cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

A217405 2-adic valuation of A217404.

Original entry on oeis.org

36, 28, 54, 10, 87, 10, 20, 1, 59, 121, 97, 134, 135, 88, 131, 175, 58, 66
Offset: 1

Views

Author

James G. Merickel, Oct 02 2012

Keywords

Comments

See the main sequence corresponding to this one, A217404.

Examples

			A217404(1)=(2^36)*(7^15), so that the exponent on 2, 36, is the first term here.

Crossrefs

Cf. A217404, A217406.

Programs

  • PARI
    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)); apply(k->valuation(k,2),vecsort(Vec(v))) \\ Charles R Greathouse IV, Sep 19 2013

A217406 7-adic valuation of A217404.

Original entry on oeis.org

15, 19, 13, 29, 3, 34, 31, 43, 24, 2, 14, 11, 11, 29, 14, 4, 55, 89
Offset: 1

Views

Author

James G. Merickel, Oct 02 2012

Keywords

Comments

See main sequence, A217404.

Examples

			A217404(1)=(2^36)*(7^15), so this sequence's first term is 15, the exponent on 7 in the main sequence.

Crossrefs

Subsequence of A215876.
Cf. A217404, A217405.

Programs

  • PARI
    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)); apply(k -> valuation(k,7), vecsort(Vec(v))) \\ Charles R Greathouse IV, Sep 19 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

Views

Author

James G. Merickel, Oct 02 2012

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

Subsequence of A215876.
Cf. A216854, A217408, A217409, A217404, A217410, A217413, A217416, A217419, A217422, A217425, A217428, A217431.

Programs

  • Maple
    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
  • PARI
    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

Formula

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

Extensions

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

Views

Author

James G. Merickel, Oct 02 2012

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Formula

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

Views

Author

James G. Merickel, Oct 05 2012

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Formula

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

Views

Author

James G. Merickel, Oct 05 2012

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Formula

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

Views

Author

James G. Merickel, Oct 05 2012

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

James G. Merickel, Oct 03 2012

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Formula

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

Views

Author

James G. Merickel, Oct 03 2012

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Formula

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

Views

Author

James G. Merickel, Oct 05 2012

Keywords

Comments

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

Examples

			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).

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

A217431(n) = 2^A217429(n) * 19^A217430(n)
Showing 1-10 of 12 results. Next

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