A373645
The smallest number whose prime factor concatenation, as well as the number itself, when written in base n, contains all digits 0,1,...,(n-1).
Original entry on oeis.org
2, 11, 114, 894, 13155, 127041, 2219826, 44489860, 1023485967, 26436195405, 755182183459, 23609378957430, 802775563829902, 29480898988179429, 1162849454580682365
Offset: 2
a(5) = 894 = 12034_5 which contains all the digits 0..4, and 894 = 2 * 3 * 149 = 2_5 * 3_5 * 1044_5, and the factors contain all digits 0..4.
a(10) = 1023485967 which contains all digits 0..9, and 1023485967 = 3 * 3 * 7 * 16245809, and the factors contain all digits 0..9.
a(15) = 29480898988179429 = 102345C86EA7BD9_15 which contains all the digits 0..E, and 29480898988179429 = 3 * 7 * 17 * 139 * 594097474723 = 3_15 * 7_15 * 12_15 * 94_15 * 106C1A8B5ED_15, and the factors contain all digits 0..E.
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.
Original entry on oeis.org
-1, 9, 4, 4, 8, 6, 15, 4, 6, 14, 30, 114, 4, 12, 10, 35, 190, 894, 4, 8, 33, 188, 377, 2355, 13155, 4, 16, 14, 66, 462, 3269, 22971, 127041, 4, 10, 66, 85, 762, 5359, 36526, 279806, 2219826, 4, 12, 39, 102, 1118, 9096, 62959, 572746, 5053742, 44489860, 4, 12, 95, 132
Offset: 2
T(2, 1) = 9 = 3^2 -> 1001_2 = 11_2^2, have the digit 1 in common, and no lesser composite has this property.
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.
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.
The array begins:
n\k:0, 1, 2, 3, 4, 5, 6,
2: -1, 9, 4;
3: 4, 8, 6, 15;
4: 4, 6, 14, 30, 114;
5: 4, 12, 10, 35, 190, 894;
6: 4, 8, 33, 188, 377, 2355, 13155;
-
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)
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)))
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