cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A235377 Positions of 10s in A235354.

Original entry on oeis.org

56, 57, 69, 78, 83, 89, 92, 95, 104, 109, 111, 123, 124, 128, 136, 139, 142, 158, 160, 171, 173, 176, 177, 178, 180, 185, 194, 200, 201, 203, 211, 214, 219, 222, 223, 228, 231, 236, 237, 241, 245, 246, 248, 256, 259, 270, 274, 280, 281, 296, 298, 302, 307, 314
Offset: 1

Views

Author

Vladimir Shevelev and Peter J. C. Moses, Jan 08 2014

Keywords

Comments

If prime(a(n)) is written in base k>=2, and the k-representation is read in decimal, then all such numbers, for k = 2,3,...,9, are composite.

Crossrefs

Cf. A052033, A235354, A235376, A236174, A236437.

Programs

  • PARI
    isok(n) = {my(p = prime(n)); for (b = 2, 9, if (isprime(subst(Pol(digits(p, b)), x, 10)), return(0));); return (1);} \\ Michel Marcus, Jan 18 2014

A235376 Positions of 9's in A235354.

Original entry on oeis.org

58, 167, 244, 300, 310, 326, 348, 361, 380, 394, 422, 493, 520, 525, 560, 574, 677, 826, 877, 935, 938, 1010, 1049, 1116, 1168, 1201, 1210, 1212, 1334, 1346, 1382, 1411, 1433, 1585, 1591, 1640, 1683, 1750, 1790, 1796, 1805, 1848, 1858, 1891, 1997, 2032, 2114
Offset: 1

Views

Author

Vladimir Shevelev and Peter J. C. Moses, Jan 08 2014

Keywords

Comments

If prime(a(n)) is written in base k, 2<=k<=9, and the k-representation is read in decimal, then it will be a prime only for k=9.

Crossrefs

Cf. A000040, A235354.

Programs

  • PARI
    isok(n) = {my(p = prime(n)); for (b = 2, 8, if (isprime(subst(Pol(digits(p, b)), x, 10)), return(0));); isprime(subst(Pol(digits(p, 9)), x, 10));} \\ Michel Marcus, Jan 18 2014

A235379 Numerator of arithmetic mean of the first n terms of A235354.

Original entry on oeis.org

3, 5, 7, 11, 3, 19, 23, 27, 29, 33, 40, 11, 49, 53, 11, 59, 66, 35, 73, 77, 27, 42, 88, 15, 94, 48, 11, 103, 107, 37, 117, 121, 43, 66, 134, 23, 140, 71, 146, 37, 150, 51, 157, 40, 164, 83, 169, 59, 181, 183, 11, 97, 198, 101, 42, 55, 230, 239, 242, 247, 250, 127
Offset: 1

Views

Author

Vladimir Shevelev and Peter J. C. Moses, Jan 08 2014

Keywords

Comments

Denominators are in A235380.
See comments in A235354.

Crossrefs

Cf. A235354.

A235380 Denominator of arithmetic mean of the first n terms of A235354.

Original entry on oeis.org

1, 2, 3, 4, 1, 6, 7, 8, 9, 10, 11, 3, 13, 14, 3, 16, 17, 9, 19, 20, 7, 11, 23, 4, 25, 13, 3, 28, 29, 10, 31, 32, 11, 17, 35, 6, 37, 19, 39, 10, 41, 14, 43, 11, 45, 23, 47, 16, 49, 50, 3, 26, 53, 27, 11, 14, 57, 58, 59, 60, 61, 31, 63, 64, 65, 22, 67, 68, 23
Offset: 1

Views

Author

Vladimir Shevelev and Peter J. C. Moses, Jan 08 2014

Keywords

Comments

Numerators are in A235379.

Crossrefs

Cf. A235354, A235379.

A052033 Primes base 10 that are never primes in any smaller base b, 2<=b<10, expansions interpreted as decimal numbers.

Original entry on oeis.org

263, 269, 347, 397, 431, 461, 479, 499, 569, 599, 607, 677, 683, 719, 769, 797, 821, 929, 941, 1019, 1031, 1049, 1051, 1061, 1069, 1103, 1181, 1223, 1229, 1237, 1297, 1307, 1367, 1399, 1409, 1439, 1453, 1487, 1489, 1523, 1553, 1559, 1571, 1619, 1637
Offset: 1

Views

Author

Patrick De Geest, Dec 15 1999

Keywords

Links

Crossrefs

Cf. A038537, A052026-A052033, A236174, A236437, A235354, A235377.

Programs

A236174 Maximal prime among the base-k representations of the n-th prime, read in decimal, for k=2,3,...,10.

Original entry on oeis.org

2, 11, 101, 13, 23, 31, 101, 103, 10111, 131, 43, 211, 131, 223, 101111, 311, 113, 331, 2111, 1013, 1021, 2221, 1103, 1011001, 1201, 1100101, 10211, 1223, 1231, 1301, 331, 2003, 211, 12011, 10010101, 2113, 10011101, 10100011, 2213, 10101101, 10110011, 20201, 2333, 21011, 3011, 11000111, 21211, 337, 3203, 11100101
Offset: 1

Views

Author

Vladimir Shevelev, Jan 19 2014

Keywords

Comments

Let p = n-th prime. Write p in base k, k=2,3,4,5,..., and stop when the result is a prime when looked at in base 10. - N. J. A. Sloane, Jan 25 2014

Examples

			Let n=10, then prime(n)=29 (in base 10). The representations of 29 in bases 2,3,4,...,10 are 11101,1002,131,...,29 respectively. In this list 131 is the first and therefore the maximal prime. Thus a(10)=131.

Links

Crossrefs

Cf. A000040, A052033, A235354, A235376, A235377, A236437.

Programs

  • Mathematica
    Map[First[First[Select[Map[{#,PrimeQ[#]}&,Map[FromDigits,IntegerDigits[Prime[#],Range[2,10]]]],#[[2]]==True&]]]&,Range[50]]
Table[SelectFirst[Table[FromDigits[IntegerDigits[Prime[n],b]],{b,2,10}],PrimeQ],{n,80}] (* Harvey P. Dale, May 17 2024 *)
  • PARI
    base_b(n, b) = {
  my(s=[], r, x);
  while(n>0,
    r = n%b;
    n = n\b;
    s = concat(r, s)
  );
  x=10;
  eval(Pol(s))
}
A236174(maxp) = {
  my(s=[], b, t);
  forprime(p=2, maxp,
    for(b=2, 10,
      t=base_b(p, b);
      if(isprime(t), s=concat(s, t); break)
    )
  );
  s
} \\ Colin Barker, Jan 23 2014
  • Python
    from sympy import prime, isprime
def A236174(n):
    p = prime(n)
    for b in range(2,11):
        x, y, z = p, 0, 1
        while x >= b:
            x, r = divmod(x,b)
            y += r*z
            z *= 10
        y += x*z
        if isprime(y):
            return y # Chai Wah Wu, Jan 03 2015

A236437 Primes which occur in their proper place in A236174.

Original entry on oeis.org

2, 263, 269, 347, 397, 431, 461, 479, 499, 569, 599, 607, 677, 683, 719, 769, 797, 821, 929, 941, 1019, 1031, 1049, 1051, 1061, 1069, 1103, 1181, 1223, 1229, 1237, 1297, 1307, 1367, 1399, 1409, 1439, 1453, 1487, 1489, 1523, 1553, 1559, 1571, 1619, 1637, 1733, 1759, 1811, 1823, 1949, 1973, 1997
Offset: 1

Views

Author

N. J. A. Sloane, Jan 25 2014

Keywords

Comments

Primes p such that A236174(k) = prime(k) for some k. The values of k are (essentially) given in A235377.
Same as A052033 if the initial 2 is omitted.

Examples

			263 is the 56th prime and is also the 56th term in A236174.

Links

Crossrefs

Cf. A052033, A236174, A235377, A235354.

Programs

  • Python
    from sympy import prime, isprime
def A236174(n):
    p = prime(n)
    for b in range(2,11):
        x, y, z = p, 0, 1
        while x >= b:
            x, r = divmod(x,b)
            y += r*z
            z *= 10
        y += x*z
        if isprime(y):
            return y
A236437_list = [prime(n) for n in range(1,10**6) if A236174(n) == prime(n)]
# Chai Wah Wu, Jan 03 2015

A235922 Minimal k > 1 such that the base-k representation of the n-th palindrome in A002113, read in decimal, is also palindrome.

Original entry on oeis.org

2, 2, 3, 2, 3, 2, 5, 2, 3, 2, 10, 10, 2, 10, 4, 10, 10, 5, 2, 10, 6, 3, 10, 6, 3, 10, 7, 10, 6, 10, 3, 10, 10, 3, 5, 10, 10, 5, 7, 10, 2, 10, 8, 6, 10, 10, 4, 10, 4, 10, 8, 10, 6, 10, 10, 9, 10, 3, 10, 10, 10, 10, 10, 10, 9, 10, 10, 2, 10, 10, 10, 5, 10, 9, 3, 4, 5, 10, 10, 10, 2, 10, 10, 10, 3
Offset: 1

Views

Author

Vladimir Shevelev, Jan 17 2014

Keywords

Comments

We conjecture that limit of average (a(1)+...+a(n))/n exists and equals 10.

Crossrefs

Cf. A002113, A235354.

A235923 Minimal k > 1 such that the base-k representation of the n-th Ramanujan prime (A104272), read in decimal, is also a Ramanujan prime.

Original entry on oeis.org

3, 10, 4, 7, 10, 2, 10, 6, 5, 6, 5, 10, 7, 10, 8, 10, 2, 10, 10, 2, 10, 7, 4, 10, 10, 8, 3, 4, 10, 3, 3, 3, 10, 10, 4, 10, 3, 10, 10, 6, 7, 10
Offset: 1

Views

Author

Vladimir Shevelev, Jan 17 2014

Keywords

Comments

Conjecture 1. Every number 2,...,10 occurs infinitely many times.
Conjecture 2. There exists limit of average (a(1)+...+a(n))/n.
However note that, already for the eleventh Ramanujan prime 101 the binary representation, read in decimal, is rather large prime: 1100101. We cannot decide, if it is a Ramanujan prime, using b-file in A104272. Therefore, the calculation of the sequence here ends off. It is a general problem: how to decide, if a large prime is a Ramanujan one?

Examples

			The third Ramanujan prime is 17. If k=2, we have 10001, if k=3, we have 122, if k=4, we have 101. In this list, read in decimal, 101 is the first prime. Since 101 is a Ramanujan prime, then a(3)=4.

Crossrefs

Cf. A104272, A235354.

Programs

  • PARI
    ramanujan_prime_list(n) = {my(L=vector(n), s=0, k=1); for(k=1, prime(3*n)-1, if(isprime(k), s++); if(k%2==0 && isprime(k/2), s--); if(sA104272
a(n, vp, pmax) = {my(p=vp[n], d, np); for (b=2, 10, d = digits(p, b); np = fromdigits(d, 10); if (np > pmax, return (0)); if (vecsearch(vp, np), return (b)););}
lista(nn) = {my(vp = ramanujan_prime_list(nn), pmax = vecmax(vp)); for (n=1, nn, my(result = a(n, vp, pmax)); if (result, print1(result, ", "), break););} \\ use nn=10^7 to get 42 terms \\ Michel Marcus, Dec 17 2018

Extensions

a(11)-a(42) from Michel Marcus, Dec 17 2018

A236205 For k=2,3,...,10, read in decimal base-k representation of n. a(n) is the sum of numbers of divisors of all read numbers.

Original entry on oeis.org

9, 20, 20, 33, 24, 42, 28, 48, 42, 38, 36, 60, 31, 63, 49, 64, 33, 88, 31, 66, 59, 58, 31, 98, 39, 58, 67, 80, 42, 79, 30, 100, 57, 56, 64, 145, 36, 76, 52, 88, 50, 146, 26, 102, 90, 53, 32, 132, 56, 80, 58, 81, 40, 140, 44, 136, 50, 67, 36, 120, 32, 64, 114
Offset: 1

Views

Author

Vladimir Shevelev, Jan 20 2014

Keywords

Comments

In particular, for prime n's we have a subsequence: 20,20,24,28,36,31,33,31,31,42,30,36,50,26,32,40,36,32,...

Crossrefs

Cf. A000005, A235354.

Extensions

More terms from Peter J. C. Moses, Jan 20 2014
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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