A377037
Position of first zero in the n-th differences of the composite numbers (A002808), or 0 if it does not appear.
Original entry on oeis.org
1, 14, 2, 65, 1, 83, 2, 7, 1, 83, 2, 424, 12, 32, 11, 733, 10, 940, 9, 1110, 8, 1110, 7, 1110, 6, 1110, 112, 1110, 111, 1110, 110, 2192, 109, 13852, 108, 13852, 107, 13852, 106, 13852, 105, 17384, 104, 17384, 103, 17384, 102, 17384, 101, 27144, 552, 28012, 551
Offset: 2
The third differences of the composite numbers are:
-1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -2, 1, 0, 0, 1, -1, -1, ...
so a(3) = 14.
The version for prime instead of composite is
A376678.
For noncomposite numbers we have
A376855.
This is the first position of 0 in row n of the array
A377033.
For squarefree instead of composite we have
A377042, nonsquarefree
A377050.
For prime-power instead of composite we have
A377055.
A377036 gives first term of the n-th differences of the composite numbers, for primes
A007442 or
A030016.
Cf.
A018252,
A064113,
A065310,
A065890,
A140119,
A173390,
A233671,
A258025,
A258026,
A350004,
A376602 (zero),
A376603 (nonzero),
A376651 (positive),
A376652 (negative),
A376680,
A377034,
A377035.
-
nn=10000;
u=Table[Differences[Select[Range[nn],CompositeQ],k],{k,2,16}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
m=Table[Position[u[[k]],0][[1,1]],{k,mnrm[Union[First/@Position[u,0]]]}]
A373404
Sum of the n-th maximal antirun of composite numbers differing by more than one.
Original entry on oeis.org
18, 9, 36, 15, 54, 21, 46, 25, 26, 27, 90, 33, 34, 35, 74, 39, 126, 45, 94, 49, 50, 51, 106, 55, 56, 57, 180, 63, 64, 65, 134, 69, 216, 75, 76, 77, 158, 81, 166, 85, 86, 87, 178, 91, 92, 93, 94, 95, 194, 99, 306, 105, 324, 111, 226, 115, 116, 117, 118, 119
Offset: 1
Row sums of:
4 6 8
9
10 12 14
15
16 18 20
21
22 24
25
26
27
28 30 32
33
34
35
36 38
39
40 42 44
Partial sums are a subset of
A053767 (partial sums of composite numbers).
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
A375740
Numbers k such that A007916(k+1) - A007916(k) = 1. In other words, the k-th non-perfect-power is 1 less than the next.
Original entry on oeis.org
1, 3, 4, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77
Offset: 1
The non-perfect-powers are 2, 3, 5, 6, 7, 10, 11, 12, 13, ... which increase by one after positions 1, 3, 4, 6, ...
Non-perfect-powers:
Non-prime-powers (exclusive):
-
radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Join@@Position[Differences[Select[Range[100],radQ]],1]
-
from itertools import count, islice
from sympy import perfect_power
def A375740_gen(): # generator of terms
a, b = -1, 0
for n in count(2):
c = not perfect_power(n)
if c:
a += 1
if b&c:
yield a
b = c
A375740_list = list(islice(A375740_gen(), 52)) # Chai Wah Wu, Sep 11 2024
A376603
Points of nonzero curvature in the sequence of composite numbers (A002808).
Original entry on oeis.org
2, 4, 6, 8, 10, 12, 13, 17, 19, 23, 24, 26, 28, 30, 31, 35, 36, 40, 42, 46, 47, 49, 51, 55, 56, 58, 59, 63, 64, 70, 71, 73, 75, 77, 79, 81, 82, 94, 95, 97, 98, 102, 104, 112, 114, 118, 119, 123, 124, 126, 127, 131, 132, 136, 138, 146, 148, 150, 152, 162, 163
Offset: 1
The composite numbers (A002808) are:
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, ...
with first differences (A073783):
2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, ...
with first differences (A073445):
0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, ...
with nonzero terms at (A376603):
2, 4, 6, 8, 10, 12, 13, 17, 19, 23, 24, 26, 28, 30, 31, 35, 36, 40, 42, 46, 47, ...
Partitions into composite numbers are counted by
A023895, factorizations
A050370.
These are the positions of nonzero terms in
A073445.
For prime instead of composite we have
A333214.
For upward concavity (instead of nonzero) we have
A376651, downward
A376652.
A379312
Positive integers whose prime indices include a unique 1 or prime number.
Original entry on oeis.org
2, 3, 5, 11, 14, 17, 21, 26, 31, 35, 38, 39, 41, 46, 57, 58, 59, 65, 67, 69, 74, 77, 83, 86, 87, 94, 95, 98, 106, 109, 111, 115, 119, 122, 127, 129, 141, 142, 143, 145, 146, 147, 157, 158, 159, 178, 179, 182, 183, 185, 191, 194, 202, 206, 209, 211, 213, 214
Offset: 1
The terms together with their prime indices begin:
2: {1}
3: {2}
5: {3}
11: {5}
14: {1,4}
17: {7}
21: {2,4}
26: {1,6}
31: {11}
35: {3,4}
38: {1,8}
39: {2,6}
41: {13}
46: {1,9}
57: {2,8}
58: {1,10}
59: {17}
65: {3,6}
67: {19}
69: {2,9}
74: {1,12}
77: {4,5}
These "old" primes are listed by
A008578.
A080339 is the characteristic function for the old prime numbers.
Other counts of prime indices:
-
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Length[Select[prix[#],#==1||PrimeQ[#]&]]==1&]
A373405
Sum of the n-th maximal antirun of odd primes differing by more than two.
Original entry on oeis.org
3, 5, 18, 30, 71, 109, 202, 199, 522, 210, 617, 288, 990, 372, 390, 860, 701, 1281, 829, 1194, 1645, 4578, 852, 2682, 4419, 3300, 2927, 2438, 1891, 2602, 14660, 1632, 1650, 3378, 3480, 18141, 2052, 3121, 2112, 4310, 8922, 13131, 6253, 3851, 3889, 3929, 13788
Offset: 1
Row-sums of:
3
5
7 11
13 17
19 23 29
31 37 41
43 47 53 59
61 67 71
73 79 83 89 97 101
The partial sums are a subset of
A071148 (partial sums of odd primes).
A375713
Indices of consecutive non-prime-powers (A361102) differing by 1. Numbers k such that the k-th and (k+1)-th non-prime-powers differ by just one.
Original entry on oeis.org
5, 8, 9, 15, 16, 17, 19, 20, 23, 24, 27, 28, 30, 31, 32, 33, 36, 38, 40, 41, 44, 45, 46, 47, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 63, 64, 67, 68, 71, 72, 74, 75, 76, 77, 78, 79, 81, 82, 85, 87, 88, 89, 90, 93, 94, 95, 96, 97, 98, 99, 100, 103, 104, 105, 106
Offset: 1
The initial non-prime-powers are 1, 6, 10, 12, 14, 15, 18, 20, 21, which first increase by one after the fifth and eighth terms.
The inclusive version is a(n) - 1.
For non-perfect-powers we have
A375740.
Prime-powers inclusive:
Non-prime-powers inclusive:
A376651
Points of upward concavity in the sequence of composite numbers (A002808).
Original entry on oeis.org
4, 8, 12, 17, 23, 26, 30, 35, 40, 46, 49, 55, 58, 63, 70, 73, 77, 81, 94, 97, 102, 112, 118, 123, 126, 131, 136, 146, 150, 162, 173, 176, 180, 185, 195, 200, 205, 210, 216, 219, 229, 242, 245, 249, 262, 267, 276, 280, 285, 292, 297, 302, 305, 310, 317, 320
Offset: 1
The composite numbers are (A002808):
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, ...
with first differences (A073783):
2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, ...
with first differences (A073445):
0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, ...
with positive terms at (A376651):
4, 8, 12, 17, 23, 26, 30, 35, 40, 46, 49, 55, 58, 63, 70, 73, 77, 81, 94, 97, ...
Partitions into composite numbers are counted by
A023895, factorizations
A050370.
These are the positions of positive terms in
A073445, negative
A376652.
For zero second differences (instead of positive) we have
A376602.
A376652
Points of downward concavity in the sequence of composite numbers (A002808).
Original entry on oeis.org
2, 6, 10, 13, 19, 24, 28, 31, 36, 42, 47, 51, 56, 59, 64, 71, 75, 79, 82, 95, 98, 104, 114, 119, 124, 127, 132, 138, 148, 152, 163, 174, 178, 181, 187, 196, 201, 206, 212, 217, 221, 230, 243, 247, 250, 263, 268, 278, 281, 286, 293, 298, 303, 306, 311, 318, 321
Offset: 1
The composite numbers are (A002808):
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, ...
with first differences (A073783):
2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, ...
with second differences (A073445):
0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, ...
with negative terms at (A376651):
2, 6, 10, 13, 19, 24, 28, 31, 36, 42, 47, 51, 56, 59, 64, 71, 75, 79, 82, 95, 98, ...
Partitions into composite numbers are counted by
A023895, factorizations
A050370.
These are the positions of negative terms in
A073445, positive
A376651.
For zero second differences instead of negative we have
A376602.
-
Comps:= remove(isprime, [seq(i,i=4..1000)]):
D1:= Comps[2..-1]-Comps[1..-2]:
D2:= D1[2..-1]-D1[1..-2]:
select(t -> D2[t] < 0, [$1..nops(D2)]); # Robert Israel, Nov 06 2024
-
Join@@Position[Sign[Differences[Select[Range[1000],CompositeQ],2]],-1]
A376680
Run-lengths of first differences of composite numbers.
Original entry on oeis.org
2, 2, 2, 2, 2, 2, 1, 4, 2, 4, 1, 2, 2, 2, 1, 4, 1, 4, 2, 4, 1, 2, 2, 4, 1, 2, 1, 4, 1, 6, 1, 2, 2, 2, 2, 2, 1, 12, 1, 2, 1, 4, 2, 8, 2, 4, 1, 4, 1, 2, 1, 4, 1, 4, 2, 8, 2, 2, 2, 10, 1, 10, 1, 2, 2, 2, 1, 4, 2, 8, 1, 4, 1, 4, 1, 4, 2, 4, 1, 2, 2, 8, 1, 12, 1, 2
Offset: 1
The composite numbers (A002808) are:
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, ...
with first differences (A073783):
2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, ...
with runs:
(2,2), (1,1), (2,2), (1,1), (2,2), (1,1), (2), (1,1,1,1), (2,2), (1,1,1,1), ...
with lengths (A376680):
2, 2, 2, 2, 2, 2, 1, 4, 2, 4, 1, 2, 2, 2, 1, 4, 1, 4, 2, 4, 1, 2, 2, 4, 1, 2, ...
For prime instead of composite we have
A333254, first appearances
A335406.
These are the first differences of
A376603.
A064113 lists positions of adjacent equal prime gaps.
Comments