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.
A371307
The smallest nonpalindromic number in base n that shares the same prime divisors as its digit reversal in base n.
Original entry on oeis.org
135, 32, 8, 8, 245, 12, 16, 16, 1089, 15, 72, 24, 468, 28, 32, 32, 108, 24, 48, 40, 98, 44, 144, 39, 1800, 52, 392, 35, 27869, 60, 64, 64, 45, 68, 216, 72, 162, 76, 400, 48, 75809, 48, 968, 88, 4590, 92, 288, 60, 238, 100, 1352, 104, 242, 63, 120, 112, 143370, 72, 1800, 120, 640, 124, 105
Offset: 2
a(2) = 135 as 135 = 3^3 * 5 = 10000111_2 whose digit reversal is 11100001_2 = 225 = 3^2 * 5^2, both of which have 3 and 5 as prime divisors.
a(10) = 1089 as 1089 = 3^2 * 11^2 whose digit reversal is 9801 = 3^4 * 11^2, both of which have 3 and 11 as prime divisors. See also A110819.
a(14) = 468 as 468 = 2^2 * 3^2 * 13 = 256_14 whose digit reversal is 652_14 = 1248 = 2^5 * 3 * 13, both of which have 2, 3, and 13 as prime divisors.
A372488
The smallest nonpalindromic number that shares n or more distinct prime factors with the prime factors of its reverse.
Original entry on oeis.org
10, 12, 24, 264, 8580, 24024, 2168166, 67897830, 2448684420
Offset: 0
a(3) = 264 as 264 = 2^3 * 3 * 11 and 264 in reverse is 462 = 2 * 3 * 7 * 11, which share three prime factors 2, 3, and 11.
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)))
Showing 1-4 of 4 results.
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