A051416 -id:A051416 - cp's OEIS frontend

A107666 Primes having only {4, 6, 9} as digits.

449, 499, 4649, 4969, 4999, 6449, 6469, 6949, 9649, 9949, 44449, 44699, 46499, 46649, 49499, 49669, 49999, 64499, 64969, 66449, 66499, 66949, 69499, 94649, 94949, 94999, 96469, 99469, 444449, 444469, 444649, 446969, 449699, 464699, 464999, 466649, 469649, 469969

Offset: 1

Rick L. Shepherd, May 19 2005

Intersection of A000040 and A107665. - K. D. Bajpai, Sep 08 2014

			From _K. D. Bajpai_, Sep 08 2014: (Start)
4649 is a term because it is a prime having only semiprime digits 4, 6 and 9.
6469 is a term because it is a prime having only semiprime digits 4, 6 and 9.
449 is the smallest prime comprising only semiprime digits 4, 6 or 9.
(End)

Jason Bard, Table of n, a(n) for n = 1..10000 (first 2069 terms from K. D. Bajpai)
Index to entries for primes with digits in a given set

Cf. A107665 (numbers with semiprime digits), A001358 (semiprimes), A051416 (primes whose digits are all composite), A020466 (primes with digits 4 and 9 only), A093402 (primes of form 44...9), A093945 (primes of form 499...).
Cf. A107342, A111494, A111496, A111697.
Cf. A385768 - A385800.

N:= 4:  Dgts:= {4, 6, 9}:  A:= NULL:
for d from 1 to N do
K:= combinat[cartprod]([Dgts minus {0}, Dgts $(d-1)]);
while not K[finished] do L:= K[nextvalue]();  x:= add(L[i]*10^(d-i), i=1..d);
if isprime(x) then A:= A, x fi od od: A;  # K. D. Bajpai, Sep 08 2014

Select[Prime[Range[50000]], Intersection[IntegerDigits[#], {0, 1, 2, 3, 5, 7, 8}] == {} &] (* K. D. Bajpai, Sep 08 2014 *)

a(35)-a(38) from K. D. Bajpai, Sep 08 2014

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

A279366 Primes whose proper substrings of consecutive digits are all composite.

Original entry on oeis.org

89, 449, 499, 4649, 4969, 6469, 6869, 6949, 8669, 8699, 8849, 9649, 9949, 44699, 46649, 48649, 48869, 49669, 64849, 84869, 86969, 88469, 94849, 94949, 98869, 99469, 444469, 444869, 446969, 466649, 468869, 469849, 469969, 494699, 496669, 496849, 498469, 644669

Offset: 1

Rodrigo de O. Leite, Dec 10 2016

All digits are composite. Each term ends with the digit '9'. Since each term is prime, it never serves as the suffix of any subsequent term; e.g., no term beyond 89 ends with the digits '89', so the only remaining allowed two-digit endings are '49', '69', and '99'; no terms beyond 449 and 499 end with '449' or '499' (and '899' is ruled out because of 89), so the only remaining allowed three-digit endings are '469', '649', '669', '699', '849', '869', '949', '969', and '999' (and each of these appears as the ending of at least one four-digit term, except '999', which doesn't appear as the ending of any term until a(75) = 4696999). - Jon E. Schoenfield, Dec 10 2016

Number of terms < 10^k, k=1,2,3,...: 0, 1, 2, 10, 13, 38, 66, 197, 410, 1053, 2542, 7159, 18182, 49388, ..., . Robert G. Wilson v, Jan 15 2017

			44699 is in the sequence because 4, 6, 9, 44, 46, 69, 99, 446, 469, 669, 4469 and 4699 are composite numbers. However, 846499 is not included because 4649 is prime.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 63 terms from Rodrigo de O. Leite)

Subsequence of A051416.

Cf. A033274, A061372, A254754.

Select[Prime@ Range[5, 10^5], Function[n, Times @@ Boole@ Map[CompositeQ, Flatten@ Map[FromDigits /@ Partition[n, #, 1] &, Range[Length@ n - 1]]] == 1]@ IntegerDigits@ # &] (* Michael De Vlieger, Dec 10 2016 *)
Select[Flatten[Table[FromDigits/@Tuples[{4,6,8,9},d],{d,6}]],PrimeQ[#]&&AllTrue[ FromDigits /@ Union[Flatten[Table[Partition[IntegerDigits[#],n,1],{n,IntegerLength[#]-1}],1]], CompositeQ]&] (* Harvey P. Dale, Jul 15 2023 *)

from sympy import isprime
from itertools import count, islice, product
def ok(n):
    s = str(n)
    if set(s) & {"1", "2", "3", "5", "7"} or not isprime(n): return False
    ss2 = set(s[i:i+l] for i in range(len(s)-1) for l in range(2, len(s)))
    return not any(isprime(int(ss)) for ss in ss2)
def agen():
    for d in count(2):
        for p in product("4689", repeat=d-1):
            t = int("".join(p)+"9")
            if ok(t): yield t
print(list(islice(agen(), 38))) # Michael S. Branicky, Oct 07 2022

A157527 Primes using only the composite digits (4, 6, 8, 9) and all of them.

Original entry on oeis.org

46489, 46889, 48649, 48869, 64489, 64849, 68449, 68489, 84649, 84869, 88469, 444869, 448969, 449689, 468499, 468869, 468889, 468899, 469849, 486449, 486869, 486949, 488689, 489689, 489869, 496849, 496889, 498469, 498689, 644489, 644869

Offset: 1

Lekraj Beedassy, Mar 02 2009, Mar 03 2009

Subsequence of A051416.

There are no 4-digit terms so each term must have at least one repeating digit. - Harvey P. Dale, Oct 05 2023

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A051416, A138165, A034844, A108419.

a := proc (n) if convert(convert(ithprime(n), base, 10), set) = {4, 6, 8, 9} then ithprime(n) else end if end proc: seq(a(n), n = 1 .. 53000); # Emeric Deutsch, Mar 03 2009
isA157527 := proc(n) local dgs ; if not isprime(n) then false; else dgs := convert(convert(n,base,10),set) ; if dgs intersect {4,6,8,9} <> {4,6,8,9} then false; elif dgs intersect {0,1,2,3,5,7} <> {} then false; else true; fi; fi; end: for n from 1 to 100000 do p := ithprime(n) ; if isA157527(p) then printf("%d,",p) ; fi; od: # R. J. Mathar, Mar 03 2009

With[{c={4,6,8,9}},Select[Flatten[Table[10 FromDigits/@Tuples[c,n]+9,{n,5}]],PrimeQ[#] && Intersection[c,IntegerDigits[#]]==c&]] (* Harvey P. Dale, Oct 05 2023 *)

Corrected and extended by numerous correspondents, Mar 04 2009

A342049 Primes formed by the concatenation of exactly two consecutive composite numbers.

Original entry on oeis.org

89, 5051, 5657, 6263, 6869, 8081, 9091, 9293, 120121, 186187, 188189, 200201, 216217, 242243, 246247, 252253, 278279, 300301, 308309, 318319, 338339, 342343, 350351, 362363, 368369, 390391, 402403, 410411, 416417, 426427, 428429, 440441, 446447, 450451, 452453, 470471, 476477, 482483

Offset: 1

Bernard Schott, Feb 26 2021

When a prime is obtained by the concatenation of exactly two consecutive composite numbers, the first one always ends with 0, 2, 6, 8 while the second one ends respectively with 1, 3, 7, 9.
a(1) = 89 is also the smallest prime whose digits are composite (A051416).
a(n) has an even number of digits. If it would have an odd number of digits then it is like 99..99100..00 but that is composite. - David A. Corneth, Feb 27 2021

			If (2,q) is the smallest term formed by the concatenation of 2 consecutive composite numbers with each q digits: (2,1) = a(1) = 89, (2,2) = a(2) = 5051, (2,3) = a(9) = 120121, (2,4) = 10021003, (2,5) = 1001010011, (2,6) = 100010100011.

G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 89.

Cf. A002808, A087341.

Subsequence of A030458 and A121608.

isc(c) = (c>1) && ! isprime(c);
isok(p) = {if (isprime(p), my(d=digits(p)); for (i=1, #d-1, my(b = fromdigits(vector(i, k, d[k]))); if (d[i+1], my(c = fromdigits(vector(#d-i, k, d[k+i]))); if (isc(b) && isc(c) && ((primepi(c) - primepi(b)) == c-b-1), return (1)); ); ); ); } \\ Michel Marcus, Feb 27 2021

first(n) = { pc = 4; my(res = vector(n)); t = 0; forcomposite(c = 6, oo, nc = pc * 10^#digits(c) + c; if(isprime(nc), t++; res[t] = nc; if(t >= n, return(res) ) ); pc = c; ) } \\ David A. Corneth, Feb 27 2021

is(n) = { my(d = digits(n)); if(#d % 2 == 1, return(0) ); fc = fromdigits(vector(#d \ 2, i, d[i])); lc = fromdigits(vector(#d \ 2, i, d[i+#d\2])); lc - fc == 1 && !isprime(fc) && !isprime(lc) && nextprime(fc)==nextprime(lc) && isprime(n) } \\ David A. Corneth, Feb 27 2021

from sympy import isprime
def agento(lim):
  digs, pow10 = 1, 10
  while True:
    for c2 in range(max(pow10//10+1, 3), pow10, 2):
      if not isprime(c2) and not isprime(c2-1):
        c1c2 = (c2-1)*pow10+c2
        if c1c2 > lim: return
        if isprime(c1c2): yield c1c2
    digs, pow10 = digs+1, pow10*10
print([an for an in agento(482483)]) # Michael S. Branicky, Feb 27 2021

A386193 Primes having only {4, 6, 7, 8} as digits.

Original entry on oeis.org

7, 47, 67, 467, 487, 647, 677, 787, 877, 887, 4447, 4787, 4877, 7477, 7487, 7687, 7867, 7877, 8447, 8467, 8647, 8677, 8747, 8867, 8887, 44647, 44687, 44777, 44867, 44887, 46447, 46477, 46687, 46747, 46867, 46877, 47777, 48487, 48647, 48677, 48767, 48787, 48847

Offset: 1

Jason Bard, Jul 18 2025

Supersequence of A385794, A385795, A385799.

Cf. A000040, A030432, A051416, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [4, 6, 7, 8]];

Flatten[Table[Select[FromDigits /@ Tuples[{4, 6, 7, 8}, n], PrimeQ], {n, 7}]]

primes_with(, 1, [4, 6, 7, 8]) \\ uses function in A385776

print(list(islice(primes_with("4678"), 41))) # uses function/imports in A385776

A386194 Primes having only {4, 6, 7, 9} as digits.

Original entry on oeis.org

7, 47, 67, 79, 97, 449, 467, 479, 499, 647, 677, 769, 797, 947, 967, 977, 997, 4447, 4649, 4679, 4799, 4967, 4969, 4999, 6449, 6469, 6679, 6779, 6947, 6949, 6967, 6977, 6997, 7477, 7499, 7649, 7669, 7699, 7949, 9467, 9479, 9497, 9649, 9677, 9679, 9697, 9749, 9767

Offset: 1

Jason Bard, Jul 18 2025

Supersequence of A107666, A261183, A261184, A385794.

Cf. A000040, A051416, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [4, 6, 7, 9]];

Flatten[Table[Select[FromDigits /@ Tuples[{4, 6, 7, 9}, n], PrimeQ], {n, 7}]]

primes_with(, 1, [4, 6, 7, 9]) \\ uses function in A385776

print(list(islice(primes_with("4679"), 41))) # uses function/imports in A385776

A386195 Primes having only {4, 7, 8, 9} as digits.

Original entry on oeis.org

7, 47, 79, 89, 97, 449, 479, 487, 499, 787, 797, 877, 887, 947, 977, 997, 4447, 4787, 4789, 4799, 4877, 4889, 4987, 4999, 7477, 7487, 7489, 7499, 7789, 7877, 7879, 7949, 8447, 8747, 8779, 8849, 8887, 8999, 9479, 9497, 9749, 9787, 9887, 9949, 44449, 44497, 44777

Offset: 1

Jason Bard, Jul 18 2025

Supersequence of A106110, A261183, A385795, A385796.

Cf. A000040, A051416, A385776.

[p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [4, 7, 8, 9]];

Flatten[Table[Select[FromDigits /@ Tuples[{4, 7, 8, 9}, n], PrimeQ], {n, 7}]]

primes_with(, 1, [4, 7, 8, 9]) \\ uses function in A385776

print(list(islice(primes_with("4789"), 41))) # uses function/imports in A385776

cp's OEIS Frontend

A107666 Primes having only {4, 6, 9} as digits.

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

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A279366 Primes whose proper substrings of consecutive digits are all composite.

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A157527 Primes using only the composite digits (4, 6, 8, 9) and all of them.

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Maple

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A342049 Primes formed by the concatenation of exactly two consecutive composite numbers.

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PARI

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A386193 Primes having only {4, 6, 7, 8} as digits.

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Magma

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Python

A386194 Primes having only {4, 6, 7, 9} as digits.

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Author

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Programs

Magma

Mathematica

PARI