A262728 -id:A262728 - cp's OEIS frontend

A262729 Strong (2,3,5,7)-primes. (See Comments for precise definition.)

2, 171472673, 343808687, 1364225981, 1469999801, 1871684753, 2110769237, 2227044401, 2411201729, 2485782361, 2545607453, 3795488227, 3946237717, 4213334953, 4395443513, 5308651577, 5770033901, 5832097819, 6385775491, 6694883219, 7064806421, 7235208829

Offset: 1

Clark Kimberling, Oct 03 2015

Let V = (b(1), b(2), ..., b(k)), where k > 1 and b(i) are distinct integers > 1 for j = 1..k. Call p a V-prime if the digits of p in base b(1) spell a prime in each of the bases b(2), ..., b(k). Call p a strong V-prime if p is a (b(j), ..., b(k))-prime for each of the tuples (b(j), ..., b(k)), for j = 1..k-1.

a(157) > 10^11. - Hiroaki Yamanouchi, Oct 25 2015

			Let p = 171472673. Confirmation that p is a strong (2,3,5,7)-prime follows.
Base-2 for p: u = (1,0,1,0,0,0,1,1,1,0,0,0,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0,1);
u in base 3 spells the prime 8488002487771;
u in base 5 spells the prime 7749195106457425001;
u is base 7 spells the prime 67054080721013093290423.
Base-3 for p: v = (1, 0, 2, 2, 2, 1, 1, 2, 2, 2, 0, 1, 0, 2, 1, 2, 0, 2);
v in base 5 spells the prime 838940251427;
v in base 7 spells the prime 243692337097757.
Base-5 for p: w = (3, 2, 2, 3, 4, 4, 1, 1, 1, 1, 4, 3);
w in base 7 spells the prime 6598716743.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..156

Cf. A000040, A262727, A262728.

{b1, b2, b3, b4} = {2, 3, 5, 7}; z = 10000000;
Select[Prime[Range[z]],
PrimeQ[FromDigits[IntegerDigits[#, b1], b2]] &&
PrimeQ[FromDigits[IntegerDigits[#, b1], b3]] &&
PrimeQ[FromDigits[IntegerDigits[#, b1], b4]] &&
PrimeQ[FromDigits[IntegerDigits[#, b2], b3]] &&
PrimeQ[FromDigits[IntegerDigits[#, b2], b4]] &&
PrimeQ[FromDigits[IntegerDigits[#, b3], b4]] &]
(* Peter J. C. Moses, Sep 27 2015 *)

a(4)-a(22) from Hiroaki Yamanouchi, Oct 25 2015

A262727 Strong (2,3,5)-primes (see comments).

Original entry on oeis.org

2, 7, 13, 41, 151, 173, 181, 223, 331, 641, 1373, 1759, 2011, 3061, 4507, 5867, 9601, 13537, 14533, 14591, 14821, 15101, 15161, 30557, 32707, 37657, 38653, 45361, 46687, 48463, 54331, 54773, 59197, 63853, 70321, 76031, 77041, 78101, 87133, 91541, 95083

Offset: 1

Clark Kimberling, Nov 07 2015

Let V = (b(1), b(2),...,b(k)), where k > 1 and b(i) are distinct integers > 1 for j = 1..k. Call p a V-prime if the digits of p in base b(1) spell a prime in each of the bases b(2), ..., b(k). Call p a strong V-prime if p is a (b(j),...,b(k))-prime for each of the vectors (b(j),...,b(k)), for j = 1..k.

Cf. A000040, A235266, A262728.

{b1, b2, b3} = {2, 3, 5}; z = 50000;
u = Select[Prime[Range[z]],
PrimeQ[FromDigits[IntegerDigits[#, b1], b2]] &&
PrimeQ[FromDigits[IntegerDigits[#, b1], b3]] &&
PrimeQ[FromDigits[IntegerDigits[#, b2], b3]] &]
(* Peter J. C. Moses, Sep 27 2015 *)

A261967 {2,3,5}-primes. (See comments.)

Original entry on oeis.org

2, 151, 3061, 9517861, 11903341, 15344551, 15460771, 19975771, 37935091, 42234271, 52312411, 199938421, 228523501, 237049321, 270798991, 315266641, 315522931, 327445201, 354600601, 423223741, 466801171, 498309631, 499063711, 547916791, 585381361, 621504721

Offset: 1

Clark Kimberling, Nov 09 2015

Let S = {b(1), b(2), ..., b(k)}, where k > 1 and b(i) are distinct integers > 1 for i = 1..k. Call p an S-prime if the digits of p in base b(i) spell a prime in each of the bases b(j) in S, for i = 1..k and j = 1..k. Equivalently, p is an S-prime if p is a strong-V prime (defined at A262729) for every permutation of the vector V = (b(1), b(2), ..., b(k)). Note that strong (2,3,5)-primes (A262727) form a proper subset of {2,3,5}-primes. It may be of interest to consider the sets of {2,3,5,7}-primes, {2,3,5,7,11}-primes, etc. Is every such set infinite?

Cf. A000040, A235266, A262727, A262841, A262728, A262729.

{b1, b2, b3} = {2, 3, 5}; z = 10000000;
Select[Prime[Range[z]],
PrimeQ[FromDigits[IntegerDigits[#, b1], b2]] &&
PrimeQ[FromDigits[IntegerDigits[#, b1], b3]] &&
PrimeQ[FromDigits[IntegerDigits[#, b2], b1]] &&
PrimeQ[FromDigits[IntegerDigits[#, b2], b3]] &&
PrimeQ[FromDigits[IntegerDigits[#, b3], b1]] &&
PrimeQ[FromDigits[IntegerDigits[#, b3], b2]] &]
(* Peter J. C. Moses, Sep 27 2015 *)

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A262729 Strong (2,3,5,7)-primes. (See Comments for precise definition.)

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A262727 Strong (2,3,5)-primes (see comments).

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A261967 {2,3,5}-primes. (See comments.)

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