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A374225 Irregular triangle read by rows: T(n,k), n > 1 and k <= n, is the smallest composite number x whose set of digits and the set of digits in all prime factors of x, when written in base n, contain exactly k digits in common, or -1 if no such number exists.

%I A374225 #36 May 20 2025 14:54:52
%S A374225 -1,9,4,4,8,6,15,4,6,14,30,114,4,12,10,35,190,894,4,8,33,188,377,2355,
%T A374225 13155,4,16,14,66,462,3269,22971,127041,4,10,66,85,762,5359,36526,
%U A374225 279806,2219826,4,12,39,102,1118,9096,62959,572746,5053742,44489860,4,12,95,132
%N A374225 Irregular triangle read by rows: T(n,k), n > 1 and k <= n, is the smallest composite number x whose set of digits and the set of digits in all prime factors of x, when written in base n, contain exactly k digits in common, or -1 if no such number exists.
%e A374225 T(2, 1) = 9 = 3^2 -> 1001_2 = 11_2^2, have the digit 1 in common, and no lesser composite has this property.
%e A374225 T(6, 2) = 33 = 3 * 11 -> 53_6 = 3_6 * 15_6, have this 2 digits 3 and 5 in common, and no lesser composite has this property.
%e A374225 T(11, 6) = 174752 = 2^5 * 43 * 127 -> 10A326_11 = 2_11^5 * 3A_11 * 106_11, have the 6 digits 0, 1, 2, 3, 6 and A in common, and no lesser composite has this property.
%e A374225 The array begins:
%e A374225   n\k:0,  1,  2,   3,    4,     5,    6,
%e A374225   2: -1,  9,  4;
%e A374225   3:  4,  8,  6,  15;
%e A374225   4:  4,  6, 14,  30,  114;
%e A374225   5:  4, 12, 10,  35,  190,   894;
%e A374225   6:  4,  8, 33, 188,  377,  2355, 13155;
%o A374225 (PARI) card(base,x)=my(m=factor(x),u=[],v=[],w=[]);my(u=Set(digits(x,base)));for(i=1,#m~,w=Set(digits(m[i,1],base));v=setunion(v,w));#setintersect(u,v)
%o A374225 T(n,k)=my(x);if(k>n,return(0));if(n==2&&k==0,return(-1));forcomposite(m=max(2,n^(k-1)),oo,x=card(n,m);if(x==k,return(m)))
%Y A374225 Cf. A358003, A372249, A372280, A372295, A372384, A373645.
%K A374225 sign,base,tabf
%O A374225 2,2
%A A374225 _Jean-Marc Rebert_, Jul 01 2024