Displaying 1-10 of 13 results found.
a(n) = largest member of the n-th term in S(3) (defined in Comments).
+10
12
2, 5, 7, 17, 13, 19, 23, 53, 31, 43, 41, 59, 79, 67, 71, 137, 151, 157, 127, 131, 149, 181, 167, 233, 197, 211, 199, 241, 229, 239, 479, 419, 457, 449, 283, 293, 313, 349, 337, 401, 359, 367, 373, 397, 389, 463, 421, 727, 653, 661, 719, 701, 523, 647, 571, 631, 617, 619, 607, 643, 659, 691, 1453, 739, 1283, 1429, 761, 769
COMMENTS
Let H(L,b) be the Hamming graph whose vertices are the sequences of length L over the alphabet {0,1,...,b-1} with adjacency being defined by having Hamming distance 1. Let P(L,b) be the subgraph of H(L,b) induced by the set of vertices which are base b representations of primes with L digits (not allowing leading 0 digits). Let S(b) be the sequence of all components of the graphs P(L,b), L>0, sorted by the smallest prime in a component.
a(n) = number of components of the graph P(n,10) (defined in Comments).
+10
5
1, 1, 1, 1, 1, 7, 38, 365
COMMENTS
Let H(n,b) be the Hamming graph whose vertices are the sequences of length n over the alphabet {0,1,...,b-1} with adjacency being defined by having Hamming distance 1. Let P(n,b) be the subgraph of H(n,b) induced by the set of vertices which are base b representations of primes with n digits (not allowing leading 0 digits).
For 6 and 7 digit primes there is a single large component and the remaining components have size 1. For 8 digit primes there is a single large component, the size 2 component {89391959, 89591959} and the remaining components have size 1. [ W. Edwin Clark, Mar 31 2009]
EXAMPLE
The 6-digit primes 294001, 505447, 584141, 604171, 929573, 971767 (cf. A050249) have the property that changing any single digit always gives a composite number, so these are isolated nodes in the graph P(6,10) (which also has one large connected component).
a(n) = largest member of the n-th term in S(10) (defined in Comments).
+10
4
7, 97, 997, 9973, 99991, 999983, 9999991, 294001, 505447, 584141, 604171, 929573, 971767, 99999989, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3070663, 3085553, 3326489, 4393139, 5152507, 5285767, 5564453, 5575259
COMMENTS
Let H(L,b) be the Hamming graph whose vertices are the sequences of length L over the alphabet {0,1,...,b-1} with adjacency being defined by having Hamming distance 1. Let P(L,b) be the subgraph of H(L,b) induced by the set of vertices which are base b representations of primes with L digits (not allowing leading 0 digits). Let S(b) be the sequence of all components of the graphs P(L,b), L>0, sorted by the smallest prime in a component.
a(n) = size of the n-th term in S(2) (defined in Comments).
+10
2
2, 2, 1, 1, 5, 3, 4, 9, 2, 1, 1, 7, 4, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 12, 1, 20, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 29, 19, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 3, 1, 75, 2, 19, 4, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 23, 1, 82, 76, 1, 1, 3, 1, 1, 3, 3, 4, 2, 3, 3, 1, 2, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 9, 1, 2, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1
COMMENTS
Let H(L,b) be the Hamming graph whose vertices are the sequences of length L over the alphabet {0,1,...,b-1} with adjacency being defined by having Hamming distance 1. Let P(L,b) be the subgraph of H(L,b) induced by the set of vertices which are base b representations of primes with L digits (not allowing leading 0 digits). Let S(b) be the sequence of all components of the graphs P(L,b), L>0, sorted by the smallest prime in a component.
a(n) = smallest member of the n-th term in S(10) (defined in Comments).
+10
2
2, 11, 101, 1009, 10007, 100003, 1000003, 294001, 505447, 584141, 604171, 929573, 971767, 10000019, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3070663, 3085553, 3326489, 4393139, 5152507, 5285767, 5564453, 5575259
COMMENTS
Let H(L,b) be the Hamming graph whose vertices are the sequences of length L over the alphabet {0,1,...,b-1} with adjacency being defined by having Hamming distance 1. Let P(L,b) be the subgraph of H(L,b) induced by the set of vertices which are base b representations of primes with L digits (not allowing leading 0 digits). Let S(b) be the sequence of all components of the graphs P(L,b), L>0, sorted by the smallest prime in a component.
a(n) = smallest member of the n-th term in S(2) (defined in Comments).
+10
1
2, 5, 11, 13, 17, 37, 41, 67, 73, 107, 127, 131, 149, 173, 191, 193, 211, 223, 233, 239, 241, 251, 257, 263, 277, 281, 337, 349, 353, 373, 419, 431, 443, 491, 509, 521, 541, 547, 557, 613, 653, 661, 683, 701, 709, 719, 733, 761, 769, 787, 853, 877, 907, 1019, 1031, 1091, 1093, 1153, 1163, 1187, 1193, 1201, 1259, 1381, 1433, 1451, 1453, 1553, 1597, 1637, 1657, 1709, 1721, 1753, 1759, 1777, 1783, 1811, 1889, 1907, 1931, 1973, 2027
COMMENTS
Let H(L,b) be the Hamming graph whose vertices are the sequences of length L over the alphabet {0,1,...,b-1} with adjacency being defined by having Hamming distance 1. Let P(L,b) be the subgraph of H(L,b) induced by the set of vertices which are base b representations of primes with L digits (not allowing leading 0 digits). Let S(b) be the sequence of all components of the graphs P(L,b), L>0, sorted by the smallest prime in a component.
a(n) = largest member of the n-th term in S(2) (defined in Comments).
+10
1
3, 7, 11, 13, 31, 61, 59, 113, 89, 107, 127, 227, 181, 173, 191, 229, 211, 223, 233, 239, 241, 251, 257, 479, 277, 503, 337, 349, 353, 373, 419, 431, 443, 491, 509, 619, 1021, 953, 557, 613, 653, 661, 683, 701, 709, 751, 733, 761, 773, 787, 853, 877, 971, 1019, 2029, 1123, 1879, 1409, 1163, 1699, 1193, 1201, 1259, 1381, 1433, 1451, 1453, 1553, 1597, 1637, 1913, 1709, 1979, 1753, 1759, 1777, 2039, 1811, 2017, 1907, 1931, 1973, 2027
COMMENTS
Let H(L,b) be the Hamming graph whose vertices are the sequences of length L over the alphabet {0,1,...,b-1} with adjacency being defined by having Hamming distance 1. Let P(L,b) be the subgraph of H(L,b) induced by the set of vertices which are base b representations of primes with L digits (not allowing leading 0 digits). Let S(b) be the sequence of all components of the graphs P(L,b), L>0, sorted by the smallest prime in a component.
a(n) = number of components of the graph P(n,3) (defined in Comments).
+10
1
1, 2, 4, 8, 15, 32, 88, 209, 539, 1403, 3698, 9962, 26447, 71579, 196590, 541473, 1501720, 4186566, 11737617
COMMENTS
Let H(n,b) be the Hamming graph whose vertices are the sequences of length n over the alphabet {0,1,...,b-1} with adjacency being defined by having Hamming distance 1. Let P(n,b) be the subgraph of H(n,b) induced by the set of vertices which are base b representations of primes with n digits (not allowing leading 0 digits).
a(n) = size of the n-th term in S(3) (defined in Comments).
+10
1
1, 2, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 3, 1, 1, 5, 2, 3, 2, 2, 1, 2, 1, 4, 2, 2, 1, 2, 1, 1, 8, 3, 5, 3, 1, 1, 2, 2, 1, 3, 2, 1, 1, 2, 1, 2, 1, 6, 4, 2, 3, 4, 1, 5, 1, 2, 2, 3, 1, 1, 1, 1, 9, 1, 4, 5, 1, 1, 2, 11, 6, 6, 2, 3, 1, 1, 4, 1, 1, 1, 3, 4, 1, 6, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 7, 1
COMMENTS
Let H(L,b) be the Hamming graph whose vertices are the sequences of length L over the alphabet {0,1,...,b-1} with adjacency being defined by having Hamming distance 1. Let P(L,b) be the subgraph of H(L,b) induced by the set of vertices which are base b representations of primes with L digits (not allowing leading 0 digits). Let S(b) be the sequence of all components of the graphs P(L,b), L>0, sorted by the smallest prime in a component.
a(n) = smallest member of the n-th term in S(3) (defined in Comments).
+10
1
2, 3, 7, 11, 13, 19, 23, 29, 31, 37, 41, 59, 61, 67, 71, 83, 97, 103, 109, 113, 149, 163, 167, 173, 191, 193, 199, 223, 229, 239, 251, 257, 271, 281, 283, 293, 307, 331, 337, 347, 353, 367, 373, 379, 389, 409, 421, 487, 491, 499, 503, 521, 523, 569, 571, 577, 599, 601, 607, 643, 659, 691, 733, 739, 743, 757, 761, 769, 773
COMMENTS
Let H(L,b) be the Hamming graph whose vertices are the sequences of length L over the alphabet {0,1,...,b-1} with adjacency being defined by having Hamming distance 1. Let P(L,b) be the subgraph of H(L,b) induced by the set of vertices which are base b representations of primes with L digits (not allowing leading 0 digits). Let S(b) be the sequence of all components of the graphs P(L,b), L>0, sorted by the smallest prime in a component.
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