A334100 -id:A334100 - cp's OEIS frontend

A147545 Primes of the form p*2^k+1 with k>0 and p=1 or p in this sequence.

3, 5, 7, 11, 13, 17, 23, 29, 41, 47, 53, 59, 83, 89, 97, 107, 113, 137, 167, 179, 193, 227, 233, 257, 353, 359, 389, 449, 467, 641, 719, 769, 773, 857, 929, 1097, 1283, 1409, 1433, 1439, 1553, 1697, 1889, 2657, 2819, 2879, 3089, 3329, 3593, 3617, 3779, 5639

Offset: 1

T. D. Noe, Nov 07 2008

nonn

This sequence starts like A074781 but grows much faster. Observe that there can be large differences between consecutive terms. Can it be shown that there is always such a prime between consecutive powers of 2? Or that this sequence is infinite? By theorem 1 of the Noe paper, this sequence is a subsequence of A135832, primes in Section I of the phi iteration.
From Antti Karttunen, Apr 19 2020: (Start)
Sequence can be considered as a generalization of Fermat primes, A019434, which is a subsequence of this sequence.
All terms with binary weight k (A000120, at least 2 for these terms) can be found as a subset of primes found on the row k-1 of array A334100. E.g. primes with weight 2 are Fermat primes (A019434), those with weight 3 are A334092 (which doesn't contain any other primes), those with weight 4 are in A334093 (among also other kind of primes), those with weights 5, 6, 7 are included as (proper) subsets in A334094, A334095 and A334096 respectively. (End)

T. D. Noe, Table of n, a(n) for n=1..2000
T. D. Noe, Primes in classes of the iterated totient function, J. Integer Sequences, 11 (2008), Article 08.1.2.

Cf. A000265, A329697, A334100.
Subsequence of A074781, and of A135832.
Subsequences: A019434, A334092 (including A039687, A050526, A300407).

nn=2^13; t={1}; i=1; While[q=t[[i]]; k=1; While[p=1+q*2^k; p

A000265(n) = (n>>valuation(n,2));
isA147454(n) = ((n>2)&&isprime(n)&&((1==(n=A000265(n-1)))||isA147454(n))); \\ Antti Karttunen, Apr 19 2020

A329697(a(n)) = A000120(a(n)) - 1. - Antti Karttunen, Apr 19 2020

A335910 Square array where row n lists all numbers k for which A335885(k) = n, read by falling antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 8, 6, 11, 27, 16, 7, 13, 33, 81, 32, 10, 15, 37, 99, 243, 64, 12, 18, 39, 107, 297, 729, 128, 14, 19, 43, 109, 321, 891, 2187, 256, 17, 21, 45, 111, 327, 963, 2673, 6561, 512, 20, 22, 53, 117, 333, 981, 2889, 8019, 19683, 1024, 24, 23, 54, 121, 351, 999, 2943, 8667, 24057, 59049, 2048, 28, 25, 55, 129, 363, 1053, 2997, 8829, 26001, 72171, 177147

Offset: 0

Antti Karttunen, Jul 01 2020

Array is read by descending antidiagonals with (n,k) = (0,0), (0,1), (1,0), (0,2), (1,1), (2,0), ... where A(n,k) is the (k+1)-th solution x to A335885(x) = n. The row indexing (n) starts from 0, and column indexing (k) also from 0.
For any odd prime p that appears on row n, either p-1 or p+1 appears on row n-1.
The e-th powers of the terms on row n form a subset of terms on row (e*n). More generally, a product of terms that occur on rows i_1, i_2, ..., i_k can be found at row (i_1 + i_2 + ... + i_k), because A335885 is completely additive.

			The top left corner of the array:
n\k |     0      1      2      3      4      5      6      7      8      9
----+--------------------------------------------------------------------------
  0 |     1,     2,     4,     8,    16,    32,    64,   128,   256,   512, ...
  1 |     3,     5,     6,     7,    10,    12,    14,    17,    20,    24, ...
  2 |     9,    11,    13,    15,    18,    19,    21,    22,    23,    25, ...
  3 |    27,    33,    37,    39,    43,    45,    53,    54,    55,    57, ...
  4 |    81,    99,   107,   109,   111,   117,   121,   129,   131,   135, ...
  5 |   243,   297,   321,   327,   333,   351,   363,   387,   393,   405, ...
  6 |   729,   891,   963,   981,   999,  1053,  1089,  1161,  1177,  1179, ...
  7 |  2187,  2673,  2889,  2943,  2997,  3159,  3267,  3483,  3531,  3537, ...
  8 |  6561,  8019,  8667,  8829,  8991,  9477,  9801, 10449, 10593, 10611, ...
  9 | 19683, 24057, 26001, 26487, 26973, 28431, 29403, 31347, 31779, 31833, ...

Index entries for sequences that are permutations of the natural numbers

Cf. A335885.
Cf. A000079, A335911, A335912 (rows 0-2), A000244 (is very like the leftmost column).
Cf. also arrays A334100, A335430.

up_to = 78-1; \\ = binomial(12+1,2)-1.
memoA335885 = Map();
A335885(n) = if(1==n,0,my(v=0); if(mapisdefined(memoA335885,n,&v), v, my(f=factor(n)); v = sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+min(A335885(f[k,1]-1),A335885(f[k,1]+1))))); mapput(memoA335885,n,v); (v)));
memoA335910sq = Map();
A335910sq(n, k) = { my(v=0); if((0==k), v = -1, if(!mapisdefined(memoA335910sq,[n,k-1],&v), v = A335910sq(n, k-1))); for(i=1+v,oo,if(A335885(1+i)==n,mapput(memoA335910sq,[n,k],i); return(1+i))); };
A335910list(up_to) = { my(v = vector(1+up_to), i=0); for(a=0,oo, for(col=0,a, i++; if(i > #v, return(v)); v[i] = A335910sq(col,(a-(col))))); (v); };
v335910 = A335910list(up_to);
A335910(n) = v335910[1+n];
for(n=0,up_to,print1(A335910(n),", "));

A334099 The least k for which A329697(k) = n; Position of first occurrence of n (and also records) in A329697.

Original entry on oeis.org

1, 3, 7, 19, 43, 127, 283, 659, 1319, 3957, 9227, 21599, 50123, 129263, 258527, 775581, 1551163, 4340087, 9750239, 27353747, 65148847, 156067127, 340997113, 955523423

Offset: 0

Antti Karttunen, Apr 14 2020

Note that although most of the terms after 1 are primes, we also have a few composites: a(9) = a(1)*a(8) = 3*1319 = 3957, a(15) = a(1)*a(14) = 3*258527 = 775581, a(22) = a(8)*a(14) = 340997113.
a(n) <= 3^n and in particular, a(n+1) <= 3*a(n), n > 0 and more generally a(n + m) <= a(n) * a(m) where m, n >= 0. - David A. Corneth, Apr 15 2020
The above follows because A329697 is totally additive.

The leftmost column of A334100.
Cf. A329697 (a left inverse).
Cf. A067513.
Cf. A007755, A105017, and also A329662 (analogous sequence when using the map k -> k + k/p).

With[{s = Array[Length@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, #, # != 2^IntegerExponent[#, 2] &] - 1 &, 10^6]}, {1}~Join~Array[FirstPosition[s, #][[1]] &, Max@ s]] (* Michael De Vlieger, Apr 30 2020 *)

A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
m=-1; k=0; for(n=1,2^32, t=A329697(n); if(t>m, m=t; write("b334099.txt", k, " ", n); k++));

For all n >= 0, A329697(a(n)) = n.

A359584 Positions of odd terms in A329697.

Original entry on oeis.org

3, 5, 6, 10, 12, 17, 19, 20, 21, 23, 24, 27, 29, 31, 33, 34, 35, 37, 38, 39, 40, 42, 45, 46, 48, 53, 54, 55, 58, 61, 62, 65, 66, 68, 70, 73, 74, 75, 76, 78, 80, 83, 84, 89, 90, 92, 96, 101, 103, 106, 108, 110, 113, 116, 119, 122, 123, 124, 125, 127, 129, 130, 132, 133, 136, 139, 140, 141, 146, 147

Offset: 1

Antti Karttunen, Jan 07 2023

nonn

Numbers that occur on odd-indexed rows of array A334100.

Positions of -1's in A359581, positions of 1's in A359583 (characteristic function).
Cf. A329697, A334100, A359585 (complement).
Cf. A334101, A334103, A334105 (subsequences).

PARI
```
isA359584(n) = A359583(n);
```

A359585 Positions of even terms in A329697.

Original entry on oeis.org

1, 2, 4, 7, 8, 9, 11, 13, 14, 15, 16, 18, 22, 25, 26, 28, 30, 32, 36, 41, 43, 44, 47, 49, 50, 51, 52, 56, 57, 59, 60, 63, 64, 67, 69, 71, 72, 77, 79, 81, 82, 85, 86, 87, 88, 91, 93, 94, 95, 97, 98, 99, 100, 102, 104, 105, 107, 109, 111, 112, 114, 115, 117, 118, 120, 121, 126, 128, 131, 134, 135, 137

Offset: 1

Antti Karttunen, Jan 07 2023

nonn

Numbers that occur on even-indexed rows of array A334100.

Positions of +1's in A359581, positions of 0's in A359583.
Cf. A329697, A334100, A359584 (complement).
Cf. A000079, A334102, A334104, A334106 (subsequences).

PARI
```
isA359585(n) = !A359583(n);
```

A364042 a(n) = 2^A329697(n) - A000120(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 0, 0, 2, 5, 0, 5, 1, 4, 0, 1, 1, 4, 1, 4, 0, 3, 0, 6, 0, 5, 2, 5, 5, 4, 0, 1, 5, 12, 1, 4, 4, 11, 0, 13, 1, 0, 1, 4, 4, 3, 1, 12, 4, 11, 0, 3, 3, 10, 0, 6, 6, 13, 0, 13, 5, 12, 2, 5, 5, 4, 5, 12, 4, 11, 0, 13, 1, 4, 5, 0, 12, 11, 1, 4, 4, 11, 4, 11, 11, 10, 0, 1

Offset: 1

Antti Karttunen, Jul 03 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000120, A003401 (positions of 0's), A329697, A334100, A336477.

A364042(n) = ((2^A329697(n))-hammingweight(n));

cp's OEIS Frontend

A147545 Primes of the form p*2^k+1 with k>0 and p=1 or p in this sequence.

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A335910 Square array where row n lists all numbers k for which A335885(k) = n, read by falling antidiagonals.

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A334099 The least k for which A329697(k) = n; Position of first occurrence of n (and also records) in A329697.

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A359584 Positions of odd terms in A329697.

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A359585 Positions of even terms in A329697.

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A364042 a(n) = 2^A329697(n) - A000120(n).

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