A217302 -id:A217302 - cp's OEIS frontend

A217115 Greatest number (in decimal representation) with n nonprime substrings in base-5 representation (substrings with leading zeros are considered to be nonprime).

67, 88, 442, 567, 2213, 2837, 3067, 11068, 14713, 15337, 15338, 57943, 73568, 77213, 76697, 289717, 280338, 370443, 386068, 386587, 389713, 1852217, 1524067, 1898442, 1930342, 1932943, 1948568, 7242943, 9261088, 9664717, 9586567, 9654712, 9710942, 9742849, 46305443

Offset: 0

Hieronymus Fischer, Dec 20 2012

The sequence is well-defined in that for each n the set of numbers with n nonprime substrings is not empty and finite. Proof of existence: Define m(n):=2*sum_{j=i..k} 5^j, where k:=floor((sqrt(8n+1)-1)/2), i:= n-(k(k+1)/2). For n=0,1,2,3,... the m(n) in base-5 representation are 2, 22, 20, 222, 220, 200, 2222, 2220, 2200, 2000, 22222, 22220, .... m(n) has k+1 digits and (k-i+1) 2’s. Thus, the number of nonprime substrings of m(n) is ((k+1)(k+2)/2)-k-1+i=(k(k+1)/2)+i=n. This proves the statement of existence. Proof of finiteness: Each 4-digit base-5 number has at least 2 nonprime substrings. Hence, each m := 4*floor((n+2)/2)-digit number has at least 2*(m/4) = 2*floor((n+2)/2) >= n+1 nonprime substrings. Consequently, there is a boundary b < 5^(m-1) such that all numbers > b have more than n nonprime substrings. It follows, that the set of numbers with n nonprime substrings is finite.

			a(0) = 67, since 67 = 232_5 (base-5) is the greatest number with zero nonprime substrings in base-5 representation.
a(1) = 88 = 323_5 has 6 substrings in base-5 representation (2, 2, 3, 23, 32, 323), the only nonprime substring is 323_5. 88 is the greatest number with 1 nonprime substring.
a(2) = 442 = 3232_5 has 10 substrings in base-5 representation (2, 2, 3, 3, 23, 32, 32, 232, 323 and 3232), exactly 2 of them are nonprime substrings (323_5=88 and 3232_5=442), and there is no greater number with 2 nonprime substrings in base-5 representation.
a(5) = 2837 = 42322_5 has 5 nonprime substrings in base-5 representation, these are 4, 22, 42, 322 and 4232, all the other substrings are prime. There is no greater number with 5 nonprime substrings in base-5 representation.

Hieronymus Fischer, Table of n, a(n) for n = 0..70

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
Cf. A035244, A079397, A213300 - A213321.
Cf. A217102 - A217109, A217112 - A217119.
Cf. A217302 - A217309.

a(n) >= A217105(n).
a(n) >= A217305(A000217(A110592(a(n)))-n).
a(n) <= 5^(n+3).
a(n) <= 5^(4*floor(n/2)), n>1.
a(n) <= 5^min((n + 6)/2, 9*floor((n+20)/21)).
a(n) <= 125*5^(n/2).
With m := floor(log_5(a(n))) + 1:
a(n+m+1) >= 5*a(n), if a(n)!=1 (mod 5).
a(n+m) >= 5*a(n), if a(n)=1 (mod 5).

A217116 Greatest number (in decimal representation) with n nonprime substrings in base-6 representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

23, 839, 5039, 6983, 7127, 42743, 45863, 46199, 275183, 279143, 277199, 1088531, 1674863, 1651103, 1674859, 6713711, 9906599, 10045559, 10072943, 39190247, 40278647, 60273359, 60295079, 60294239, 60437659, 241671887, 342609527, 359245007, 361640159, 362625959

Offset: 0

Hieronymus Fischer, Dec 20 2012

The sequence is well-defined in that for each n the set of numbers with n nonprime substrings is not empty and finite. Proof of existence: Define m(n):=2*sum_{j=i..k} 6^j, where k:=floor((sqrt(8n+1)-1)/2), i:= n-(k(k+1)/2). For n=0,1,2,3,... the m(n) in base-6 representation are 2, 22, 20, 222, 220, 200, 2222, 2220, 2200, 2000, 22222, 22220, .... m(n) has k+1 digits and (k-i+1) 2’s. Thus, the number of nonprime substrings of m(n) is ((k+1)(k+2)/2)-k-1+i=(k(k+1)/2)+i=n. This proves the statement of existence. Proof of finiteness: Each 3-digit base-6 number has at least 1 nonprime substring. Hence, each 3(n+1)-digit number has at least n+1 nonprime substrings. Consequently, there is a boundary b < 6^(3n+2) such that all numbers > b have more than n nonprime substrings. It follows, that the set of numbers with n nonprime substrings is finite.

			(0) = 23, since 23 = 35_6 (base-6) is the greatest number with zero nonprime substrings in base-6 representation.
a(1) = 839 = 3515_6 has 1 nonprime substring in base-6 representation (= 1). All the other base-6 substrings (3, 5, 15_6=11, 35_6=23, 51_6=31, 351_6=139, 515_6=191 and 3515_6=839) are prime substrings. 839 is the greatest number with 1 nonprime substring.
a(2) = 5039 = 35155_6 has 2 nonprime substrings in base-6 representation (1 and 55_6=35), and there is no greater number with 2 nonprime substrings in base-6 representation.
a(5) = 42743 = 525515_6 has 5 nonprime substrings in base-6 representation, these are 1, 52_6=32, 55_6=35, 5515_6=1271 and 52551_6=7123, and there is no greater number with 5 nonprime substrings in base-6 representation.

Hieronymus Fischer, Table of n, a(n) for n = 0..55

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
Cf. A035244, A079397, A213300 - A213321.
Cf. A217102 - A217109, A217112 - A217119.
Cf. A217302 - A217309.

a(n) >= A217106(n).
a(n) >= A217306(A000217(num_digits_6(a(n)))-n), where num_digits_6(x) is the number of digits of the base-6 representation of x.
a(n) <= 6^min(n+3, 7*floor((n+7)/8)).
a(n) <= 216*6^n.
a(n+m+1) >= 6*a(n), where m := floor(log_6(a(n))) + 1.

A217117 Greatest number (in decimal representation) with n nonprime substrings in base-7 representation (substrings with leading zeros are considered to be nonprime).

Original entry on oeis.org

37, 331, 317, 2322, 2389, 15259, 16260, 16728, 100291, 113825, 116101, 117109, 796777, 796781, 819719, 823003, 4753901, 5577444, 5738035, 5738039, 5761027, 31150219, 39041113, 39336580, 40166250, 40326841, 40336249, 218051538, 273271861

Offset: 0

Hieronymus Fischer, Dec 20 2012

The sequence is well-defined in that for each n the set of numbers with n nonprime substrings is not empty and finite. Proof of existence: Define m(n):=2*sum_{j=i..k} 7^j, where k:=floor((sqrt(8n+1)-1)/2), i:= n-(k(k+1)/2). For n=0,1,2,3,... the m(n) in base-7 representation are 2, 22, 20, 222, 220, 200, 2222, 2220, 2200, 2000, 22222, 22220, .... m(n) has k+1 digits and (k-i+1) 2’s. Thus, the number of nonprime substrings of m(n) is ((k+1)(k+2)/2)-k-1+i=(k(k+1)/2)+i=n. This proves the statement of existence. Proof of finiteness: Each 3-digit base-7 number has at least 1 nonprime substring. Hence, each 3(n+1)-digit number has at least n+1 nonprime substrings. Consequently, there is a boundary b < 7^(3n+2) such that all numbers > b have more than n nonprime substrings. It follows, that the set of numbers with n nonprime substrings is finite.

			a(0) = 37, since 37 = 52_7 (base-7) is the greatest number with zero nonprime substrings in base-7 representation.
a(1) = 331 = 652_7 has 1 nonprime substring in base-7 representation (= 6). All the other base-7 substrings (2, 5, 52_7=37, 65_7=47 and 652_7=331) are prime substrings. 331 is the greatest number with 1 nonprime substring.
a(2) = 317 = 632_7 has 6 substrings in base-7 representation (2, 3, 6, 32, 63 and 632), exactly 2 of them are nonprime substrings (6 and 32_6=20), and there is no greater number with 2 nonprime substrings in base-7 representation.
a(8) = 100291 = 565252_3 has 8 nonprime substrings in base-7 representation, these are 6, 252_7, 525_7, 565_7, 5252_7, 5652_7, 6525_7 and 65252_7. There is no greater number with 8 nonprime substrings in base-7 representation.

Hieronymus Fischer, Table of n, a(n) for n = 0..50

Cf. A019546, A035232, A039996, A046034, A069489, A085823, A211681, A211682, A211684, A211685.
Cf. A035244, A079397, A213300 - A213321.
Cf. A217102 - A217109, A217112 - A217119.
Cf. A217302 - A217309.

a(n) >= A217107(n).
a(n) >= A217307(A000217(num_digits_7(a(n)))-n), where num_digits_7(x) is the number of digits of the base-7 representation of x.
a(n) <= 7^min(n+2, 5*floor((n+4)/5)).
a(n) <= 7^(n+2).
a(n) <= 7^min(3 + n/2, 8*floor((n+15)/16)).
a(n) <= 343*7^(n/2).
With m := floor(log_7(a(n))) + 1:
a(n+m+1) >= 7*a(n), if a(n)!=1 (mod 7).
a(n+m) >= 7*a(n), if a(n)=1 (mod 7).

cp's OEIS Frontend

A217115 Greatest number (in decimal representation) with n nonprime substrings in base-5 representation (substrings with leading zeros are considered to be nonprime).

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A217116 Greatest number (in decimal representation) with n nonprime substrings in base-6 representation (substrings with leading zeros are considered to be nonprime).

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A217117 Greatest number (in decimal representation) with n nonprime substrings in base-7 representation (substrings with leading zeros are considered to be nonprime).

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