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Lexicographically earliest sequence of positive integers such that for any t and k, with k>=1, where t = a(n) = a(n+k) = a(n+2*k), only one occurrence of k, for a given t, appears anywhere in the sequence.
+0
0
1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 3, 2, 3, 2, 1, 3, 1, 2, 3, 2, 2, 1, 3, 3, 2, 1, 2, 3, 3, 3, 1, 1, 4, 1, 2, 3, 4, 3, 1, 3, 1, 4, 4, 2, 3, 2, 2, 4, 3, 4, 2, 4, 4, 2, 1, 4, 1, 3, 2, 2, 4, 5, 3, 1, 3, 3, 1, 4, 4, 2, 4, 4, 3, 1, 1, 2, 3, 3, 2, 5, 5, 3, 5, 2, 1, 3, 4, 5, 4, 1, 5, 4, 3, 1, 2, 4, 1, 4, 1, 5, 2, 2, 3, 3, 5, 5, 5, 4, 5, 1, 4, 3, 2, 5
OFFSET
1,4
COMMENTS
See A381599 for the index where n first appears, and A381598 for the index where three consecutive n's appears.
LINKS
EXAMPLE
a(1) = a(2) = a(3) = 1, which is the first appearance of three 1's separated by one term.
a(4) = 2 as 1 cannot be chosen as that would form a(2) = a(3) = a(4) = 1, but three 1's separated by one term has already appeared.
a(5) = 1, which also forms three 1's separated by two terms, a(1) = a(3) = a(5) = 1.
a(17) = 3 as 1 cannot be chosen as that would form a(15) = a(16) = a(17) = 1, but three 1's separated by one term has already appeared, while choosing 2 would form a(11) = a(14) = a(17) = 2, but three 2's separated by three terms has already appeared at a(4) = a(7) = a(10) = 2.
CROSSREFS
Cf. A381598 (triplets), A381599 (where n first appears), A370708 (indices where 1's appear), A281511, A229037.
KEYWORD
nonn,new
AUTHOR
Scott R. Shannon, Mar 01 2025
STATUS
approved
Index of first term of three consecutive n's in A381597.
+0
0
1, 9, 34, 147, 111, 359, 437, 389, 594, 826, 1102, 83317, 1789, 5142, 2931, 12671
OFFSET
1,2
COMMENTS
The terms vary greatly in size - after 5.2 million terms of A381597 no three consecutive 17's or 18's have appeared, although three consecutive 19's appear at index 6474. The largest known term is a(192) = 5135798.
CROSSREFS
KEYWORD
nonn,more,new
AUTHOR
Scott R. Shannon, Mar 01 2025
STATUS
approved
Index where n first appears in A381597.
+0
0
1, 4, 17, 39, 68, 124, 191, 286, 441, 577, 776, 1043, 1192, 1556, 1736, 2214, 2744, 3221, 3519, 4248, 5028, 5542, 6574, 7013, 8093, 8945, 10110, 11043, 12413, 13223, 14476, 15923, 17430, 18617, 20027, 21991, 24016, 25364, 27414, 29356, 31392, 32614, 35743, 37888, 40301, 42620, 45696, 47776, 51109, 53264, 56429, 58471, 61676, 64468, 69437, 72011, 75626
OFFSET
1,2
CROSSREFS
KEYWORD
nonn,new
AUTHOR
Scott R. Shannon, Mar 01 2025
STATUS
approved
G.f. A(x) satisfies A(x) = 1/(1 - x * A(x*A(x)^2)^2).
+0
0
1, 1, 3, 16, 113, 955, 9178, 97427, 1121705, 13836694, 181295019, 2507119320, 36416096984, 553461581406, 8774534872463, 144744539399484, 2479088917439527, 44004108702467428, 808171916050540308, 15335535608825061803, 300272362335527090277, 6059534345675248667550
OFFSET
0,3
FORMULA
Let a(n,k) = [x^n] A(x)^k.
a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(2*n-j+k,j)/(2*n-j+k) * a(n-j,2*j).
PROG
(PARI) a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n-j+k, j)/(2*n-j+k)*a(n-j, 2*j)));
CROSSREFS
KEYWORD
nonn,new
AUTHOR
Seiichi Manyama, Mar 01 2025
STATUS
approved
G.f. A(x) satisfies A(x) = 1/(1 - x * A(x*A(x)^3)^3).
+0
0
1, 1, 4, 31, 320, 3969, 56080, 876204, 14860614, 270231265, 5223002719, 106613106181, 2287120272173, 51367948203527, 1204141944566399, 29385603693050274, 744943334951904519, 19580887642660810193, 532781828387893449124, 14984377196395037979472
OFFSET
0,3
FORMULA
Let a(n,k) = [x^n] A(x)^k.
a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(3*n-2*j+k,j)/(3*n-2*j+k) * a(n-j,3*j).
PROG
(PARI) a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-2*j+k, j)/(3*n-2*j+k)*a(n-j, 3*j)));
CROSSREFS
KEYWORD
nonn,new
AUTHOR
Seiichi Manyama, Mar 01 2025
STATUS
approved
First prime of cousin prime pairs which differ, in their binary representation, by a single bit.
+0
0
3, 19, 43, 67, 97, 163, 193, 307, 313, 379, 457, 499, 643, 673, 739, 769, 859, 883, 907, 937, 1009, 1297, 1483, 1489, 1579, 1609, 1867, 1873, 1993, 2083, 2137, 2203, 2347, 2377, 2473, 2539, 2617, 2659, 2683, 2689, 2707, 2833, 2857, 2953
OFFSET
1,1
COMMENTS
The first prime of a cousin prime pair is a prime p for which p+4 is also prime.
The only way for p and p+4 to differ at a single bit position is when p has a 0 bit at its "4" position, so p == {0,1,2,3} (mod 8), and so this sequence is the intersection of A023200 and A047471.
EXAMPLE
3 is a term since it's a cousin prime with 7 and their respective binary representations 011 and 111 differ at a single bit position.
13 is not a term since, although it's a cousin prime with 17, their respective binary representations 1101 and 10001 differ at more than a single bit position.
PROG
(Python)
import sympy
def ok(n): return (n&5)==1 and sympy.isprime(n) and sympy.isprime(n+4)
CROSSREFS
Cf. A023200 (cousin primes), A047471, A071695.
KEYWORD
nonn,base,new
AUTHOR
James S. DeArmon, Jan 01 2025
STATUS
approved
Concatenate the sequences S(k) = [0, 1, -1, ..., k, -k] for k = 0, 1, ...
+0
0
0, 0, 1, -1, 0, 1, -1, 2, -2, 0, 1, -1, 2, -2, 3, -3, 0, 1, -1, 2, -2, 3, -3, 4, -4, 0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 7, -7, 0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 7, -7, 8, -8, 0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 7, -7, 8, -8, 9, -9
OFFSET
0,8
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..10200 (k = 0..100)
MATHEMATICA
Flatten[Table[(-1)^j*Floor[j/2], {k, 0, 10}, {j, 2*k + 1}]] (* Paolo Xausa, Mar 01 2025 *)
CROSSREFS
KEYWORD
sign,easy,new
AUTHOR
N. J. A. Sloane, Mar 01 2025 [Suggested by Franklin T. Adams-Watters, Sep 21 2011]
STATUS
approved
Count down from k to -k for k = 0, 1, 2, ... .
+0
0
0, 1, 0, -1, 2, 1, 0, -1, -2, 3, 2, 1, 0, -1, -2, -3, 4, 3, 2, 1, 0, -1, -2, -3, -4, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, 8, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9
OFFSET
0,5
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..10200 (k = 0..100)
MATHEMATICA
Flatten[Table[Range[k, -k, -1], {k, 0, 10}]] (* Paolo Xausa, Mar 01 2025 *)
CROSSREFS
KEYWORD
sign,easy,new
AUTHOR
N. J. A. Sloane, Mar 01 2025 [Suggested by Franklin T. Adams-Watters, Sep 21 2011]
STATUS
approved
a(n) is the second element of the sorted multiset of bases and exponents (including exponents = 1) in the prime factorization of n.
+0
0
2, 3, 2, 5, 1, 7, 3, 3, 1, 11, 2, 13, 1, 1, 4, 17, 2, 19, 2, 1, 1, 23, 2, 5, 1, 3, 2, 29, 1, 31, 5, 1, 1, 1, 2, 37, 1, 1, 2, 41, 1, 43, 2, 2, 1, 47, 2, 7, 2, 1, 2, 53, 2, 1, 2, 1, 1, 59, 1, 61, 1, 2, 6, 1, 1, 67, 2, 1, 1, 71, 2, 73, 1, 2, 2, 1, 1, 79, 2, 4, 1, 83, 1, 1, 1
OFFSET
2,1
FORMULA
a(p) = p, for p prime.
EXAMPLE
a(10) = 1 because the prime factorization of 10 is 2^1*5^1, the multiset of these bases and exponents is {1, 1, 2, 5} and its second element is 1.
a(18) = 2 because the prime factorization of 18 is 2^1*3^2, the multiset of these bases and exponents is {1, 2, 2, 3} and its second element is 2.
MATHEMATICA
A381576[n_] := Sort[Flatten[FactorInteger[n]]][[2]];
Array[A381576, 100, 2]
CROSSREFS
Second column of A381178.
KEYWORD
nonn,easy,new
AUTHOR
Paolo Xausa, Feb 28 2025
STATUS
approved
a(n) is the mode of the multiset of bases and exponents (including exponents = 1) in the prime factorization of n (using largest mode if multimodal).
+0
0
2, 3, 2, 5, 1, 7, 3, 3, 1, 11, 2, 13, 1, 1, 4, 17, 2, 19, 2, 1, 1, 23, 3, 5, 1, 3, 2, 29, 1, 31, 5, 1, 1, 1, 2, 37, 1, 1, 5, 41, 1, 43, 2, 5, 1, 47, 4, 7, 2, 1, 2, 53, 3, 1, 7, 1, 1, 59, 2, 61, 1, 7, 6, 1, 1, 67, 2, 1, 1, 71, 3, 73, 1, 5, 2, 1, 1, 79, 5, 4, 1, 83, 2, 1, 1
OFFSET
2,1
FORMULA
a(p) = p, for p prime.
EXAMPLE
The prime factorization of 132 is 2^2*3^1*11^1, the multiset of these bases and exponents is {1, 1, 2, 2, 3, 11} and its largest most frequent element is 2.
MATHEMATICA
A381404[n_] := Max[Commonest[Flatten[FactorInteger[n]]]];
Array[A381404, 100, 2]
CROSSREFS
Cf. A000040 (fixed points).
KEYWORD
nonn,easy,new
AUTHOR
Paolo Xausa, Feb 27 2025
STATUS
approved

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