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A269230 Indices for which A037053(n), the smallest prime with n digits '0', does not have n consecutive digits '0'. 4
32, 43, 46, 49, 50, 60, 69, 72, 73, 74, 78, 82, 84, 86, 88, 90, 91, 93, 94, 95, 98, 101, 107, 108, 110, 115, 116, 121, 123, 124, 125, 126, 130, 132, 136, 137, 139, 144, 147, 149, 152, 153, 154, 156, 158, 159, 160, 161, 163, 164, 166, 169, 170, 171, 172, 173, 176, 177, 178, 179 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Sequence A085824 lists the indices n for which A037053(n) has only two nonzereo digits, i.e., A037053(n) = a*10^(n+1) + b, with 1 <= a,b <= 9.

It is conjectured that, apart from A037053(0) = 2, all other terms have three nonzero digits and are therefore of the form A037053(n) = a*10^(n+2) + b*10^k + c, where 1 <= a,b,c <= 9 and 1 <= k <= n+1.

Whenever 1 < k < n+1, the n digits '0' are not consecutive but separated in two "chunks" of length n-k+1 and k-1, respectively. These indices n are listed here.

I conjecture that k < n+1 for all n (where k is function of n, of course).

For most indices n listed here, the smallest prime with n consecutive digits '0' is of the above form with k = n+1, i.e., of the form ab0...0c = (10a+b)*10^(n+1) + c.

The first index n for which this is not the case remains to be found. It can be expected that for this index n, the least prime with n consecutive digits '0' is either of the form a0...0b0c = a*10^(n+3) + b*100 + c (in which case it equals A037053(n+1)) or of the form a0...0bc with a > 9 (in which case it equals A037053(n+1) if a = 0 (mod 10)).

Sequence A269260 lists the values a > 9 such that the least prime with (at least) n consecutive '0's equals nextprime(a*10^(n+1)), for the numbers n listed here. - M. F. Hasler, Feb 22 2016

The first two values of n that do not satisfy the above forms are 192 and 213. The least prime with 192 consecutive 0's is 11100...0007. The least prime with 213 consecutive 0's is 100...000499. - Chai Wah Wu, Mar 11 2018

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..771

PROG

(PARI) for(n=1, 999, n+2<#(t=digits(A037053(n))) && !t[#t-2] && print1(n", "))

(PARI) a269230=[32]; A269230(n)={my(t); while(n>#a269230, for(k=vecmax(a269230)+1, 9e9, (t=A037053(k))>10^(k+2) && t%10^(k+2)>99 && (a269230=concat(a269230, k)) && break)); a269230[n]} \\ M. F. Hasler, Feb 22 2016

CROSSREFS

Sequence in context: A302881 A303529 A167528 * A229115 A035112 A308765

Adjacent sequences:  A269227 A269228 A269229 * A269231 A269232 A269233

KEYWORD

nonn,base

AUTHOR

M. F. Hasler, Feb 20 2016

STATUS

approved

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Last modified September 21 07:08 EDT 2019. Contains 327253 sequences. (Running on oeis4.)