A107665 -id:A107665 - cp's OEIS frontend

A107342 Semiprimes with semiprime digits (digits 4, 6, 9 only).

4, 6, 9, 46, 49, 69, 94, 446, 466, 469, 649, 669, 694, 699, 949, 4449, 4469, 4499, 4666, 4694, 4699, 4946, 6499, 6646, 6649, 6694, 6999, 9446, 9449, 9466, 9469, 9946, 9969, 44494, 44669, 44949, 44966, 44969, 44999, 46469, 46666, 46946, 46969, 46994

Offset: 1

Jonathan Vos Post, May 22 2005

Numbers n such that all digits of n are elements of A001358 and n is an element of A001358.
Numbers n such that n is an element of A107665 and n is an element of A001358.
Conjecture: almost all terms (asymptotic density 1) end with 9 and have either 3k+1 or 3k+2 occurrences of the digit 4 for some nonnegative k. (Otherwise they'd be divisible by 2 or 3 and these semiprimes would be expected to be rare; the sequence is too thin to prove this directly.) - Charles R Greathouse IV, Nov 12 2021

			4 = 2^2
6 = 2 * 3
9 = 3^2
46 = 2 * 23
49 = 7^2
69 = 3 * 23
94 = 2 * 47

Vincenzo Librandi, Table of n, a(n) for n = 1..600

Intersection of A001358 and A107665.

Cf. A029581, A107666, A111494, A111496, A111697, A254715.

fQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2 && Union[ Join[{4, 6, 9}, IntegerDigits[n]]] == {4, 6, 9}; Select[ Range[ 47000], fQ[ # ] &] (* Robert G. Wilson v, May 27 2005 *)
Flatten[Table[Select[FromDigits/@Tuples[{4,6,9},n],PrimeOmega[#]==2&],{n,5}]] (* Harvey P. Dale, Jun 14 2015 *)

is(n)=bigomega(n)==2 && #setminus(Set(digits(n)),[4,6,9])==0 \\ Charles R Greathouse IV, Nov 12 2021

More terms from Robert G. Wilson v, May 27 2005

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

Original entry on oeis.org

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

A111494 3-almost primes with semiprime digits (digits 4, 6, 9 only).

Original entry on oeis.org

44, 66, 99, 494, 646, 946, 964, 969, 994, 4646, 4669, 4949, 4966, 4994, 4996, 6446, 6466, 6494, 6946, 6964, 6969, 6994, 9494, 9499, 9644, 9669, 9694, 9699, 9994, 44446, 44466, 44644, 44666, 44994, 46446, 46466, 46646, 46669, 46694, 46699, 46949

Offset: 1

Jonathan Vos Post, Nov 15 2005

			a(1) = 44 = 2^2 * 11, a(2) = 66 = 2 * 3 * 11, a(3) = 99 = 3^2 * 11, a(4) = 494 = 2 * 13 * 19, a(5) = 646 = 2 * 17 * 19, a(6) = 946 = 2 * 11 * 43, a(7) = 964 = 2^2 * 241, a(8) = 969 = 3 * 17 * 19, a(9) = 994 = 2 * 7 * 71, a(10) = 4646 = 2 * 23 * 101, a(11) = 4669 = 7 * 23 * 29.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001358, A014612, A107665, A107666, A107342, A111496, A111697.

Table[Select[FromDigits/@Tuples[{4,6,9},n],PrimeOmega[#]==3&],{n,5}]// Flatten (* Harvey P. Dale, Jan 08 2019 *)

do(N)=my(v=List(),a=[4,6,9]); for(d=1,N, forvec(u=vector(d,i,[1,3]), t=fromdigits(apply(n->a[n],u)); if(bigomega(t)==3, listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Feb 01 2017

Corrected by Ray Chandler, Nov 19 2005

A111496 4-almost primes with semiprime digits (digits 4, 6, 9 only).

Original entry on oeis.org

444, 644, 664, 666, 966, 996, 999, 4444, 4466, 4494, 4696, 4964, 6644, 6666, 6699, 9444, 9496, 9646, 9964, 9966, 9999, 44444, 44499, 44646, 44996, 46444, 46449, 46964, 46996, 49444, 49494, 49946, 49966, 49996, 64446, 64494, 64644, 64666

Offset: 1

Jonathan Vos Post, Nov 16 2005

			a(1) = 444 = 2^2 * 3 * 37, a(2) = 644 = 2^2 * 7 * 23, a(3) = 664 = 2^3 * 83, a(4) = 666 = 2 * 3^2 * 37, a(5) = 966 = 2 * 3 * 7 * 23, a(6) = 996 = 2^2 * 3 * 83, a(7) = 999 = 3^3 * 37, a(8) = 4466 = 2 * 7 * 11 * 29, a(9) = 4494 = 2 * 3 * 7 * 107, a(10) = 4696 = 2^3 * 587.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001358, A014613, A107665, A107666, A107342, A111494, A111697.

do(N)=my(v=List(), a=[4, 6, 9]); for(d=1, N, forvec(u=vector(d, i, [1, 3]), t=fromdigits(apply(n->a[n], u)); if(bigomega(t)==4, listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Feb 01 2017

A111697 5-almost primes with semiprime digits (digits 4, 6, 9 only).

Original entry on oeis.org

464, 496, 696, 944, 4446, 4496, 4664, 6444, 6669, 6996, 9666, 9944, 44649, 44664, 44694, 44696, 44946, 44964, 46664, 46696, 49446, 49496, 49944, 64664, 66664, 66996, 69464, 69944, 69996, 94996, 96464, 96664, 96996, 99664, 99946, 99996

Offset: 1

Jonathan Vos Post, Nov 17 2005

			a(1) = 464 = 2^4 x 29, a(2) = 496 = 2^4 * 31, a(3) = 696 = 2^3 * 3 * 29, a(4) = 944 = 2^4 * 59, a(5) = 4446 = 2 * 3^2 * 13 * 19, a(6) = 4496 = 2^4 * 281, a(7) = 4664 = 2^3 * 11 * 53, a(8) = 6444 = 2^2 * 3^2 * 179, a(9) = 6669 = 3^3 * 13 * 19, a(10) = 6996 = 2^2 * 3 * 11 * 53.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001358, A014614, A107665, A107666, A107342, A111494, A111496.

Table[Select[FromDigits/@Tuples[{4,6,9},n],PrimeOmega[#]==5&],{n,3,5}]//Flatten (* Harvey P. Dale, Dec 16 2024 *)

do(N)=my(v=List(), a=[4, 6, 9]); for(d=1, N, forvec(u=vector(d, i, [1, 3]), t=fromdigits(apply(n->a[n], u)); if(bigomega(t)==5, listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Feb 01 2017

Corrected by Ray Chandler, Nov 19 2005

A053961 Squares composed of digits {4,6,9}.

Original entry on oeis.org

4, 9, 49, 64, 44944, 69696, 4999696, 9696996, 9966649, 999444996, 4649466969, 64944444964, 6469646694964996, 69446644696999969, 9446966946446994496, 44499464666496696649, 446646664496496964644, 466994666666446696996, 9469699669494449499664, 494494499966694999949669969

Offset: 1

Patrick De Geest, Mar 15 2000

P. De Geest, Squares containing at most three distinct digits, Index entries for related sequences
Author?, Source (txt)
Hisanori Mishima, Sporadic tridigital solutions

Cf. A053960, A107665, A107666.

fQ[n_] := Union[ Join[{4, 6, 9}, IntegerDigits[n]]] == {4, 6, 9}; lst = {}; Do[ If[ fQ[n^2], AppendTo[lst, n^2]], {n, 2*10^9}]; lst (* Robert G. Wilson v, Jun 01 2005 *)

a(n) = A053960(n)^2.

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 03 2005

Missing term inserted by Sean A. Irvine, Apr 28 2022

A108614 Semiprimes with non-semiprimes digits (no digits 4,6,9 in semiprimes).

Original entry on oeis.org

10, 15, 21, 22, 25, 33, 35, 38, 51, 55, 57, 58, 77, 82, 85, 87, 111, 115, 118, 121, 122, 123, 133, 155, 158, 177, 178, 183, 185, 187, 201, 202, 203, 205, 213, 215, 217, 218, 221, 235, 237, 253, 278, 287, 301, 302, 303, 305, 321, 323, 327, 335, 355, 358, 371

Offset: 1

Zak Seidov, Jun 13 2005

Complement of A107342 in the class of semiprimes.

This is to semiprimes A001358 as A034844 (Primes with nonprime digits) is to primes A000040. [Jonathan Vos Post, Jul 15 2010]

Cf. A107342, A137167.

Cf. A107665, A107666, A111494, A111496, A111697, A179336. [Jonathan Vos Post, Jul 15 2010]

cnd[n_]:=Plus@@Last/@FactorInteger[n]==2&&Union[FreeQ[IntegerDigits[n], # ]&/@{4, 6, 9}]=={True};Select[Range[600], cnd[ # ]&]

{j in A001358 and j not in A179463}. [Jonathan Vos Post, Jul 15 2010]

A179463 Semiprimes A001358 containing at least one semiprime digit in base 10.

Original entry on oeis.org

4, 6, 9, 14, 26, 34, 39, 46, 49, 62, 65, 69, 74, 86, 91, 93, 94, 95, 106, 119, 129, 134, 141, 142, 143, 145, 146, 159, 161, 166, 169, 194, 206, 209, 214, 219, 226, 247, 249, 254, 259, 262, 265, 267, 274, 289, 291, 295, 298, 299, 309, 314, 319, 326, 329, 334, 339, 341

Offset: 1

Jonathan Vos Post, Jul 15 2010

Semiprimes containing at least one 4, 6, or 9 digit base 10.
This is to semiprimes A001358 as A179336 is to primes A000040.
This properly includes the subset A107342 Semiprimes with semiprime digits.

Cf. A107342 Semiprimes with semiprime digits (digits 4, 6, 9 only), A107665 Numbers with semiprime digits (digits 4, 6, 9 only), A107666 Primes with semiprime digits (digits 4, 6, 9 only), A111494, A111496, A111697, A108614 Semiprimes with non-semiprimes digits (no digits 4, 6, 9 in semiprimes), A179336.

spdQ[n_]:=Module[{idn=IntegerDigits[n]},MemberQ[idn,4] || MemberQ[ idn,6] || MemberQ[ idn,9]]; Select[Select[Range[350],PrimeOmega[#]==2&],spdQ] (* Harvey P. Dale, Jun 24 2013 *)

Corrected (a(37) added) by Harvey P. Dale, Jun 24 2013

A111730 6-almost primes with semiprime digits (digits 4, 6, 9 only).

Original entry on oeis.org

64, 96, 4644, 4944, 6664, 6966, 9464, 9996, 44464, 44944, 46496, 46644, 49644, 49696, 64449, 64496, 66444, 66696, 69444, 69496, 69966, 94496, 94644, 94696, 96496, 96944, 99666, 99944, 444496, 444664, 444696, 444996, 446664, 446944, 446964, 449469, 449694, 449964, 464496, 464646, 464664, 464994, 469464, 494494, 494944, 494949, 494964, 496464, 499446, 499944, 644464, 644944

Offset: 1

Jonathan Vos Post, Nov 18 2005

			64 = 2^6
96 = 2^5 * 3
4644 = 2^2 * 3^3 * 43
4944 = 2^4 * 3 * 103
6664 = 2^3 * 7^2 * 17
6966 = 2 * 3^4 * 43
9464 = 2^3 * 7 * 13^2
9996 = 2^2 * 3 * 7^2 * 17
44464 = 2^4 * 7 * 397
44944 = 2^4 * 53^2 = 212^2
46496 = 2^5 * 1453

Intersection of A046306 and A107665.

Cf. A001358, A107342, A111494, A111496, A111697.

Select[Range[645000],ContainsOnly[IntegerDigits[#],{4,6,9}]&&PrimeOmega[#]==6&] (* James C. McMahon, Jun 05 2024 *)

isok(k) = (bigomega(k) == 6) && (#setminus(Set(digits(k)), Set([4,6,9])) == 0); \\ Michel Marcus, Apr 13 2022

Missing a(1)=64 prepended and several terms corrected by Georg Fischer and Michel Marcus, Apr 13 2022

cp's OEIS Frontend

A107342 Semiprimes with semiprime digits (digits 4, 6, 9 only).

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

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A111494 3-almost primes with semiprime digits (digits 4, 6, 9 only).

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A111496 4-almost primes with semiprime digits (digits 4, 6, 9 only).

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A111697 5-almost primes with semiprime digits (digits 4, 6, 9 only).

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A053961 Squares composed of digits {4,6,9}.

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A108614 Semiprimes with non-semiprimes digits (no digits 4,6,9 in semiprimes).

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A179463 Semiprimes A001358 containing at least one semiprime digit in base 10.

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