A058430 -id:A058430 - cp's OEIS frontend

A199340 Primes having only {0, 3, 4} as digits.

3, 43, 433, 443, 3343, 3433, 4003, 30403, 33343, 33403, 34033, 34303, 34403, 40343, 40433, 43003, 43403, 300043, 300343, 304033, 304303, 304433, 330433, 333433, 334043, 334333, 334403, 343303, 343333, 343433, 400033, 403003, 403043, 403433, 430303, 430333

Offset: 1

M. F. Hasler, Nov 05 2011

All terms end in '3'. This could be used to speed up the given program.

A020461 is a subsequence. - Vincenzo Librandi, Jul 23 2015

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Index to entries about primes with digits in a given set.

Cf. Primes that contain only the digits (3,4,k): this sequence (k=0), A199341 (k=1), A199342 (k=2), A199345 (k=5), A199346 (k=6), A199347 (k=7), A199348 (k=8), A199349 (k=9).
Cf. A020449 - A020472, A199325 - A199329.
Cf. A058429, A058430.

[p: p in PrimesUpTo(5*10^5) | Set(Intseq(p)) subset [3, 4, 0]]; // Vincenzo Librandi, Jul 23 2015

Select[Prime[Range[5 10^4]], Complement[IntegerDigits[#], {3, 4, 0}]=={} &] (* Vincenzo Librandi, Jul 23 2015 *)
Select[FromDigits/@Tuples[{0,3,4},6],PrimeQ] (* Harvey P. Dale, Mar 21 2020 *)
Select[10#+3&/@FromDigits/@Tuples[{0,3,4},5],PrimeQ] (* Harvey P. Dale, May 02 2022 *)

a(n, list=0, L=[0, 3, 4], reqpal=0)={my(t); for(d=1, 1e9, u=vector(d, i, 10^(d-i))~; forvec(v=vector(d, i, [1+(i==1&!L[1]), #L]), isprime(t=vector(d, i, L[v[i]])*u)||next; reqpal && !isprime(A004086(t)) && next; list && print1(t", "); n--||return(t)))} \\ Syntax updated for current PARI version. - M. F. Hasler, Jul 25 2015

{forprime(p=3,1e6,p%10==3&&!setminus(Set(digits(p)),[3,4])&&print1(p","))} \\ [0] evaluates to false. - M. F. Hasler, Jul 25 2015

A058429 Numbers k such that k^2 contains only digits {0,3,4}, not ending with zero.

Original entry on oeis.org

2, 548, 585688, 58591812, 200824138, 5773251280200207952, 20832739723817975138362

Offset: 1

Patrick De Geest, Nov 15 2000

P. De Geest, Index to related sequences
Hisanori Mishima, Sporadic tridigital solutions

Cf. A058430.

\\ From M. F. Hasler, May 14 2007, edited Nov 14 2017: (Start)
admissibleMod(LIM=10000)={ local( t=[4], tt=1 ); while( LIM > tt*=10, t=concat([t,t+vector(#t,i,3)*tt,t+vector(#t,i,4)*tt])); /*print("t="t);*/ t=Set(t); tt=[]; for(i=1,LIM, setsearch(t,i^2%LIM) && tt=concat(tt,i)); concat(tt,LIM+tt[1])}
A058429(Nmax=1e19,N=2,addMod=100000,debug=1)={my( a=[], Nnext, N2, numDigits, place, addNext=admissibleMod(addMod=round(addMod)), d=1, add=vector(addMod,i,if(i-1>addNext[d],d++);addNext[d]-i+1), nextOK=[0,2,1,0,0,5,4,3,2,1], pow10 = vector( d=#Str((Nmax=round(Nmax))^2), i, 10^(i-1)) ); nextOK = vector( #nextOK, i, if( nextOK[i],nextOK[i]*pow10)); N=round(N); while( Nmax >= N, numDigits = #Str(N2=N^2); 3 > N2 \ pow10[numDigits] && N = sqrtint( 3*pow10[numDigits]+3 )+1; Nnext = min( Nmax, sqrtint(pow10[numDigits]*10)\3*2); if( debug, print( "checking from "N" to "Nnext": <= ",1+max(0,Nnext-N)*(#addNext-1) \ addMod," candidates.")); N += add[1+ N%addMod]; place=1; while( Nnext >= N, dr = divrem( N2=N^2, pow10[ place=numDigits ] ); while( place-- && !d=nextOK[1+ (dr = divrem( dr[2], pow10[ place ] ))[1]], ); place || break; N = sqrtint( N2 - dr[2] + d[ place ])+1; N+=add[1+N%addMod]); if( !place, debug && print( N, "^2 = ", N^2); a=concat(a,N); N=Nnext+1 ); N=Nnext*5\2);a } \\ (End)

a(6) from M. F. Hasler, May 14 2007

a(7) from Mishima's page added by Max Alekseyev, Jul 13 2009

A119059 Triangular numbers composed of digits {0,3,4}.

Original entry on oeis.org

3, 300, 3003, 3403, 3444000, 344334403, 433430403, 3434340003, 4033034303403, 3430000044444403, 4333004330434300, 300034434334030003, 333343033034040030, 44400033434030304430, 3340004334430334340000, 30340003004344033440000, 4430044300033330340443003

Offset: 1

Giovanni Resta, May 10 2006

Giovanni Resta, Tridigital Triangular Numbers

Cf. A000217, A058430, A119060. See A119033 for a table of cross-references.

a(n) = A000217(A119060(n)). - Michel Marcus, Mar 22 2023

a(16)-a(17) from Tyler Busby, Mar 22 2023

A294660 Least nonnegative integer not occurring earlier whose square has no digit in common with the square of the previous term, a(0) = 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 8, 10, 15, 12, 16, 20, 11, 22, 13, 18, 14, 28, 19, 17, 21, 23, 26, 29, 24, 30, 25, 33, 58, 27, 34, 47, 38, 45, 31, 48, 41, 50, 37, 52, 44, 65, 40, 57, 76, 32, 63, 35, 60, 39, 62, 36, 88, 46, 67, 51, 183, 75, 43, 55, 42, 53, 56, 70, 61, 64, 85, 59, 77, 69, 73, 78, 89

Offset: 0

M. F. Hasler, Nov 08 2017

This is not a permutation of the nonnegative integers, since numbers whose square has all digits '1' through '9' (cf. A294661, e.g., 11826 with 11826^2 = 139854276) can never appear - and these numbers have asymptotic density 1.

Will all integers whose square does not have all of the digits 1-9, eventually appear? Or might the sequence be finite? Since a(n)^2 has no digits in common with a(n-1)^2, it is sufficient for a(n+1) to exist, to find a number whose square has a subset of the digits of a(n-1)^2. Is this always possible? This problem sometimes has only "sporadic k-digital solutions", see, e.g., A058430, A030175, ... and the link to De Geest's page.

			Since a(7)^2 = 7^2 = 49, the subsequent term cannot be 8, since 8^2 = 64 has the digit 4 in common with 49. Therefore, a(8) = 9, with 9^2 = 81 having no common digit with 49.
a(1201) = 1037. So the square of the next term must not have any of the digits in {0, 1, 3, 5, 6, 7, 9}, only 2, 4, 8 are allowed. The least such number that has not been used before is a(1202) = 210912978, with a(1202)^2 = 210912978^2 = 44484284288828484. - _Alois P. Heinz_, Nov 09 2017

M. F. Hasler, Table of n, a(n) for n = 0..4829
P. De Geest, Squares containing at most three distinct digits, Index entries for related sequences

Cf. A030287 (strictly increasing), A067581 (do not take squares).

{u=a=0; for(n=0, 99, print1(a", "); u+=1<

{u=[a=0]; for(n=0, 99, print1(a", "); D=Set(if(a, digits(a^2))); for(k=u[1]+1, oo, setsearch(u, k)&&next; #setintersect(D, Set(digits(k^2)))&&next; u=setunion(u,[a=k]); break); while(#u>1&&u[2]==u[1]+1,u=u[^1])); a}

A136927 Numbers k such that k and k^2 use only the digits 0, 3, 4, 5 and 6.

Original entry on oeis.org

0, 6, 60, 66, 600, 656, 660, 666, 6000, 6560, 6600, 6660, 6666, 60000, 63566, 65600, 66000, 66600, 66660, 66666, 600000, 603656, 604434, 635660, 656000, 660000, 660656, 666000, 666600, 666660, 666666, 6000000, 6036560, 6044340, 6356600, 6560000, 6600000, 6606560, 6660000, 6666000, 6666600, 6666660, 6666666, 60000000, 60054434, 60365600

Offset: 1

Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008

Generated with DrScheme.

See also A058429-A058430. - M. F. Hasler, Jan 24 2008

			665605465350506^2 = 443030635504463643353434456036.

Jonathan Wellons, Table of n, a(n) for n = 1..495
J. Wellons, Tables of Shared Digits [archived]

With[{c={0,3,4,5,6},nn=8},Select[FromDigits/@Tuples[c,nn],SubsetQ[ c,IntegerDigits[ #^2]]&]] (* Harvey P. Dale, Apr 27 2022 *)

A136928 Numbers k such that k and k^2 use only the digits 0, 3, 4, 5 and 8.

Original entry on oeis.org

0, 548, 5480, 54800, 55548, 548000, 555388, 555480, 588048, 5480000, 5553880, 5554548, 5554800, 5834388, 5880480, 54800000, 55538800, 55545480, 55548000, 58343880, 58804800, 548000000, 550500548, 555388000, 555454800, 555480000, 583438800, 588048000, 5480000000, 5505005480, 5553880000, 5554548000, 5554800000, 5834388000

Offset: 1

Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008

Generated with DrScheme.

			583854308054548^2 = 340885853033855033588543484304.

Jonathan Wellons, Table of n, a(n) for n = 1..102
J. Wellons, Tables of Shared Digits [archived]

See also A058429-A058430. - M. F. Hasler, Jan 24 2008

A136929 Numbers k such that k and k^2 use only the digits 0, 3, 4, 5 and 9.

Original entry on oeis.org

0, 3, 30, 300, 3000, 30000, 30503, 30903, 300000, 305030, 309030, 550503, 950503, 953903, 3000000, 3005003, 3009003, 3050300, 3055003, 3090300, 5505030, 9505030, 9539030, 30000000, 30050030, 30090030, 30503000, 30550030, 30903000, 55050300, 95050300, 95055003, 95390300, 300000000, 300050003, 300050503, 300090003, 300500300

Offset: 1

Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008

Generated with DrScheme.

See also A058429-A058430. - M. F. Hasler, Jan 24 2008

			950439335440903^2 = 903334930353345333433405455409.

Jonathan Wellons, Table of n, a(n) for n = 1..352
J. Wellons, Tables of Shared Digits [archived]

A136930 Numbers k such that k and k^2 use only the digits 0, 3, 4, 6 and 7.

Original entry on oeis.org

0, 6, 60, 600, 6000, 60000, 66076, 600000, 660760, 6000000, 6607600, 60000000, 60003706, 60036706, 66036076, 66066376, 66076000, 600000000, 600037060, 600306344, 600367060, 600373006, 606336074, 660360760, 660663760, 660760000, 660760474, 6000000000, 6000036706, 6000370600, 6000637006, 6003063440, 6003670600, 6003730060

Offset: 1

Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008

Generated with DrScheme.

See also A058429-A058430. - M. F. Hasler, Jan 24 2008

			660760703770306^2 = 436604707647030077763607333636.

Jonathan Wellons, Table of n, a(n) for n = 1..181
J. Wellons, Tables of Shared Digits [archived]

A136931 Numbers k such that k and k^2 use only the digits 0, 3, 4, 6 and 8.

Original entry on oeis.org

0, 6, 8, 60, 80, 600, 800, 6000, 6638, 6808, 6844, 8000, 60000, 60406, 60688, 66380, 66808, 68080, 68440, 80000, 600000, 604060, 606880, 663800, 668080, 680800, 684400, 800000, 6000000, 6003406, 6004006, 6040600, 6068800, 6638000, 6680800, 6808000, 6844000, 8000000, 60000000, 60034060, 60040060, 60406000, 60688000, 66363008, 66380000

Offset: 1

Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008

Generated with DrScheme.

			683084686436344^2 = 466604688843838404646364086336.

Jonathan Wellons, Table of n, a(n) for n = 1..378
J. Wellons, Tables of Shared Digits [archived]

See also A058429-A058430. - M. F. Hasler, Jan 24 2008

s={0,3,4,6,8}; Select[Range[0,10^6], SubsetQ[s, Sort[DeleteDuplicates[IntegerDigits[#]]]] && SubsetQ[s, Sort[DeleteDuplicates[IntegerDigits[#^2]]]] &] (* Stefano Spezia, Aug 11 2025 *)

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A199340 Primes having only {0, 3, 4} as digits.

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A058429 Numbers k such that k^2 contains only digits {0,3,4}, not ending with zero.

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A119059 Triangular numbers composed of digits {0,3,4}.

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A294660 Least nonnegative integer not occurring earlier whose square has no digit in common with the square of the previous term, a(0) = 0.

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A136927 Numbers k such that k and k^2 use only the digits 0, 3, 4, 5 and 6.

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A136928 Numbers k such that k and k^2 use only the digits 0, 3, 4, 5 and 8.

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A136929 Numbers k such that k and k^2 use only the digits 0, 3, 4, 5 and 9.

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A136930 Numbers k such that k and k^2 use only the digits 0, 3, 4, 6 and 7.

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A136931 Numbers k such that k and k^2 use only the digits 0, 3, 4, 6 and 8.

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