A343590 -id:A343590 - cp's OEIS frontend

A343675 Undulating alternating palindromic primes.

2, 3, 5, 7, 101, 181, 383, 727, 787, 929, 10301, 10501, 14341, 16361, 16561, 18181, 30103, 30703, 32323, 36563, 38183, 38783, 70507, 72727, 74747, 78787, 90709, 94949, 96769, 1074701, 1092901, 1212121, 1218121, 1412141, 1616161, 1658561, 1856581, 1878781, 3072703

Offset: 1

Chai Wah Wu, Apr 25 2021

All terms have an odd number of decimal digits.
For n > 3, a(n) is odd and not divisible by 5.
Intersection of A002385, A030144 and A059168.
Subsequence of A343590.

			16361 is a term as it is a palindromic prime, the digits 1, 6, 3, 6 and 1 have odd and even parity alternately, and also alternately rise and fall.

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

Cf. A002385, A030144, A059168, A343590.

Union@Flatten[{{2,3,5,7},Array[Select[FromDigits/@Riffle@@@Tuples[{Tuples[{1,3,5,7,9},#],Tuples[{0,2,4,6,8},#-1]}],(s=Union@Partition[Sign@Differences@IntegerDigits@#,2];(s=={{1,-1}}||s=={{-1,1}})&&PrimeQ@#&&PalindromeQ@#)&]&,4]}] (* Giorgos Kalogeropoulos, Apr 26 2021 *)

from sympy import isprime
def f(w):
    for s in w:
        for t in range(int(s[-1])+1,10,2):
            yield s+str(t)
def g(w):
    for s in w:
        for t in range(1-int(s[-1])%2,int(s[-1]),2):
            yield s+str(t)
A343675_list = [2,3,5,7]
for l in range(1,9):
    for d in '1379':
        x = d
        for i in range(1,l+1):
            x = g(x) if i % 2 else f(x)
        A343675_list.extend([int(p+p[-2::-1]) for p in x if isprime(int(p+p[-2::-1]))])
        y = d
        for i in range(1,l+1):
            y = f(y) if i % 2 else g(y)
        A343675_list.extend([int(p+p[-2::-1]) for p in y if isprime(int(p+p[-2::-1]))])

A343591 Smoothly undulating alternating primes.

Original entry on oeis.org

2, 3, 5, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 181, 383, 727, 787, 929, 18181, 32323, 72727, 74747, 78787, 94949, 1212121, 1616161, 323232323, 383838383, 727272727, 929292929, 989898989, 12121212121, 14141414141, 32323232323, 383838383838383, 38383838383838383, 72727272727272727

Offset: 1

Bernard Schott, Apr 21 2021

Equivalently, numbers that are primes, smoothly undulating = in which the digits alternate: ababab... with a <> b (A032758) and alternating = in which parity of the digits alternates (A030144).
Charles W. Trigg was the first to use the word 'smoothly' for these integers.
If we note a(ba) the terms where the substring (ba) is repeated in their decimal expansion, there exist only 16 possibilities with a odd <> 5 and b even <> 0 to get such primes. Indeed, there exist primes of the form 1(21), 1(41), 1(61), 1(81), 3(23), 3(83), 7(27), 7(47), 7(87), 9(29), 9(49), 9(89). There do not exist terms of the form 3(63), 7(67), 9(69), as they are always composite.
Now, what about possible terms of the form 3(43)? If (43) is repeated 3k times, 3(43) is divisible by 3; if (43) is repeated 3k+1 times, 3(43) is divisible by 7; so if such a prime exists, then the substring (43) must be repeated 3k+2 times, but it is not known if such smoothly undulating prime 3(43) exists and if it exists, (43) must be repeated at least 9302 times, so k >= 3100 (link).
Some properties:
-> Every term has two digits or an odd number of digits.
-> All terms with an odd number of digits are palindromic (A059758).
-> Only 2 and the nine 2-digit terms begin with an even digit.

			1616161 is a term as it is prime and the digits 1 and 6 have odd and even parity and alternate.

Charles W. Trigg, Special Palindromic Primes, Journal of Recreational Mathematics, 4 (July 1971) 169-170.

Robert Israel, Table of n, a(n) for n = 1..69
Mathematics Stack Exchange, (Unsolved) In this infinite sequence, no term is a prime: prove/disprove.

Intersection of A030144 and A032758.
Subsequence of A343590.
Cf. A059758, A242541.

f:= proc(n) local i,a,b,c,d;
  c:= add(10^i,i=1..n-1,2);
  d:= add(10^i,i=0..n-1,2);
  if n = 2 then op(select(isprime,[seq(seq(a*c+b*d, b=[1,3,7,9]),a=[2,4,6,8])]))
    else op(select(isprime, [seq(seq(a*c+b*d, a=[0,2,4,6,8]),b=[1,3,7,9])]))
  fi
end proc:
f(1):= (2,3,5,7):
map(f, [1,2,seq(i,i=3..17,2)]); # Robert Israel, Nov 09 2023

f[o_,e_,n_,m_] := FromDigits @ Riffle[ConstantArray[o,n], ConstantArray[e,n-m]]; seq[n_,m_] := Module[{o = Range[1,9,2], e = Range[0,8,2]}, Select[Union[f @@@ Join[Tuples[{o, e, {n}, {m}}], Tuples[{Rest @ e, o, {n}, {m}}]]], PrimeQ]]; s = seq[1, 1]; Do[s = Join[s, seq[m, Boole[m > 1]]], {m, 1, 10}]; s (* Amiram Eldar, Apr 21 2021 *)

from sympy import isprime
def agenthru(maxdigits):
  if maxdigits >= 1: yield from [2, 3, 5, 7]
  for digits in [2]*(maxdigits >= 2) + list(range(3, maxdigits+1, 2)):
    hlf, odd = (digits+1)//2, digits%2
    d1range = "1379" if digits%2 == 1 else "123456789"
    d2range = "1379" if digits%2 == 0 else "0123456789"
    for d1 in d1range:
      for d2 in d2range:
        if int(d1)%2 == int(d2)%2: continue
        t = int("".join([*sum(zip(d1*hlf, d2*(digits-hlf)), ())]+[d1*odd]))
        if isprime(t): yield t
print([p for p in agenthru(17)]) # Michael S. Branicky, Apr 21 2021

A343676 Number of n-digit undulating alternating primes.

Original entry on oeis.org

4, 9, 23, 49, 175, 321, 1189, 3025, 12111, 28492, 113409, 251513, 1068440, 2629980, 11690210, 28498852, 128871588, 298890814, 1309837146

Offset: 1

Chai Wah Wu, Apr 25 2021

a(n) is the number of n-digit terms in A343590.

a(n) <= A057333(n).

Cf. A002385, A030144, A057333, A059168, A343590, A343675.

def f(w):
    for s in w:
        for t in range(int(s[-1])+1,10,2):
            yield s+str(t)
def g(w):
    for s in w:
        for t in range(1-int(s[-1])%2,int(s[-1]),2):
            yield s+str(t)
def A343676(n):
    c = 0
    for d in '123456789':
        x = d
        for i in range(1,n):
            x = g(x) if i % 2 else f(x)
        c += sum(1 for p in x if isprime(int(p)))
        if n > 1:
            y = d
            for i in range(1,n):
                y = f(y) if i % 2 else g(y)
            c += sum(1 for p in y if isprime(int(p)))
    return c

a(18)-a(19) from Martin Ehrenstein, Apr 27 2021

A343677 Number of (2n+1)-digit undulating alternating palindromic primes.

Original entry on oeis.org

4, 6, 19, 34, 100, 241, 697, 1779, 6590, 16585, 57237, 179291, 591325, 1707010, 6520756, 18271423, 65212230, 210339179, 706823539

Offset: 0

Chai Wah Wu, Apr 25 2021

a(n) is the number of (2n+1)-digit terms in A343675.

a(n) <= A057332(n).

Cf. A002385, A030144, A057332, A059168, A343590.

from sympy import isprime
def f(w):
    for s in w:
        for t in range(int(s[-1])+1,10,2):
            yield s+str(t)
def g(w):
    for s in w:
        for t in range(1-int(s[-1])%2,int(s[-1]),2):
            yield s+str(t)
def A343677(n):
    if n == 0:
        return 4
    c = 0
    for d in '1379':
        x = d
        for i in range(1,n+1):
            x = g(x) if i % 2 else f(x)
        c += sum(1 for p in x if isprime(int(p+p[-2::-1])))
        y = d
        for i in range(1,n+1):
            y = f(y) if i % 2 else g(y)
        c += sum(1 for p in y if isprime(int(p+p[-2::-1])))
    return c

a(17)-a(18) from Chai Wah Wu, May 02 2021

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A343675 Undulating alternating palindromic primes.

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A343591 Smoothly undulating alternating primes.

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A343676 Number of n-digit undulating alternating primes.

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A343677 Number of (2n+1)-digit undulating alternating palindromic primes.

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Python

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