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.

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

Original entry on oeis.org

326249942735257021186048, 3059867626981844171358208, 1745397244661045235955007488, 3297183493952696040281709568, 53076679184360679286299951104, 55415762982862962349014692709376
Offset: 1

Views

Author

James G. Merickel, Oct 02 2012

Keywords

Comments

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

Examples

			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).

Links

Crossrefs

Subsequence of A215876.
Cf. A216854, A217405, A217406, A217407, 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(2^r*7^s, r=0..floor(log[2](N/7^s))),s=0..floor(log[7](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(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

Formula

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

Extensions

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

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).

A019550 a(n) is the concatenation of n and 2n.

Original entry on oeis.org

12, 24, 36, 48, 510, 612, 714, 816, 918, 1020, 1122, 1224, 1326, 1428, 1530, 1632, 1734, 1836, 1938, 2040, 2142, 2244, 2346, 2448, 2550, 2652, 2754, 2856, 2958, 3060, 3162, 3264, 3366, 3468, 3570, 3672, 3774, 3876, 3978, 4080, 4182, 4284, 4386, 4488, 4590
Offset: 1

Views

Author

R. Muller

Keywords

Comments

Concatenation of digits of n and 2*n. - Harvey P. Dale, Sep 13 2011
All terms are divisible by 6. - Robert Israel, Sep 21 2015

Links

Crossrefs

Cf. concatenation of n and k*n: A020338 (k=1), this sequence (k=2), A019551 (k=3), A019552 (k=4), A019553 (k=5), A009440 (k=6), A009441 (k=7), A009470 (k=8), A009474 (k=9).
Cf. A235497.
Supersequence of A117304.

Programs

  • Magma
    [Seqint(Intseq(2*n) cat Intseq(n)): n in [1..50]]; // Vincenzo Librandi, Feb 04 2014
  • Maple
    seq(n*(10^(1+ilog10(2*n))+2),n=1..100); # Robert Israel, Sep 21 2015
  • Mathematica
    nxt[n_]:=Module[{idn=IntegerDigits[n],idn2=IntegerDigits[2n]}, FromDigits[ Join[ idn,idn2]]]; Array[nxt,40] (* Harvey P. Dale, Sep 13 2011 *)
  • PARI
    a(n) = eval(Str(n, 2*n)); \\ Michel Marcus, Sep 21 2015
  • Python
    def a(n): return int(str(n) + str(2*n))
print([a(n) for n in range(1, 46)]) # Michael S. Branicky, Dec 24 2021

Formula

From Robert Israel, Sep 21 2015 (Start)
G.f.: (6*(2*x+75*x^5-60*x^6) + 90*Sum_{k>=1} 10^k*x^(5*10^k)*(5*10^k - (5*10^k-1)*x))/(1-x)^2.
a(n+2) - 2*a(n+1) + a(n) = 45*10^(2*k+1) if n = 5*10^k-2, 90*10^k-450*10^(2*k) if n = 5*10^k-1, 0 otherwise. (End)

Extensions

Offset changed from 0 to 1 by Vincenzo Librandi, Feb 04 2014

A031287 Position of n-th 0 in A007376.

Original entry on oeis.org

0, 11, 31, 51, 71, 91, 111, 131, 151, 171, 191, 192, 194, 197, 200, 203, 206, 209, 212, 215, 218, 222, 252, 282, 312, 342, 372, 402, 432, 462, 491, 492, 494, 497, 500, 503, 506, 509, 512, 515, 518, 522, 552, 582, 612, 642, 672, 702
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

A007376(a(n)) = 0;
a(n) = Max{A031287(n), A031288(n), A031289(n), A031290(n), A031291(n), A031292(n), A031293(n), A031294(n), A031295(n), A031296(n)}. [Reinhard Zumkeller, Jul 28 2011]

Links

Crossrefs

Cf. A193428.

Programs

  • Haskell
    import Data.List (elemIndices)
a031287 n = a031287_list !! (n-1)
a031287_list = map (+ 1) $ elemIndices 0 a007376_list
-- Reinhard Zumkeller, Jul 28 2011

Extensions

a(1)=0 inserted for consistency with change in A007376 by Sean A. Irvine, Apr 21 2020

A031288 Position of n-th 1 in A007376.

Original entry on oeis.org

1, 10, 12, 13, 14, 16, 18, 20, 22, 24, 26, 28, 33, 53, 73, 93, 113, 133, 153, 173, 190, 193, 195, 196, 199, 202, 205, 208, 211, 214, 217, 220, 221, 223, 224, 225, 226, 227, 229, 230, 232, 233, 235, 236, 238, 239, 241, 242, 244, 245
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

A007376(a(n)) = 1;
a(n) = Min{A031287(n), A031288(n), A031289(n), A031290(n), A031291(n), A031292(n), A031293(n), A031294(n), A031295(n), A031296(n)}. [Reinhard Zumkeller, Jul 28 2011]

Links

Crossrefs

Cf. A193428.

Programs

  • Haskell
    import Data.List (elemIndices)
a031288 n = a031288_list !! (n-1)
a031288_list = map (+ 1) $ elemIndices 1 a007376_list
-- Reinhard Zumkeller, Jul 28 2011

A031289 Position of n-th 2 in A007376.

Original entry on oeis.org

2, 15, 30, 32, 34, 35, 36, 38, 40, 42, 44, 46, 48, 55, 75, 95, 115, 135, 155, 175, 198, 228, 251, 254, 257, 258, 260, 263, 266, 269, 272, 275, 278, 288, 318, 348, 378, 408, 438, 468, 490, 493, 496, 498, 499, 502, 505, 508, 511, 514
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

A007376(a(n)) = 2.

Links

Crossrefs

Cf. A193428; A031287, A031288, A031290, A031291, A031292, A031293, A031294, A031295, A031296.

Programs

  • Haskell
    import Data.List (elemIndices)
a031289 n = a031289_list !! (n-1)
a031289_list = map (+ 1) $ elemIndices 2 a007376_list
-- Reinhard Zumkeller, Jul 28 2011
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