A030000 -id:A030000 - cp's OEIS frontend

A322919 Numbers k such that k and k-1 both first appear in the same power of 2 (in base 10).

59, 74, 111, 785, 793, 914, 957, 985, 1070, 1467, 2019, 2099, 2332, 2610, 2934, 3028, 3083, 3311, 3334, 3973, 4198, 4208, 4334, 4590, 4689, 4785, 5247, 5350, 5535, 6166, 6335, 6669, 6761, 7167, 7340, 7707, 7980, 8668, 8990, 9180, 9840, 11110, 13096, 16285, 17418, 18091, 18361, 19219, 20522, 21494, 21827

Offset: 1

Keith F. Lynch, Dec 30 2018

			For instance 2019 is in the sequence since 2018 and 2019 both appear in 2^212 and neither appear in any smaller power of two.

David A. Corneth, Table of n, a(n) for n = 1..3106

Indices of consecutive repeats in A030000.

#include 
int main() {
  int n = 1000001;  /* Highest term */
  int p = 2;        /* Powers of two.  Test throughly if you change it. */
  int r = 10;       /* Base ten.  Test throughly if you change it. */
  char a[n];
  int b,c,i,j,k,k2,l,lk,m,ok,ok2,u,d[7],f[n],g[n],v[n];
  u = n;
  for (j=0;jM. F. Hasler, Jul 05 2021*/
    f[j] = g[j] = -1;
  }
  a[0] = 1;
  for (m=0;(m-1;l--) {
          ok = 1;
          for (j=lk-1;j>-1;j--) if (a[l+j] != d[j]) ok = 0;
          if (ok) ok2 = 1;
        }
        if (ok2) {
          f[k] = m;
          u--;
        }
      }
      if ((g[k]==-1) && (lk<=b+1)) {
        ok = 1;
        for (j=lk-1;j>-1;j--) if (a[b-lk+j+1] != d[j]) ok = 0;
        if (ok) g[k] = m;
      }
    }
    c = 0;
    for (j=0;j r-1) {
        c = a[j] / r;
        a[j] %= r;
      }
    }
  }
  for (i=1;i

uptoQdigits(n) = {v = vector(10^n); p = 1/2; todo = 10^n; my(res = List());
for(i = 1, oo, p<<=1; process(p, n); if(todo <= 0, break)); for(i = 1, #v - 1,
if(v[i] == v[i+1], listput(res, i))); res}
process(p, n) = {my(dp = digits(p), vd, lp = logint(p, 2)); qdp = #dp; my(t = min(n, qdp)); for(qd = 1, t, for(j = 1, qdp - qd + 1, vd = fromdigits(vector(qd, i, dp[j+i-1])); if(v[vd + 1] == 0, v[vd + 1] = lp; todo--)))} \\ David A. Corneth, Dec 31 2018

A328375 Numbers k such that the decimal expansion of 2^k contains the substring 777.

Original entry on oeis.org

24, 40, 75, 152, 166, 179, 181, 191, 194, 199, 214, 230, 235, 260, 282, 296, 304, 311, 317, 323, 326, 332, 342, 345, 363, 370, 374, 390, 417, 424, 426, 443, 455, 468, 471, 474, 475, 483, 489, 490, 505, 512, 523, 524, 536, 540, 559, 567, 581, 584, 585, 588, 593

Offset: 1

Eder Vanzei, Oct 14 2019

The decimal expansion of 2^k ends in 7776 iff k == 40 (mod 500), so the sequence is infinite. - Jon E. Schoenfield, Oct 14 2019

Conjecture: if n > 30536, then a(n) = n + 3623. - Chai Wah Wu, Oct 26 2019

			16777216 = 2^24.

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

Cf. A007356 (contains 666), A030000 (contains n).

q:= n-> searchtext("777", cat(2^n))>0:
select(q, [$1..600])[];  # Alois P. Heinz, Oct 26 2019

aQ[n_] := SequenceCount[IntegerDigits[2^n], {7, 7, 7}] > 0; Select[Range[660], aQ] (* Amiram Eldar, Oct 26 2019 *)

A328375_list = [k for k in range(1000) if '777' in str(2**k)] # Chai Wah Wu, Oct 26 2019

A346203 a(n) is the smallest nonnegative number k such that the decimal expansion of the product of the first k primes contains the string n.

Original entry on oeis.org

3, 0, 1, 3, 10, 7, 2, 9, 9, 8, 4, 18, 17, 11, 15, 16, 14, 18, 24, 16, 11, 4, 9, 5, 21, 13, 13, 13, 9, 21, 3, 5, 10, 14, 12, 13, 26, 24, 12, 17, 18, 15, 12, 26, 16, 22, 10, 16, 12, 11, 13, 7, 13, 20, 17, 19, 11, 20, 15, 18, 11, 14, 21, 13, 10, 24, 20, 14, 21, 8, 9

Offset: 0

Ilya Gutkovskiy, Jul 10 2021

			a(5) = 7 since 5 occurs in prime(7)# = 2 * 3 * 5 * 7 * 11 * 13 * 17 = 510510, but not in prime(0)#, prime(1)#, prime(2)#, ..., prime(6)#.

Ilya Gutkovskiy, Scatterplot of a(n) up to n=100000
Index entries for sequences related to primorial numbers

Cf. A002110, A030000, A062584, A082058, A346120.

primorial[n_] := Product[Prime[j], {j, 1, n}]; a[n_] := (k = 0; While[! MatchQ[IntegerDigits[primorial[k]], {_, Sequence @@ IntegerDigits[n], _}], k++]; k); Table[a[n], {n, 0, 70}]

a(n) = my(k=0, p=1, q=1, sn=Str(n)); while (#strsplit(Str(q), sn)==1, k++; p=nextprime(p+1); q*=p); k; \\ Michel Marcus, Jul 13 2021; corrected Jun 15 2022

from sympy import nextprime
def A346203(n):
    m, k, p, s = 1, 0, 1, str(n)
    while s not in str(m):
        k += 1
        p = nextprime(p)
        m *= p
    return k # Chai Wah Wu, Jul 12 2021

A102387 a(n) is the smallest number m such that 2^m contains the first n decimal digits of Pi.

Original entry on oeis.org

5, 17, 74, 144, 144, 2003, 2003, 37929, 82810, 161449, 712201, 2401519, 7339199, 33662541

Offset: 1

Sergei Bernstein (sergeibernstein(AT)gmail.com), Feb 22 2005

			The first number is 5 because 2^5 is 32, which contains 3, the first digit of Pi.

Cf. A030000, A011545.

a(n) = A030000(A011545(n)).

a(11)-a(14) from Giovanni Resta, Nov 14 2013

A176761 Partial sums of A030001, starting at n=1.

Original entry on oeis.org

1, 3, 35, 39, 295, 311, 33079, 33087, 37183, 38207, 1099511665983, 1099511666111, 1099511797183, 1099512059327, 1099514156479, 1099514156495, 1099648374223, 1100722116047, 1100722124239, 1100722126287, 1100722388431, 9896815410639

Offset: 1

Jonathan Vos Post, Apr 25 2010

Partial sums of smallest power of 2 whose decimal expansion contains n. One may see this sequence expressed in binary, rather than decimal, for clarity, though "carries" obscure the characteristic function aspects of structure. The subsequence of primes in this partial sum begins: 3, 311, 1100722116047, 1100722388431.

			a(6) = 1 + 2 + 32 + 4 + 256 + 16 = 311 is prime.

Cf. A030000, A030001.

a(n) = SUM[i=1..n] A030001(i).

A248018 Least number k > 0 such that n^k contains n*R_n in its decimal representation, or 0 if no such k exists.

Original entry on oeis.org

1, 43, 119, 96, 186, 1740, 6177, 8421, 104191, 0, 946417

Offset: 1

Talha Ali, Sep 29 2014

R_n is the repunit of length n, i.e., R_n = (10^n-1)/9, A002275.
a(10^n) = 0 for all n > 0. - Derek Orr, Sep 29 2014
a(9) > 86000. - Derek Orr, Sep 29 2014
Note that a(2) = A030000(22), and a(3) = A063566(333), and that sequence is also related in a similar way to sequences from A063567 up to A063572. - Michel Marcus, Sep 30 2014

			a(2) = 43 because 2^43 = 8796093022208 has the string '22' in it and 43 is the smallest power of 2 that produces such a result.
a(3) = 119 because 3^119 = 599003433304810403471059943169868346577158542512617035467 contains the string '333', and 119 is the smallest power of 3 that gives us such a result.

Cf. A002275.

def a(n):
  s = str(n)
  p = len(s)
  if s.count('1') == 1 and s.count('0') == p - 1:
    return 0
  k = 1
  while not str(n**k).count(n*s):
    k += 1
  return k
n = 1
while n < 10:
  print(a(n),end=', ')
  n += 1
# Derek Orr, Sep 29 2014

a(3) and a(5) corrected, a(6)-a(8) added by Derek Orr, Sep 29 2014

a(4) corrected and a(9)-a(11) added by Hiroaki Yamanouchi, Oct 01 2014

cp's OEIS Frontend

A322919 Numbers k such that k and k-1 both first appear in the same power of 2 (in base 10).

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A328375 Numbers k such that the decimal expansion of 2^k contains the substring 777.

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A346203 a(n) is the smallest nonnegative number k such that the decimal expansion of the product of the first k primes contains the string n.

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A102387 a(n) is the smallest number m such that 2^m contains the first n decimal digits of Pi.

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A176761 Partial sums of A030001, starting at n=1.

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A248018 Least number k > 0 such that n^k contains n*R_n in its decimal representation, or 0 if no such k exists.

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