keyword:new

A381597

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

(list; graph; refs; listen; history; text; internal format)

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

Scott R. Shannon, Table of n, a(n) for n = 1..10000

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

A381598

Index of first term of three consecutive n's in A381597.

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1, 9, 34, 147, 111, 359, 437, 389, 594, 826, 1102, 83317, 1789, 5142, 2931, 12671

(list; graph; refs; listen; history; text; internal format)

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.

LINKS

Table of n, a(n) for n=1..16.

CROSSREFS

Cf. A381597, A381599, A370708, A281511, A229037.

KEYWORD

nonn,more,new

AUTHOR

Scott R. Shannon, Mar 01 2025

STATUS

approved

A381599

Index where n first appears in A381597.

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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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..57.

CROSSREFS

Cf. A381597, A381598, A370708, A281511, A229037.

KEYWORD

nonn,new

AUTHOR

Scott R. Shannon, Mar 01 2025

STATUS

approved

A381029

G.f. A(x) satisfies A(x) = 1/(1 - x * A(x*A(x)^2)^2).

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1, 1, 3, 16, 113, 955, 9178, 97427, 1121705, 13836694, 181295019, 2507119320, 36416096984, 553461581406, 8774534872463, 144744539399484, 2479088917439527, 44004108702467428, 808171916050540308, 15335535608825061803, 300272362335527090277, 6059534345675248667550

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..21.

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

Cf. A088714, A381615.

Cf. A120971, A143508, A381600.

Cf. A381572.

KEYWORD

nonn,new

AUTHOR

Seiichi Manyama, Mar 01 2025

STATUS

approved

A381615

G.f. A(x) satisfies A(x) = 1/(1 - x * A(x*A(x)^3)^3).

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1, 1, 4, 31, 320, 3969, 56080, 876204, 14860614, 270231265, 5223002719, 106613106181, 2287120272173, 51367948203527, 1204141944566399, 29385603693050274, 744943334951904519, 19580887642660810193, 532781828387893449124, 14984377196395037979472

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..19.

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

Cf. A088714, A381029.

Cf. A120973, A212029, A381601.

Cf. A381574.

KEYWORD

nonn,new

AUTHOR

Seiichi Manyama, Mar 01 2025

STATUS

approved

A379750

First prime of cousin prime pairs which differ, in their binary representation, by a single bit.

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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

(list; graph; refs; listen; history; text; internal format)

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.

LINKS

Table of n, a(n) for n=1..44.

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

A381233

Concatenate the sequences S(k) = [0, 1, -1, ..., k, -k] for k = 0, 1, ...

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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

(list; graph; refs; listen; history; text; internal format)

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

Cf. A196199, A381232.

KEYWORD

sign,easy,new

AUTHOR

N. J. A. Sloane, Mar 01 2025 [Suggested by Franklin T. Adams-Watters, Sep 21 2011]

STATUS

approved

A381232

Count down from k to -k for k = 0, 1, 2, ... .

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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

(list; graph; refs; listen; history; text; internal format)

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

Cf. A196199, A381233.

KEYWORD

sign,easy,new

AUTHOR

N. J. A. Sloane, Mar 01 2025 [Suggested by Franklin T. Adams-Watters, Sep 21 2011]

STATUS

approved

A381576

a(n) is the second element of the sorted multiset of bases and exponents (including exponents = 1) in the prime factorization of n.

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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

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,1

LINKS

Paolo Xausa, Table of n, a(n) for n = 2..10000

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

A381404

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).

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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

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,1

LINKS

Paolo Xausa, Table of n, a(n) for n = 2..10000

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. A035306, A381178, A381403.

Cf. A000040 (fixed points).

KEYWORD

nonn,easy,new

AUTHOR

Paolo Xausa, Feb 27 2025

STATUS

approved