732, A140371 and A140373. - R. J. Mathar, Oct 29 2009

			11 mod 13 = 11, 29 mod 13 = 3, 31 mod 13 = 5, hence 11, 29 and 31 are in the sequence.

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

Cf. A003627, A045326, A003631, A045309, A045314, A042987, A078403, A042993, A167134, A167135: primes p such that p mod k is prime, for k = 3..12 resp.

[ p: p in PrimesUpTo(740) | p mod 13 in {2, 3, 5, 7, 11} ]; // Klaus Brockhaus, Oct 28 2009

f[n_]:=PrimeQ[Mod[n,13]]; lst={};Do[p=Prime[n];If[f[p],AppendTo[lst,p]],{n,6,6!}];lst
Select[Prime[Range[4000]],MemberQ[{2, 3, 5, 7, 11},Mod[#,13]]&] (* Vincenzo Librandi, Aug 05 2012 *)

{forprime(p=2, 740, if(isprime(p%13), print1(p, ",")))} \\ Klaus Brockhaus, Oct 28 2009

Edited by Klaus Brockhaus and R. J. Mathar, Oct 28 2009 and Oct 29 2009

A079130 Primes such that iterated sum-of-digits (A038194) is a square.

Original entry on oeis.org

13, 19, 31, 37, 67, 73, 103, 109, 127, 139, 157, 163, 181, 193, 199, 211, 229, 271, 283, 307, 337, 373, 379, 397, 409, 433, 463, 487, 499, 523, 541, 571, 577, 607, 613, 631, 643, 661, 733, 739, 751, 757, 769, 787, 811, 823, 829, 859, 877, 883, 919, 937, 967

Offset: 1

Klaus Brockhaus, Dec 28 2002

Primes which are 1 or 4 mod 9. - Charles R Greathouse IV, Sep 04 2014

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A078403, A079131, A079132.

select(isprime,map(t -> (9*t+1,9*t+4),[$1..1000]));  # Robert Israel, Sep 04 2014

sQ[n_]:=MemberQ[{1,4,9},NestWhile[Total[IntegerDigits[#]]&,n,#>9&]]; Select[Prime[Range[300]],sQ] (* Harvey P. Dale, Dec 06 2012 *)

forprime(p=2,1000,if(issquare(p%9),print1(p,",")))

a(n) ~ 3n log n. - Charles R Greathouse IV, Sep 04 2014

A079131 Primes such that iterated sum-of-digits (A038194) is odd.

Original entry on oeis.org

3, 5, 7, 19, 23, 37, 41, 43, 59, 61, 73, 79, 97, 109, 113, 127, 131, 149, 151, 163, 167, 181, 199, 223, 239, 241, 257, 271, 277, 293, 307, 311, 313, 331, 347, 349, 367, 379, 383, 397, 401, 419, 421, 433, 439, 457, 487, 491, 509, 523, 541, 547, 563, 577, 599

Offset: 1

Klaus Brockhaus, Dec 28 2002

Subsequence of primes of A187318. - Michel Marcus, Jun 08 2015

Primes congruent to 1, 3, 5, 7 mod 18. - Robert Israel, Jun 08 2015

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

Cf. A038194, A078403, A079130, A079132, A187318.

[p: p in PrimesUpTo(600) | p mod 18 in [1,3,5,7]]; // Vincenzo Librandi, Jun 07 2015

[a: n in [0..1000] | IsPrime(a) where a is Floor(9*n/5)]; // Vincenzo Librandi, Jun 08 2015

select(isprime, [3, seq(seq(i*18+j, j=[1,5,7]),i=0..100)]); # Robert Israel, Jun 08 2015

Select[Prime[Range[120]], OddQ[Mod[#, 9]] &] (* Bruno Berselli, Aug 31 2012 *)

forprime(p=2,600,if((p%9)%2==1,print1(p,",")))

A079132 Primes such that iterated sum-of-digits (A038194) is even.

Original entry on oeis.org

2, 11, 13, 17, 29, 31, 47, 53, 67, 71, 83, 89, 101, 103, 107, 137, 139, 157, 173, 179, 191, 193, 197, 211, 227, 229, 233, 251, 263, 269, 281, 283, 317, 337, 353, 359, 373, 389, 409, 431, 443, 449, 461, 463, 467, 479, 499, 503, 521, 557, 569, 571, 587, 593, 607

Offset: 1

Klaus Brockhaus, Dec 28 2002

Cf. A078403, A079130, A079131.

Select[Prime[Range[120]], EvenQ[Mod[#, 9]] &] (* Bruno Berselli, Aug 31 2012 *)

forprime(p=2,600,if((p%9)%2==0,print1(p,",")))

A157868 Palindromic primes with prime digital roots.

Original entry on oeis.org

2, 3, 5, 7, 11, 101, 131, 151, 191, 313, 353, 383, 727, 797, 929, 10301, 10501, 11311, 12821, 13331, 13831, 15451, 16061, 16661, 17471, 17971, 19391, 19991, 30103, 30803, 32423, 35053, 35753, 36263, 36563, 37573, 38183, 38783, 70207, 70607

Offset: 1

Lekraj Beedassy, Mar 08 2009

A002385, A078403

Corrected and extended by Ray Chandler, Mar 12 2009

A079155 The number of primes less than 10^n whose digital root (A038194) is also prime.

Original entry on oeis.org

4, 15, 85, 619, 4800, 39266, 332276, 2880818, 25423985, 227527467

Offset: 1

Robert G. Wilson v, Dec 27 2002

			a(2) = 15 because the only primes less than 100 whose have digital roots are also prime are {2,3,5,7,11,23,29,41,43,47,59,61,79,83,97}.

The primes are in A078403, their digital roots are in A078400.

c = 0; k = 1; Do[ While[ k < 10^n, If[ PrimeQ[k] && PrimeQ[ Mod[k, 9]], c++ ]; k++ ]; Print[c], {n, 1, 8}]

# use primerange (slower) vs. sieve.primerange (>> memory) for larger terms
from sympy import isprime, sieve
def afind(terms):
  s = 0
  for n in range(1, terms+1):
    s += sum(isprime(p%9) for p in sieve.primerange(10**(n-1), 10**n))
    print(s, end=", ")
afind(7) # Michael S. Branicky, Apr 15 2021

a(9)-a(10) from Michael S. Branicky, Apr 15 2021

A277995 Primes with prime subscripts whose digits are primes, whose digital root is prime, whose sum of digits is prime and whose reversal is also prime.

Original entry on oeis.org

3, 5, 353, 32732237, 35225327, 75527537, 75535277, 75557723, 75737723, 75755257, 77322233, 77752733, 322375577, 322775737, 325725577, 325773727, 337735553, 352272233, 355322777, 357333377, 357735773, 372577727, 372753727, 375577733, 375722377, 375727237, 377725723, 377752723

Offset: 1

Ilya Gutkovskiy, Nov 08 2016

Intersection of A006450, A007500, A019546, A028834 and A078403.

			32732237 is in the sequence because 32732237 is the 2016197-th prime number, 2016197 is prime, digits 2, 3 and 7 are primes, 32732237 -> 3 + 2 + 7 + 3 + 2 + 2 + 3 + 7 = 29 (is prime) -> 2 + 9 = 11 -> 1 + 1 = 2, 2 is prime and 73223723 is also prime.

Cf. A000040, A006450, A007500, A019546, A028834, A078403, A087368.

Select[Table[Prime[Prime[n]], {n, 1500000}], Complement[IntegerDigits[#1], {2, 3, 5, 7}] == {} && PrimeQ[#1 - 9 Floor[(#1 - 1)/9]] && PrimeQ[Total[IntegerDigits[#1]]] && PrimeQ[FromDigits[Reverse[IntegerDigits[#1]]]] & ]

cp's OEIS Frontend

A078400 Iterated sum-of-digits of A078403(n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions

A167134 Primes congruent to {2, 3, 5, 7} mod 11.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

A167135 Primes congruent to {2, 3, 5, 7, 11} mod 12.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Magma

Maple

Mathematica

A167119 Primes congruent to 2, 3, 5, 7 or 11 (mod 13).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A079130 Primes such that iterated sum-of-digits (A038194) is a square.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A079131 Primes such that iterated sum-of-digits (A038194) is odd.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Magma

Maple

Mathematica

PARI

A079132 Primes such that iterated sum-of-digits (A038194) is even.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

A157868 Palindromic primes with prime digital roots.

Views