Pierandrea Formusa - cp's OEIS frontend

A323026 Zeroless pandigital numbers that are between two twin primes.

123457968, 123459768, 123946578, 124397658, 124936578, 124953678, 125347698, 125437968, 125463798, 125674398, 126345978, 126495738, 126593478, 126597348, 126945738, 127394568, 127396458, 127453968, 127459638, 127659438, 129357648, 129635478, 129673548, 132564978, 132594768, 132769458

Offset: 1

Pierandrea Formusa, Jan 02 2019

Intersection of A050289 and A014574.
The definition permits repeated digits. - N. J. A. Sloane, May 16 2023
a(n) mod 10 is either 2 or 8. First term with a(n) mod 10 = 2 is a(27) = 134756982. - Chai Wah Wu, Jan 27 2019
There are 2595 terms that are pandigital without repeating any digit. - Harvey P. Dale, Jan 25 2021

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

Cf. A050289 (zeroless pandigital numbers), A014574 (average of twin primes).

isok(n) = my(d=digits(n)); vecmin(d) && (#Set(d)==9) && isprime(n-1) && isprime(n+1);
for (n=123456789, 133000000, if (isok(n), print1(n, ", "))) \\ Michel Marcus, Jan 04 2019

import itertools
from sympy import isprime
nmax=pow(10,10)
r=""
li=[]
def is_pandigit_easy(n):
    l=[]
    s=str(n)
    if '0' in s: return(False)
    for ch in s:
        if ch not in l: l.append(ch)
    l=list(set(l))
    if len(l)==9:
        return(True)
    else:
        return(False)
t=0
tmax=50
for i in range(123456789,nmax):
    if is_pandigit_easy(i):
        if isprime(i-1) and isprime(i+1):
            li.append(i)
            t+=1
            if t>tmax: break
first_elem=26
for k in li[:first_elem]:
      r=r+","+str(k)
print(r[1:])

from itertools import permutations
from sympy import isprime
A323026_list = [n for n in (int(''.join(s)) for s in permutations('123456789')) if isprime(n-1) and isprime(n+1)] # Chai Wah Wu, Jan 27 2019

A322525 Numbers such that the list of exponents of their factorization is a palindromic list of primes.

Original entry on oeis.org

2700, 5292, 9000, 13068, 18252, 24300, 24500, 24696, 31212, 38988, 47628, 55125, 57132, 60500, 68600, 84500, 90828, 95832, 103788, 117612, 136125, 144500, 147852, 158184, 164268, 166012, 180500, 181548, 190125, 199692, 218700, 231525, 231868, 238572, 243000, 264500, 266200, 280908, 303372, 325125

Offset: 1

Pierandrea Formusa, Dec 13 2018

nonn

I mean nontrivial palindrome: more than one number and not all equal numbers.

Factorization is meant to produce p1^e1*...*pk^ek, with pi in increasing order.

			9000 is a term as 9000=2^3*3^2*5^3 and the correspondent exponents list [3,2,3] is a palindromic list of primes.

Subsequence of A242414.

aQ[s_] := Length[Union[s]]>1 && AllTrue[s, PrimeQ] && PalindromeQ[s]; Select[Range[1000], aQ[FactorInteger[#][[;;,2]]] &] (* Amiram Eldar, Dec 14 2018 *)

isok(n) = (ve=factor(n)[,2]~) && (Vecrev(ve)==ve) && (#ve>1) && (#Set(ve)>1) && (#select(x->(!isprime(x)), ve) == 0); \\ Michel Marcus, Dec 14 2018

from sympy.ntheory import factorint,isprime
def all_prime(l):
    for i in l:
        if not(isprime(i)): return(False)
    return(True)
def all_equal(l):
    ll=len(l)
    set_l=set(l)
    lsl=list(set_l)
    llsl=len(lsl)
    return(llsl==1)
def pal(l):
    return(l == l[::-1])
n=350000
r=""
lp=[]
lexp=[]
def calc(n):
    global lp,lexp
    a=factorint(n)
    lp=[]
    for p in a.keys():
        lp.append(p)
    lexp=[]
    for exp in a.values():
        lexp.append(exp)
    return
for i in range(4,n):
    calc(i)
    if len(lexp)>1:
        if all_prime(lexp):
            if not(all_equal(lexp)):
                if pal(lexp):
                    r += ","+str(i)
print(r[1:])

A321890 Primes of the form p^2 + 16 where p is prime.

Original entry on oeis.org

41, 137, 857, 977, 1697, 6257, 7937, 11897, 22817, 32057, 36497, 44537, 52457, 78977, 96737, 151337, 160817, 177257, 192737, 249017, 326057, 361217, 434297, 477497, 491417, 516977, 546137, 564017, 591377, 674057, 737897, 776177, 885497, 942857, 982097, 1018097

Offset: 1

Pierandrea Formusa, Nov 20 2018

Also primes of the form p^2 + q^4 where p and q are primes. The proof of the equivalence of the set of primes p^2 + q^4, where p and q are primes, with respect to this sequence, is the following: Exactly 1 of (p, q) giving a term is 2. q^4 + 4 is divisible by 5 and/or composite and gives no terms. - David A. Corneth, Nov 21 2018

			41 is prime and 41 = 5^2 + 16, where 5 is prime, therefore 41 is a term.

select(isprime,[ithprime(p)^2+16$p=1..180]); # Muniru A Asiru, Nov 24 2018

Select[Prime[Range[100]]^2+16, PrimeQ] (* Amiram Eldar, Nov 21 2018 *)

include "globals.mzn";
int: n = 1;
int: max_val = 1200000;
array[1..n+1] of var 2..max_val: x;
% primes between 2..max_valset of int:
prime = 2..max_val diff { i | i in 2..max_val, j in 2..ceil(sqrt(i)) where i mod j = 0} ;
set of int: primes;primes = prime union {2};
solve satisfy;
constraint all_different(x) /\ x[1] in primes /\ x[2] in primes /\
pow(x[1],2)+16= x[2] ;
output [ show(x)]

upto(n) = my(res = List()); forprime(p = 3, sqrtint(n-16), if(isprime(p^2 + 16), listput(res, p^2 + 16))); res \\ David A. Corneth, Nov 21 2018

More terms from Amiram Eldar, Nov 21 2018

A320342 Maximum term in Cunningham chain of the first kind generated by the n-th prime.

Original entry on oeis.org

47, 7, 47, 7, 47, 13, 17, 19, 47, 59, 31, 37, 167, 43, 47, 107, 59, 61, 67, 71, 73, 79, 167, 2879, 97, 101, 103, 107, 109, 227, 127, 263, 137, 139, 149, 151, 157, 163, 167, 347, 2879, 181, 383, 193, 197, 199, 211, 223, 227, 229, 467, 479, 241, 503, 257, 263, 269, 271, 277, 563, 283, 587, 307, 311, 313, 317, 331, 337, 347, 349

Offset: 1

Pierandrea Formusa, Dec 10 2018

nonn

No term is a Sophie Germain prime.

A181697 is the sequence of the lengths of the chains in the name.

			a(1)=47 as prime(1)=2 and the Cunningham chain generated by 2 is (2,5,11,23,47), with maximum item 47.

Cf. A181697.

a[n_] := NestWhile[2#+1&, n, PrimeQ, 1, Infinity, -1]; a/@Prime@Range@70  (* Amiram Eldar, Dec 11 2018 *)

def cunningham_chain(p,t):
    # returns the Cunningham chain generated by p of type t (1 or 2)
    from sympy.ntheory import isprime
    if not(isprime(p)):
        raise Exception("Invalid starting number! It must be prime")
    if t!=1 and t!=2:
        raise Exception("Invalid type! It must be 1 or 2")
    elif t==1: k=t
    else: k=-1
    cunn_ch=[]
    cunn_ch.append(p)
    while isprime(2*p+k):
        p=2*p+k
        cunn_ch.append(p)
    return(cunn_ch)
from sympy import prime
n=71
r=""
for i in range(1,n):
    cunn_ch=(cunningham_chain(prime(i),1))
    last_item=cunn_ch[-1]
    r += ","+str(last_item)
print(r[1:])

A320393 First members of the Cunningham chains of the first kind whose length is a prime.

Original entry on oeis.org

2, 3, 11, 23, 29, 41, 53, 83, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 419, 431, 443, 491, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, 1013, 1019, 1031, 1049, 1103, 1223, 1289, 1439, 1451, 1481, 1499, 1511, 1559, 1583, 1601, 1733, 1811, 1889, 1901, 1931, 1973, 2003, 2039, 2063, 2069, 2129, 2141

Offset: 1

Pierandrea Formusa, Dec 10 2018

nonn

			41 is an item as it generates the Cunningham chain (41, 83, 167), of length 3, that is prime.

Cf. A059761, A059762, A059764.

aQ[n_] := PrimeQ[Length[NestWhileList[2#+1&, n, PrimeQ]] - 1]; Select[Range[2200], aQ] (* Amiram Eldar, Dec 11 2018 *)

from sympy.ntheory import isprime
def cunningham_chain(p,t):
    #it returns the cunningham chain generated by p of type t (1 or 2)
    if not(isprime(p)):
        raise Exception("Invalid starting number! It must be prime")
    if t!=1 and t!=2:
        raise Exception("Invalid type! It must be 1 or 2")
    elif t==1: k=t
    else: k=-1
    cunn_ch=[]
    cunn_ch.append(p)
    while isprime(2*p+k):
        p=2*p+k
        cunn_ch.append(p)
    return(cunn_ch)
from sympy import prime
n=350
r=""
for i in range(1,n):
    cunn_ch=(cunningham_chain(prime(i),1))
    lcunn_ch=len(cunn_ch)
    if isprime(lcunn_ch):
       r += ","+str(prime(i))
print(r[1:])

A321891 Prime numbers of the form p^3 + q, where p and q are primes.

Original entry on oeis.org

11, 13, 19, 29, 31, 37, 61, 67, 79, 97, 109, 127, 139, 157, 181, 199, 241, 271, 277, 367, 397, 409, 439, 457, 487, 499, 571, 577, 601, 607, 661, 691, 709, 727, 751, 769, 829, 919, 937, 991, 1021, 1039, 1069, 1117, 1171, 1201, 1231, 1237, 1291, 1297, 1327, 1381

Offset: 1

Pierandrea Formusa, Nov 20 2018

nonn

For reasons of parity, either p or q must be equal to 2, so this actually is the union of (mostly) "primes of the form p + 8" (A092402) and (rarely) "primes of the form p^3 + 2" (A048636 = 29, 127, 24391, 357913, ...). - M. F. Hasler, Jan 13 2025

Except for 13, these primes are the minimum or maximum prime numbers of the respective decade. - Davide Rotondo, Jan 31 2025

			37 is prime and 37 = 2^3 + 29, where 2 and 29 are primes, therefore 37 is a term.

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

Union of A048636 and A092402. - Michel Marcus, Nov 21 2018

N:= 2000: # to get terms <= N
A1:= select(t -> isprime(t) and isprime(t-8), {11,seq(i,i=13 ..N,6)}):
v:= floor((N-2)^(1/3)):
B:= select(t -> isprime(t) and isprime(t^3+2), {3,seq(i,i=5..v,6)}):
sort(convert(A1 union map(t -> t^3+2,B), list)); # Robert Israel, Mar 05 2020

nmax=4; Select[Union[Prime[Range[nmax]]^3 + 2, Prime[Range[Prime[nmax]^3]] + 8], PrimeQ] (* Amiram Eldar, Nov 21 2018 *)

include "globals.mzn";
int: n = 2;
int: max_val = 1200000;
array[1..n+1] of var 2..max_val: x;
% primes between 2..max_valset of int:
prime = 2..max_val diff { i | i in 2..max_val, j in 2..ceil(sqrt(i)) where i mod j = 0} ;
set of int: primes; primes = prime union {2};
solve satisfy;
constraint all_different(x) /\ x[1] in primes /\ x[2] in primes /\ x[3] in primes /\
pow(x[1], 3)+pow(x[2], 1)= x[3] ;
output [ show(x)]

list(lim)=my(v=List()); forprime(p=3,sqrtnint((lim\=1)-2,3), if(isprime(p^3+2), listput(v,p^3+2))); forprime(p=11,lim+8, if(isprime(p-8), listput(v,p))); Set(v) \\ Charles R Greathouse IV, Jan 13 2025

select( {is_A321891(n)=isprime(n)&& (isprime(n-8)|| (ispower(n-2, 3, &n)&&isprime(n)))}, [1..1234]) \\ M. F. Hasler, Jan 13 2025

More terms from Amiram Eldar, Nov 21 2018

A321456 Numbers k that are divisible by sum(pi)^2+sum(ei) where k=p1^e1...pj^ej with pi primes.

Original entry on oeis.org

16, 192, 288, 704, 1470, 2112, 2160, 3168, 3240, 3872, 4096, 4608, 4752, 4860, 5400, 6912, 7128, 7245, 8100, 9295, 10368, 11616, 13500, 15552, 15900, 17424, 21296, 23328, 23850, 26136, 27720, 32830, 34992, 35960, 39600, 39750, 41536, 45584, 52250, 52488, 59400, 62920, 63888, 67200, 78732, 81920, 86430

Offset: 1

Pierandrea Formusa, Nov 18 2018

nonn

Numbers k that are divisible by A001222(k)+A235323(k).

			704 is an item as its prime factorization is 2^6+11^1, sum(pi)=2+11=13, sum(e1)=6+1=7, sum(pi)^2+sum(e1)=13^2+7=169+7=176, finally 704=c*176 for c=4.

Cf. A001222, A235323.

fun[n_] := Module[{f = FactorInteger[n]}, Total@f[[;;, 1]]^2 + Total@f[[;;, 2]]]; aQ[n_] := Divisible[n, fun[n]]; Select[Range[100000], aQ] (* Amiram Eldar, Nov 18 2018 *)

ok(k)={my(f=factor(k)); k > 1 && k % (vecsum(f[,2]) + vecsum(f[,1])^2) == 0} \\ Andrew Howroyd, Nov 19 2018

from sympy.ntheory import factorint, isprime
n=100000
r=""
def calc(n):
    global r
    a=factorint(n)
    lp=[]
    for p in a.keys():
        lp.append(p)
    lexp=[]
    for exp in a.values():
        lexp.append(exp)
    if n%((sum(lp))**2+sum(lexp))==0:
       r += ","
       r += str(n)
    return
for i in range(4,n):
    calc(i)
print(r[1:])

A319371 Numbers k such that the characteristic polynomial of a wheel graph of k nodes has exactly one monomial with vanishing coefficient.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 14, 15, 16, 18, 19, 20, 22, 23, 24, 26, 27, 28, 30, 31, 32, 34, 35, 36, 38, 39, 40, 42, 43, 44, 46, 47, 48, 50, 51, 52, 54, 55, 56, 58, 59, 60, 62, 63, 64, 66, 67, 68, 70, 71, 72, 74, 75, 76, 78, 79, 80, 82, 83, 84, 86, 87, 88

Offset: 1

Pierandrea Formusa, Sep 17 2018

nonn

For the spectrum of W_n see, e.g., the Wikipedia link. - Wolfdieter Lang, Oct 30 2018

			4 is a term as the characteristic polynomial of the wheel graph of 4 nodes is x^4 - 6*x^2 - 8*x - 3, in which the monomial of x^3 has null coefficient and no other ones, so this polynomial has exactly one monomial with vanishing coefficient.
5 is not member of this sequence because the eigenvalues of A(W_5) (the adjacency matrix of W_5) has eigenvalues 0, 0, 2, 1 + sqrt(5), 1 - sqrt(5), and the monic characteristic polynomial is x^5 - 8*x^3 - 8*x^2 with three missing monomials x^0, x^1 and x^4. - _Wolfdieter Lang_, Oct 30 2018

Wikipedia, Wheel graph

Cf. A004772.

def how_many_zeros(v):
    t=0
    for el in v:
        if el==0: t += 1
    return t
r=""
for i in range(1,100):
        p = graphs.WheelGraph(i)
        cp=p.characteristic_polynomial()
        vcp=(cp.coefficients(sparse=False))
        if how_many_zeros(vcp)==1:
            r=r+","+str(i)
print(r)

Conjecture: a(n) = A004772(n) for n> 1. [clarified by Michel Marcus, Apr 16 2019]
Conjectures from Colin Barker, Nov 02 2020: (Start)
G.f.: x*(1 + x + x^2 + x^4) / ((1 - x)^2*(1 + x + x^2)).
a(n) = a(n-1) + a(n-3) - a(n-4) for n>5.
(End)

A319017 Prime numbers which satisfy the regex m1{1,m1}m2{1,m2}m3{1,m3}m4{1,m4}m5{1,m5} where mi are one-digit Lucas numbers.

Original entry on oeis.org

21347, 2213347, 2213447, 21334447, 21344447, 21344777, 22133477, 213334477, 221344477, 2134444777, 22133344447, 221344477777, 2133344447777, 2133347777777, 2213447777777, 22133344447777, 213344447777777, 221333447777777, 22133344447777777

Offset: 1

Pierandrea Formusa, Sep 07 2018

mi=Lucas numbers sequence(i), i.e., m1=2, m2=1, m3=3 and so on.

There are 168 = 2*1*3*4*7 numbers matching this regular expression, of which 19 are prime. - Charles R Greathouse IV, Jan 21 2025

			The prime number 2213447 belongs to this sequence as 2, 1, 3, 4 and 7 are the one-digit Lucas numbers (ordered as in Lucas number sequence) and each digit k is repeated at most k times (including itself), in accordance with the regex described in the sequence name (2 repeated two times, 1 one time, 3 repeated one time, 4 repeated two times and 7 one time).
The prime number 22133344447777777 is the best example, as each one-digit Lucas number k is repeated exactly k times.

Subsequence of A178550.

Cf. A000032.

A319141 Prime numbers p such that p squared + p reversed is also prime.

Original entry on oeis.org

211, 223, 271, 283, 433, 463, 691, 823, 859, 2017, 2029, 2251, 2269, 2293, 2341, 2347, 2593, 2647, 2833, 2851, 2857, 2887, 4153, 4327, 4507, 4513, 4903, 6091, 6361, 6421, 6481, 6529, 6871, 6949, 8011, 8059, 8161, 8209, 8287, 8419, 8467, 8707, 8803, 8929, 8971

Offset: 1

Pierandrea Formusa, Sep 11 2018

All terms == 1 (mod 6). - Robert Israel, Sep 13 2018

			271 belongs to this sequence as 271 squared is 73441 and 271 reversed is 172 and the sum of 73441 and 172 is 73613, which is prime.

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

Cf. A304390.

revdigs:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L));
end proc:
filter:= t -> isprime(t) and isprime(t^2+revdigs(t)):
select(filter, [seq(t,t=1..10^4,6)]); # Robert Israel, Sep 13 2018

Select[Prime@Range@1120, PrimeQ[#^2 + FromDigits[Reverse@IntegerDigits@#]] &] (* Vincenzo Librandi, Sep 14 2018 *)

forprime(p=1, 9000, if(ispseudoprime(p^2 + eval(concat(Vecrev(Str(p))))), print1(p, ", "))) \\ Felix Fröhlich, Sep 12 2018

nmax=10000
def is_prime(num):
    if num == 0 or num == 1: return(0)
    for k in range(2, num):
       if (num % k) == 0:
           return(0)
    return(1)
ris = ""
for i in range(nmax):
    if is_prime(i):
       r=int((str(i)[::-1]))
       t=pow(i,2)+r
       if is_prime(t):
          ris = ris+str(i)+","
print(ris)

cp's OEIS Frontend

User: Pierandrea Formusa

A323026 Zeroless pandigital numbers that are between two twin primes.

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A322525 Numbers such that the list of exponents of their factorization is a palindromic list of primes.

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Mathematica

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A321890 Primes of the form p^2 + 16 where p is prime.

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Maple

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MiniZinc

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Extensions

A320342 Maximum term in Cunningham chain of the first kind generated by the n-th prime.

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Python

A320393 First members of the Cunningham chains of the first kind whose length is a prime.

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Python

A321891 Prime numbers of the form p^3 + q, where p and q are primes.

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Examples

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Crossrefs

Programs

Maple

Mathematica

MiniZinc

PARI

PARI

Extensions

A321456 Numbers k that are divisible by sum(pi)^2+sum(ei) where k=p1^e1*...*pj^ej with pi primes.

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Mathematica

PARI

A321456 Numbers k that are divisible by sum(pi)^2+sum(ei) where k=p1^e1...pj^ej with pi primes.