A217419 -id:A217419 - cp's OEIS frontend

A217404 Numbers of the form 2^r * 7^s whose decimal representation has a prime number of each digit 0-9.

326249942735257021186048, 3059867626981844171358208, 1745397244661045235955007488, 3297183493952696040281709568, 53076679184360679286299951104, 55415762982862962349014692709376

Offset: 1

James G. Merickel, Oct 02 2012

This sequence's prior, erroneous, title, was 'Numbers with squarefree part 14 whose decimal representations have a prime number of copies of each digit 0-9'. James G. Merickel, Sep 19 2013

			A217405(1)=36 and A217406(1)=15, giving this sequence's first value as (2^36)*(7^15). Its decimal representation can be seen to have two each of 0's, 1's, 3's, 5's, 6's, 7's, 8's and 9's; and three each of 2's and 3's (prime number counts of each digit).

Charles R Greathouse IV, Table of n, a(n) for n = 1..18

Subsequence of A215876.

Cf. A216854, A217405, A217406, A217407, A217410, A217413, A217416, A217419, A217422, A217425, A217428, A217431.

N:= 10^100: # to get all terms <= N
filter:= proc(n) local L,P,d;
  L:= convert(n,base,10);
  P:= Vector(10);
  for d in L do P[d+1]:= P[d+1]+1 od:
  andmap(isprime,P);
end proc:
sort(select(filter, [seq(seq(2^r*7^s, r=0..floor(log[2](N/7^s))),s=0..floor(log[7](N)))])); # Robert Israel, May 08 2017

prDigits(n)=my(d=digits(n),v=vector(10));for(i=1,#d,v[d[i]+1]++);for(i=1,10,if(!isprime(v[i]),return(0))); 1
list(lim)=my(v=List(),t); for(a=0,log(lim+.5)\log(7), t=7^a; while(t<=lim, if(prDigits(t), listput(v,t)); t<<=1)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Sep 19 2013

A217404(n) = 2^A217405(n) * 7^A217406(n).

Name changed to remove ambiguity by James G. Merickel, Sep 17 2013

A217407 Numbers of the form 3^r * 5^s whose decimal representation has a prime number of each digit 0-9.

Original entry on oeis.org

38171039656829610443115234375, 129892841018736362457275390625, 1766298261467341813095601383375, 83480063729486358039093017578125, 715350795894273434303718560266875, 172661884789704345166683197021484375, 65186341275865666700926353804318984375, 5280093643345119002775034658149837734375

Offset: 1

James G. Merickel, Oct 02 2012

This sequence in particular is motivated by the coincidence that both (2^41)*(3^43) and (3^43)*(5^47) have prime numbers of each digit.

			The first term here is (3^35)*(5^17), corresponding to A217408(1)=35 and A217409(1)=17. Its decimal representation has two each of 0's, 2's, 7's, 8's and 9's; three each of 4's, 5's and 6's; and 5 each of 1's and 3's.

Robert Israel, Table of n, a(n) for n = 1..11

Subsequence of A215876.

Cf. A216854, A217408, A217409, A217404, A217410, A217413, A217416, A217419, A217422, A217425, A217428, A217431.

N:= 10^100: # to get all terms <= N
filter:= proc(n) local L,P,d;
  L:= convert(n,base,10);
  P:= Vector(10);
  for d in L do P[d+1]:= P[d+1]+1 od:
  andmap(isprime,P);
end proc:
sort(select(filter, [seq(seq(3^r*5^s, r=0..floor(log[3](N/5^s))),s=0..floor(log[5](N)))])); # Robert Israel, May 08 2017

prDigits(n)=my(d=digits(n), v=vector(10)); for(i=1, #d, v[d[i]+1]++); for(i=1, 10, if(!isprime(v[i]), return(0))); 1
list(lim)=my(v=List(), t); for(a=0, log(lim+.5)\log(5), t=5^a; while(t<=lim, if(prDigits(t), listput(v, t)); t*=3)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Sep 19 2013

a(n) = 3^A217408(n) * 5^A217409(n).

More terms from Robert Israel, May 08 2017

A217410 Numbers of the form 3^r*7^s whose decimal representations are such that each digit 0-9 appears a prime number of times.

Original entry on oeis.org

127194058437252046971768387, 13246352657250963177488450589, 1461157813024707015061910842923, 12415617112031938486785960616347, 147856680363717377959300292543841

Offset: 1

James G. Merickel, Oct 02 2012

See the formula section for more data, and the others in cross-reference for similar sequences and motivation.

			A217411(1)=37 and A217412(1)=10, so this sequence's first term is (3^37)*(7^10).  It is the smallest number with exactly 3 and 7 as its prime factors to have decimal representation with each digit 0-9 counted a prime number of times. The digits 0, 2, 3, 5, 6 and 9 occur two times each; 1, 4 and 8 occur three times each; and 7 occurs five times.

Cf. A216854, A217411, A217412, A217404, A217407, A217410, A217416, A217419, A217422, A217425, A217428, A217431.

a(n) = 3^A217411(n) * 7^A217412(n).

A217422 Numbers of the form 2^r*17^s whose decimal representations are such that each digit 0-9 appears a prime number of times.

Original entry on oeis.org

981750581622330147995648, 28801196957834700781586432, 835992910761480393266512789504, 7295132596707416278470844481536, 76976152675689985407324172386304

Offset: 1

James G. Merickel, Oct 05 2012

See formula section for more data. Others in cross-reference are similar and some hold more motivation in comments.

			A217423(1)=47 and A217424(1)=8, so this sequence's first term is 2^47 * 17^8.  It has in its decimal representation two copies each of the digits 0, 2, 3, 4, 6 and 7; and three copies each of 1, 5, 8 and 9.

Cf. A217854, A217423, A217424, A217404, A217407, A217410, A217413, A217416, A217419, A217425, A217428, A217431.

A217422(n) = 2^A217423(n)*17^A217424(n).

A217425 Numbers of the form 5^r*7^s whose decimal representations are such that each digit 0-9 appears a prime number of times.

Original entry on oeis.org

97402668820327149658203125, 81209257154451887573232591061530625, 13375863052949754169544537548117223100875, 4587921027161765680153776379004207523600125, 2478309849684200670569842256516437530517578125

Offset: 1

James G. Merickel, Oct 05 2012

See the formula section for more data, and other sequences in cross-reference for motivation and similar sequences.

			A217426(1)=13 and A217427(1)=20, so this sequence's first term is 5^13 * 7^20.  It has two copies each of the digits 1, 3, 4, 5, 7 and 9; three each of 0's, 6's and 8's; and five 2's.

Cf. A217854, A217426, A217427, A217404, A217407, A217410, A217413, A217416, A217419, A217422, A217428, A217431.

A217425(n) = 5^A217426(n) * 7^A217427(n).

A217431 Numbers of the form 3^r*13^s whose decimal representation has a prime number of copies of each digit 0-9.

Original entry on oeis.org

691159348276025798403, 510798409623548623605717, 5097400863986495932124683149477, 10996481542736751381410324522244489, 915432679064411834115450778445909529

Offset: 1

James G. Merickel, Oct 05 2012

See the formula section for more data, and others in cross-reference for motivation and similar.

a(6), if it exists, is larger than 10^1000. - Giovanni Resta, Jan 16 2014

			a(1) = 3^25 * 13^8 (so A217432(1)=25 and A217432(1)=8). Indeed, it contains two copies of each digit other than 9 and three copies of 9.  No smaller 21-digit number with this general character -- two copies of all but one digit -- and no 20-digit number with two copies of each digit has form 3^a*13^b with a,b > 0.

Cf. A217854, A217432, A217433, A217404, A217407, A217410, A217413, A217416, A217419, A217422, A217425, A217428.

nd = 50; mx = 10^nd; pr = Prime@ Range@ PrimePi@ nd; pQ[n_] := Union[DigitCount@n, pr] == pr; Sort@ Select[ Flatten@ Table[3^p*13^q, {p, Log[3, mx/13]}, {q, Log[13, mx/3^p]}], pQ] (* terms < 10^50, Giovanni Resta, Jan 16 2014 *)

A217431(n) = 3^A217432(n) * 13^A217433(n).

A217413 Numbers of the form 2^r*11^s whose decimal representations are such that each digit 0-9 appears a prime number of times.

Original entry on oeis.org

380270922216319615655477248, 804540132953535715766960128, 808356196450619282354367102976, 3705202243778559670466281109192704, 29084192305393678777275954501779456

Offset: 1

James G. Merickel, Oct 03 2012

See the formula section for more data, and the other sequences in cross-reference for similar sequences and motivation.

			A217414(1)=71 and A217415(1)=5, so this sequence's first term is 2^71 * 11^5.  It has two each of 0's, 3's, 4's, 6's, 8's and 9's; three each of 1's, 5's and 7's; and five 2's.

Cf. A217854, A217414, A217415, A217404, A217407, A217410, A217416, A217419, A217422, A217425, A217428, A217431.

A217413(n) = 2^A217414(n) * 11^A217415(n).

A217416 Numbers of the form 2^r*13^s whose decimal representations are such that each digit 0-9 appears a prime number of times.

Original entry on oeis.org

19461082905477938625332224, 1493825030770756769826391724130304, 1341437769548771819714842333610521088, 1888426137230419620556768936800026624

Offset: 1

James G. Merickel, Oct 03 2012

See the formula section for more data, and the other sequences in cross-reference for similar and motivation.

			A217417(1)=10 and A217418(1)=20, so this sequence's first term can be simply written as 338^10 (338=2*13^2).  It has two each of 0's, 1's, 5's, 6's, 7's and 8's; three each of 3's, 4's and 9's; and five 2's.

Cf. A217854, A217417, A217418, A217404, A217407, A217410, A217413, A217419, A217422, A217425, A217428, A217431.

a(n) = 2^A217417(n) * 13^A217418(n).

A217428 Numbers of the form 2^r*19^s whose decimal representation contains each digit 0-9 a prime number of times.

Original entry on oeis.org

611897345462967423035506688, 98653640312161724942079430255181824, 195052744758517664687371356106391552, 69210613747809022824201189555756335104, 401770033097538379880522919696423452672

Offset: 1

James G. Merickel, Oct 05 2012

See the formula section for more data, and others in cross-reference for similar and motivation.

			2^55 * 19^8 has two copies of each of the digits 0, 1, 2, 7 and 9; three copies of each of 3, 4, 5 and 8; and five copies of the digit 6.  All smaller numbers of the required type have at least a digit counted 0, 1, or a composite number of times, so this is a(1).

Cf. A217854, A217429, A217430, A217404, A217407, A217410, A217413, A217416, A217419, A217422, A217425, A217431.

nd = 50; mx = 10^nd; pr = Prime@Range@PrimePi@nd; pQ[n_] := Union[DigitCount@n, pr] == pr; Sort@ Select[Flatten@ Table[2^p 19^q, {p, Log[2, mx/19]}, {q, Log[19, mx/2^p]}], pQ] (* Giovanni Resta, Jan 16 2014 *)

A217431(n) = 2^A217429(n) * 19^A217430(n)

A218005 Nonsquare semiprimes pq (10 excluded) giving record large smallest number p^r q^s such that each decimal digit appears a prime number of times.

Original entry on oeis.org

6, 14, 15, 33, 57, 185, 237, 247, 291, 327, 403

Offset: 1

James G. Merickel, Oct 17 2012

The idea for this sequence derives from A216854 and A217404 through A217433. 10 is excluded as a special case, as it necessitates finding the smaller of powers of 2 and 5 to have no digit other than 0 not appearing a prime number of times (to then be multiplied by the first power of 10 to give prime count for this digit). Even the sparser sets of mere prime powers should have members satisfying the criterion; but the numbers can be quite large, and at time of submission the actual record value for this sequence's a(11) (13*31) is unknown. The record values to that point are: (2^56)*(3^12), (2^36)*(7^15), (3^35)*(5^17), (3^29)*(11^22), (3^24)*(19^22), (5^30)*(37^12), (3^48)*(79^9), (13^40)*(19^4), (3^16)*(97^26), and (3^248)*(109^244).

Cf. A216854, A217404, A217407, A217410, A217413, A217416, A217419, A217422, A217425, A217428, A217431.

cp's OEIS Frontend

A217404 Numbers of the form 2^r * 7^s whose decimal representation has a prime number of each digit 0-9.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

Extensions

A217407 Numbers of the form 3^r * 5^s whose decimal representation has a prime number of each digit 0-9.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

Extensions

A217410 Numbers of the form 3^r*7^s whose decimal representations are such that each digit 0-9 appears a prime number of times.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A217422 Numbers of the form 2^r*17^s whose decimal representations are such that each digit 0-9 appears a prime number of times.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A217425 Numbers of the form 5^r*7^s whose decimal representations are such that each digit 0-9 appears a prime number of times.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A217431 Numbers of the form 3^r*13^s whose decimal representation has a prime number of copies of each digit 0-9.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A217413 Numbers of the form 2^r*11^s whose decimal representations are such that each digit 0-9 appears a prime number of times.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A217416 Numbers of the form 2^r*13^s whose decimal representations are such that each digit 0-9 appears a prime number of times.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A217428 Numbers of the form 2^r*19^s whose decimal representation contains each digit 0-9 a prime number of times.

Views

Author

A218005 Nonsquare semiprimes pq (10 excluded) giving record large smallest number p^r q^s such that each decimal digit appears a prime number of times.