A046034 -id:A046034 - cp's OEIS frontend

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

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).

A342992 Smallest k such that k*n contains only prime digits, or 0 if no such k exists.

Original entry on oeis.org

2, 1, 1, 8, 1, 12, 1, 4, 3, 0, 2, 6, 4, 18, 5, 2, 15, 4, 3, 0, 12, 1, 1, 3, 1, 2, 1, 9, 8, 0, 12, 1, 1, 8, 1, 2, 1, 14, 7, 0, 13, 6, 54, 8, 5, 7, 5, 49, 15, 0, 5, 1, 1, 43, 1, 42, 1, 4, 43, 0, 12, 6, 4, 43, 5, 42, 5, 4, 8, 0, 5, 1, 1, 3, 1, 7, 1, 74, 3, 0, 93

Offset: 1

Metin Sariyar, Apr 13 2021

a(n) is 0 when n is divisible by 10, but when a(n) = 0, n is not always divisible by 10. For example, for n = 625, 1875, 3125, 4375, ... a(n) = 0 because no such k has been found yet for these numbers.
Conjecture: a(n) > 0 for all n that are not divisible by 5.
a(625*k) = 0 for k > 0 as the last four digits of (625*k), i.e., (625*k) mod 10000 always contains a nonprime digit. - David A. Corneth, Apr 21 2021

			a(4) = 8 because 8 is the smallest number k such that 8*4 = 32 contains only prime digits.

David A. Corneth, Table of n, a(n) for n = 1..10000
Chris K. Caldwell and G. L. Honaker, Jr., Prime Curios! 56479

Cf. A046034, A227922, A276729, A300289.

a(n) = if ((n % 10) && (n % 625), my(k=1); while (#select(x->!isprime(x), digits(k*n)), k++); k, 0); \\ Michel Marcus, Apr 21 2021

A365471 Numbers whose digits are not all primes.

Original entry on oeis.org

0, 1, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 26, 28, 29, 30, 31, 34, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 54, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 74, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86

Offset: 1

James C. McMahon, Sep 11 2023

Complement of A046034.

Union of A084984 and A365589.

Cf. A046034, A084984, A365589.

a[n_Integer?NonNegative] := Select[Range[0, n], Not[AllTrue[MemberQ[{2, 3, 5, 7}, #] & /@ IntegerDigits@#, Identity]] &]; a[86] (* Robert P. P. McKone, Sep 13 2023 *)
Select[Range[0,100],AnyTrue[IntegerDigits[#],!PrimeQ[#]&]&] (* Harvey P. Dale, Dec 22 2023 *)

A365472 Numbers whose digits are either all primes or all nonprimes.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 18, 19, 22, 23, 25, 27, 32, 33, 35, 37, 40, 41, 44, 46, 48, 49, 52, 53, 55, 57, 60, 61, 64, 66, 68, 69, 72, 73, 75, 77, 80, 81, 84, 86, 88, 89, 90, 91, 94, 96, 98, 99, 100, 101, 104, 106, 108, 109, 110, 111, 114

Offset: 1

James C. McMahon, Sep 11 2023

Complement of A365589.

Union of A046034 and A084984.

Cf. A046034, A084984, A365589.

a[n_Integer?NonNegative] := Select[Range[0, n], Module[{digits, primeDigits}, digits = IntegerDigits[#]; primeDigits = MemberQ[{2, 3, 5, 7}, #] & /@ digits; AllTrue[primeDigits, Identity] || AllTrue[primeDigits, Not]] &]; a[114] (* Robert P. P. McKone, Sep 13 2023 *)

A380490 Replace prime digits of n by 0's.

Original entry on oeis.org

1, 0, 0, 4, 0, 6, 0, 8, 9, 10, 11, 10, 10, 14, 10, 16, 10, 18, 19, 0, 1, 0, 0, 4, 0, 6, 0, 8, 9, 0, 1, 0, 0, 4, 0, 6, 0, 8, 9, 40, 41, 40, 40, 44, 40, 46, 40, 48, 49, 0, 1, 0, 0, 4, 0, 6, 0, 8, 9, 60, 61, 60, 60, 64, 60, 66, 60, 68, 69, 0, 1, 0, 0, 4, 0, 6, 0, 8, 9, 80, 81, 80

Offset: 1

Ctibor O. Zizka, Jan 25 2025

			n = 7: 7 --> 0, thus a(7) = 0.
n = 26: 26 --> 06, thus a(26) = 6.
n = 472: 472 --> 400, thus a(472) = 400.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A046034, A084984.

f:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(`if`(isprime(L[i]),0,L[i]) * 10^(i-1),i=1..nops(L))
end proc:
map(f, [$1..100]); # Robert Israel, May 18 2025

a[n_] := FromDigits[IntegerDigits[n] /. ?PrimeQ -> 0]; Array[a, 100] (* _Amiram Eldar, Jan 25 2025 *)

def a(n): return int(str(n).translate({50:48,51:48,53:48,55:48}))
print([a(n) for n in range(1, 83)]) # Michael S. Branicky, Jan 25 2025

a(A084984(n)) = A084984(n).

a(A046034(n)) = 0.

A084996 Numbers which can be written as the product of two distinct primes and containing only prime decimal digits.

Original entry on oeis.org

22, 33, 35, 55, 57, 77, 235, 237, 253, 323, 327, 335, 355, 377, 527, 533, 535, 537, 553, 573, 723, 737, 753, 755, 2227, 2253, 2257, 2323, 2327, 2335, 2353, 2533, 2537, 2573, 2577, 2722, 2723, 2733, 2735, 2757, 2773, 3223, 3227, 3233, 3235, 3273, 3277

Offset: 1

Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 30 2003

			E.g. 35 = 7*5 and contains the digits 7 and 5, both prime.

Intersection of A006881 and A046034.

Corrected and extended by Charles R Greathouse IV, Nov 17 2010.

A085907 7-smooth numbers using digits 2,3,5 and 7 only. Numbers using prime digits and having prime divisors < 10.

Original entry on oeis.org

2, 3, 5, 7, 25, 27, 32, 35, 72, 75, 225, 252, 375, 525, 735, 2352, 3375, 25725, 273375, 2333772, 52732233225

Offset: 1

Amarnath Murthy, Jul 09 2003

No more terms < 10^100. Probably no more terms. - David Wasserman, Feb 11 2005

No more terms < 10^1000. - Graham Holmes, Jan 07 2022

Intersection of A002473 and A046034.

More terms from David Wasserman, Feb 11 2005

A111483 Triangular numbers all of whose digits are primes.

Original entry on oeis.org

3, 55, 253, 325, 2775, 5253, 255255, 522753, 577275, 2235555, 2355535, 2532375, 3252525, 3352755, 7332535, 23273253, 25222753, 37372335, 55277355, 73235253, 222552253, 273253753, 325775575, 337337325, 577235253, 2373777253

Offset: 1

Jonathan Vos Post, Nov 15 2005

3 = T(2) and 253 = T(22) have all prime digits and all prime digits in their indices as triangular numbers. What is the next of this subsequence?

			3 = T(2), 55 = T(10), 253 = T(22), 325 = T(25), 2775 = T(74), 5253 = T(102), 255255 = T(714), 522753 = T(1022), 577275 = T(1074), 2235555 = T(2114), 2355535 = T(2170), 2532375 = T(2250).

Intersection of A000217 and A046034.

Cf. A000040.

fQ[n_] := Union@Join[{2, 3, 5, 7}, IntegerDigits[n(n + 1)/2]] == {2, 3, 5, 7}; s = Select[ Range[10^5], fQ[ # ] &]; s (s + 1)/2 (* Robert G. Wilson v *)

Corrected and extended by Robert G. Wilson v and Ray Chandler, Nov 18 2005

A173719 Sums of 2 successive primes s = prime(m) + prime(m+1) such that all digits of s are primes.

Original entry on oeis.org

5, 52, 222, 352, 372, 532, 752, 772, 2252, 2352, 2572, 3222, 3232, 5322, 7572, 22332, 22552, 22722, 22752, 23572, 25232, 25572, 27232, 27522, 27732, 32732, 33522, 33772, 35232, 35572, 35772, 37332, 52232, 52332, 52372, 53772, 55552, 57332, 72532, 72772, 75252, 75732, 77322, 222532, 222572, 223552, 223572

Offset: 1

Zak Seidov, Dec 22 2012

			5 = 2 + 3, 52 = 23 + 29.

Intersection of A001043 and A046034.

{a=2;b=3;for(n=1,12000,s=a+b;ev=eval(Vec(Str(s)));if(sum(k=1,#ev,isprime(ev[k]))==#ev,print1(s", "));a=b;b=nextprime(b+2))}

A241681 Numbers n such that the decimal digits of n are also the prime divisors of n.

Original entry on oeis.org

2, 3, 5, 7, 735, 2333772

Offset: 1

Michel Lagneau, Apr 27 2014

The sequence is given for a(n) < 10^11.
No more terms <= 10^150. Terms are of the form 2^e2 * 3^e3 * 7^e7 or of the form 3^e3 * 5^e5 * 7^e7, for which no other number <= 10^150 than those listed is a term. - David A. Corneth, Sep 28 2019
No more terms <= 10^1000. - Michael S. Branicky, May 30 2025

			735 = 3*5*7^2 is in the sequence because the digits 7, 3 and 5 are also the prime divisors of 735.

Cf. A108319, A108347.

Subsequence of A046034.

with(numtheory):nn:=1000000:for n from 1 to 10^11 do:lst:={}:x:=factorset(n):y:=convert(n,base,10):n1:=nops(x):n2:=nops(y): for j from 1 to n2 do:lst:=lst union {y[j]}:od:if x=lst then print(n):else fi:od:

cp's OEIS Frontend

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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A342992 Smallest k such that k*n contains only prime digits, or 0 if no such k exists.

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PARI

A365471 Numbers whose digits are not all primes.

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Mathematica

A365472 Numbers whose digits are either all primes or all nonprimes.

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A380490 Replace prime digits of n by 0's.

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A084996 Numbers which can be written as the product of two distinct primes and containing only prime decimal digits.

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A085907 7-smooth numbers using digits 2,3,5 and 7 only. Numbers using prime digits and having prime divisors < 10.

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A111483 Triangular numbers all of whose digits are primes.

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Mathematica

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A173719 Sums of 2 successive primes s = prime(m) + prime(m+1) such that all digits of s are primes.

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PARI