A058433 Numbers k such that k^2 contains only digits {0,3,9}, not ending with zero.
3, 969071253
Offset: 1
Links
- P. De Geest, Index to related sequences
- Hisanori Mishima, Sporadic tridigital solutions
Programs
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Mathematica
Sqrt[#]&/@Select[FromDigits/@Tuples[{0,3,9},18],Mod[#,10]!=0&&IntegerQ[Sqrt[#]]&] (* Harvey P. Dale, May 28 2025 *) -
PARI
/* helper function: */ admissibleMod(M=10^5, t=[3, 9], debug=0)={ my(p=1, v); while(M > p *= 10, t = concat([t, t + t[1]*v=vector(#t, i, p), t + t[2]*v])); debug && print("t="t); t=Set(t); v=[]; for(k=1, M, setsearch(t, k^2 % M) && v=concat(v, k)); concat(v, M+v[1])} /* optional arguments: Nmax = upper search limit, N = start / lower limit, addMod = step/chunk size, debug: 0=silent, 1=verbose */ A058433(Nmax=1e10, N=1, addMod=10^5, debug=1)={ my(a=[], d=1, addNext = admissibleMod(addMod=10^logint(addMod\/1, 10), [3, 9]), add = vector(addMod, i, i-1 > addNext[d] && d++; addNext[d]-i+1), pow10 = [10^k | k<-[0..logint((Nmax \/= 1)^2, 10)]], nextOK = [if(n, n*pow10) | n<-[0, 2, 1, 0, 5, 4, 3, 2, 1, 0]]); N \/= 1; while( Nmax >= N, my(N2 = N^2, numDigits = logint(N2, 10)+1, place = nextOK[1 + d = N2 \ pow10[numDigits]]); if( place, N = max(sqrtint(place[numDigits] + d*pow10[numDigits]), N+1); next); place = 1; my(Nnext = min(sqrtint((d+1)*pow10[numDigits]), Nmax)); debug && print("checking from "N" to "Nnext": <= ", 1+max(0, Nnext-N)*(#addNext-1)\ addMod," candidates."); while( Nnext >= N += add[1 + N % addMod], my(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; ); if( !place, debug && print(N "^2 = ", N^2); a=concat(a,N)); N = Nnext*3\2+1); a} \\ M. F. Hasler, May 14 2007
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