A061371 -id:A061371 - cp's OEIS frontend

A092631 Duplicate of A061371.

22, 25, 27, 32, 33, 35, 52, 55, 57, 72, 75, 77, 222, 225, 232, 235, 237, 252, 253, 255

Offset: 1

dead

A001744 Numbers n such that every digit contains a loop (version 2).

0, 4, 6, 8, 9, 40, 44, 46, 48, 49, 60, 64, 66, 68, 69, 80, 84, 86, 88, 89, 90, 94, 96, 98, 99, 400, 404, 406, 408, 409, 440, 444, 446, 448, 449, 460, 464, 466, 468, 469, 480, 484, 486, 488, 489, 490, 494, 496, 498, 499, 600, 604, 606, 608, 609, 640, 644, 646

Offset: 1

N. J. A. Sloane

See A001743 for the other version.

If n-1 is represented as a base-5 number (see A007091) according to n-1 = d(m)d(m-1)...d(3)d(2)d(1)d(0) then a(n)= Sum_{j=0..m} c(d(j))*10^j, where c(k)=0,4,6,8,9 for k=0..4. - Hieronymus Fischer, May 30 2012

			a(1000) = 46999.
a(10^4) = 809999.
a(10^5) = 44499999.
a(10^6) = 668999999.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Cf. A061371, A029581, A007091, A046034, A084544, A084984, A017042, A001743, A014261, A014263, A202267, A202268.

FromDigits/@Tuples[{0,4,6,8,9},3] (* Harvey P. Dale, Aug 16 2018 *)

is(n) = #setintersect(vecsort(digits(n), , 8), [1, 2, 3, 5, 7])==0 \\ Felix Fröhlich, Sep 09 2019

From Hieronymus Fischer, May 30 2012: (Start)
a(n) = ((2*b_m(n)) mod 8 + 4 + floor(b_m(n)/4) - floor((b_m(n)+1)/4))*10^m + sum_{j=0..m-1} ((2*b_j(n))) mod 10 + 2*floor((b_j(n)+4)/5) - floor((b_j(n)+1)/5) -floor(b_j(n)/5)))*10^j, where n>1, b_j(n)) = floor((n-1-5^m)/5^j), m = floor(log_5(n-1)).
a(1*5^n+1) = 4*10^n.
a(2*5^n+1) = 6*10^n.
a(3*5^n+1) = 8*10^n.
a(4*5^n+1) = 9*10^n.
a(n) = 4*10^log_5(n-1) for n=5^k+1,
a(n) < 4*10^log_5(n-1), otherwise.
a(n) > 10^log_5(n-1) n>1.
a(n) = 4*A007091(n-1), iff the digits of A007091(n-1) are 0 or 1.
G.f.: g(x) = (x/(1-x))*sum_{j>=0} 10^j*x^5^j*(1-x^5^j)*(4 + 6x^5^j + 8(x^2)^5^j + 9(x^3)^5^j)/(1-x^5^(j+1)).
Also: g(x) = (x/(1-x))*(4*h_(5,1)(x) + 2*h_(5,2)(x) + 2*h_(5,3)(x) + h_(5,4)(x) - 9*h_(5,5)(x)), where h_(5,k)(x) = sum_{j>=0} 10^j*(x^5^j)^k/(1-(x^5^j)^5). (End)

Ambiguous comment deleted by Zak Seidov, May 25 2010

Examples added by Hieronymus Fischer, May 30 2012

A051416 Primes whose digits are composite; primes having only {4, 6, 8, 9} as digits.

Original entry on oeis.org

89, 449, 499, 4649, 4889, 4969, 4999, 6449, 6469, 6689, 6869, 6899, 6949, 8669, 8689, 8699, 8849, 8969, 8999, 9649, 9689, 9949, 44449, 44699, 46489, 46499, 46649, 46889, 48449, 48649, 48869, 48889, 48989, 49499, 49669, 49999, 64489, 64499, 64849, 64969, 66449

Offset: 1

G. L. Honaker, Jr., Jan 17 2000

Primes formed by using only digits 4, 6, 8, 9. Of course, all the terms of this sequence end with 9. - Bernard Schott, Jan 31 2019

			89 is the smallest composite-digit prime and also the only composite-digit prime whose digits are distinct. - _Bernard Schott_, Jan 31 2019

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Chris Caldwell, The Prime Glossary, Composite number
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 89
Index to entries for primes with digits in a given set

Cf. A019546 (with prime digits), A030096 (with odd digits), A061246 (with square digits), A061371 (composite numbers with prime digits).

Subsequence of A061372 and of A152313.

Select[Prime@Range[6500], Intersection[IntegerDigits[ # ], {0, 1, 2, 3, 5, 7}] == {} & ] (* Ray Chandler, Mar 04 2007 *)
With[{c = {4, 6, 8, 9}}, Array[Select[Map[FromDigits@ Append[#, 9] &, Tuples[c, {#}]], PrimeQ] &, 4]] // Flatten (* Michael De Vlieger, Feb 02 2019 *)

Extended by Ray Chandler, Mar 04 2007

A061372 Primes having only 0,4,6,8,9 as digits.

Original entry on oeis.org

89, 409, 449, 499, 809, 4049, 4099, 4409, 4649, 4889, 4909, 4969, 4999, 6089, 6449, 6469, 6689, 6869, 6899, 6949, 8009, 8069, 8089, 8609, 8669, 8689, 8699, 8849, 8969, 8999, 9049, 9649, 9689, 9949, 40009, 40099, 40499, 40609, 40699, 40849, 40949

Offset: 1

Amarnath Murthy, May 02 2001

Primes having digits that are all 0 or composite.

			a(5) = 4049 is a prime and 4,0,9 are 0 or composite digits.

Jason Bard, Table of n, a(n) for n = 1..10000

Cf. A061371, A051416.

Select[FromDigits/@Tuples[{0,4,6,8,9},5],PrimeQ] (* Harvey P. Dale, May 04 2011 *)

More terms from Erich Friedman, May 08 2001

Definition corrected Dec 05 2006

A387093 Composite numbers that contain only prime digits and whose prime factors contain only prime digits.

Original entry on oeis.org

25, 27, 32, 35, 72, 75, 222, 225, 252, 322, 333, 375, 525, 552, 555, 575, 735, 777, 2352, 2553, 2555, 2775, 3357, 3375, 3552, 3577, 5222, 5352, 5575, 7252, 7322, 23253, 23373, 23532, 23535, 23552, 25275, 25725, 25737, 27232, 27252, 27375, 32352, 32375

Offset: 1

Scott R. Shannon and Ursula Ponting, Aug 16 2025

			25725 is a term as 25725 = 3 * 5^2 * 7^3, and both the number and its prime factors only contain prime digits.

Scott R. Shannon, Table of n, a(n) for n = 1..2298 (all terms < 10^12).

Cf. A061371, A027746, A000040, A019546.

Select[Select[Range[33000], CompositeQ], And[AllTrue[Union@ IntegerDigits[#], PrimeQ], AllTrue[Union@ Flatten@ Map[IntegerDigits, FactorInteger[#][[All, 1]] ], PrimeQ]] &] (* Michael De Vlieger, Aug 16 2025 *)

A385477 Composite numbers whose digits are odd prime numbers.

Original entry on oeis.org

33, 35, 55, 57, 75, 77, 333, 335, 355, 357, 375, 377, 533, 535, 537, 553, 555, 573, 575, 735, 737, 753, 755, 775, 777, 3333, 3335, 3337, 3353, 3355, 3357, 3375, 3377, 3535, 3537, 3553, 3555, 3573, 3575, 3577, 3735, 3737, 3753, 3755, 3757, 3773, 3775, 3777, 5335, 5337, 5353, 5355, 5357

Offset: 1

Enrique Navarrete, Jun 30 2025

Mia Boudreau, Table of n, a(n) for n = 1..5000

Subsequence of A061371.
Subsequence of A320062.
Cf. A061372, A087363.

A:=3,5,7: B:= [A]:
for d from 2 to 4 do B:= map(f,B); A:= A,op(B) od:
remove(isprime,[A]); # Robert Israel, Jun 30 2025

Select[Range[1, 6000, 2], CompositeQ[#] && AllTrue[IntegerDigits[#], MemberQ[{3, 5, 7}, #1] &] &] (* Amiram Eldar, Jun 30 2025 *)

from sympy import isprime
from itertools import count, islice, product
def agen(): yield from (t for d in count(2) for p in product("357", repeat=d) if not isprime(t:=int("".join(p))))
print(list(islice(agen(), 53))) # Michael S. Branicky, Jun 30 2025

A366826 Composite numbers whose proper substrings (of their decimal expansions) are all primes.

Original entry on oeis.org

4, 6, 8, 9, 22, 25, 27, 32, 33, 35, 52, 55, 57, 72, 75, 77, 237, 537, 737

Offset: 1

Kalle Siukola, Oct 25 2023

There are no terms greater than 999 because the only three-digit prime whose substrings are all primes is 373 (see A085823) and prepending or appending any prime digit to it would create a different three-digit substring.

			237 is included because it is composite and 2, 3, 7, 23 and 37 are all primes.
4 is included because it is composite and has no proper substrings.

Subsequence of A002808.
Cf. A000040.
Cf. A061371, A062115, A202262.
Cf. A033274, A068669, A085823, A279366.

from itertools import combinations
from sympy import isprime
for n in range(2, 1000):
    if not isprime(n):
        properSubstrings = set(
            int(str(n)[start:end]) for (start, end)
            in combinations(range(len(str(n)) + 1), 2)
        ) - set((n,))
        if all(isprime(s) for s in properSubstrings):
            print(n, end=', ')

A378428 Composites that become prime when any two of their digits are deleted.

Original entry on oeis.org

222, 225, 232, 235, 237, 252, 253, 255, 272, 273, 275, 322, 323, 325, 327, 332, 333, 335, 352, 355, 357, 372, 375, 377, 522, 525, 527, 532, 533, 535, 537, 552, 553, 555, 572, 573, 575, 722, 723, 725, 732, 735, 737, 752, 753, 755, 772, 775, 777, 1111, 1113, 1119, 1131, 1137, 1173, 1179, 1197, 1311, 1317, 1371

Offset: 1

Enrique Navarrete, Nov 26 2024

Any term < 1000 has exactly three digits and all digits are prime (cf. A061371).
The repunits (cf. A002275) R_21, R_25, R_319, R_1033 and R_49083, among others, are in the sequence since R_19, R_23, R_317, R_1031 and R_49081 are prime (cf. A004023).
The corresponding sequence for primes (cf. A378081) contains only 18 terms up to 10^100.

			1371 is in the sequence since upon deleting any two digits we get 13, 71, 17, 31, 11 and 37, all of which are prime.
1313 is not in the sequence since upon deleting the two 1s we get 33, which is not prime.

Cf. A002808, A002275, A004023, A061371, A378081.

q[n_] := Module[{d = IntegerDigits[n]}, AllTrue[FromDigits /@ Subsets[d, {Length[d] - 2}], PrimeQ]]; Select[Range[100, 1500], CompositeQ[#] && q[#] &] (* Amiram Eldar, Nov 26 2024 *)

from sympy import isprime
from itertools import combinations as C
def ok(n):
    if n < 100 or isprime(n): return False
    s = str(n)
    return all(isprime(int(t)) for i, j in C(range(len(s)), 2) if (t:=s[:i]+s[i+1:j]+s[j+1:])!="")
print([k for k in range(1500) if ok(k)]) # Michael S. Branicky, Nov 26 2024

cp's OEIS Frontend

A092631 Duplicate of A061371.

Views

Author

Keywords

A001744 Numbers n such that every digit contains a loop (version 2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A051416 Primes whose digits are composite; primes having only {4, 6, 8, 9} as digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A061372 Primes having only 0,4,6,8,9 as digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A387093 Composite numbers that contain only prime digits and whose prime factors contain only prime digits.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A385477 Composite numbers whose digits are odd prime numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Python

A366826 Composite numbers whose proper substrings (of their decimal expansions) are all primes.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Python

A378428 Composites that become prime when any two of their digits are deleted.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Python