A073783 -id:A073783 - cp's OEIS frontend

A121053 A sequence S describing the position of its prime terms.

2, 3, 5, 1, 7, 8, 11, 13, 10, 17, 19, 14, 23, 29, 16, 31, 37, 20, 41, 43, 22, 47, 53, 25, 59, 27, 61, 30, 67, 71, 73, 33, 79, 35, 83, 38, 89, 97, 40, 101, 103, 44, 107, 109, 46, 113, 127, 49, 131, 51, 137, 54, 139, 149, 56, 151, 58, 157, 163, 62, 167, 173, 64, 179, 66, 181, 191, 69, 193, 72, 197, 199, 211, 75, 223, 77, 227, 80, 229

Offset: 1

Eric Angelini, Aug 10 2006

S reads like this:
"At position 2, there is a prime in S" [indeed, this is 3]
"At position 3, there is a prime in S" [indeed, this is 5]
"At position 5, there is a prime in S" [indeed, this is 7]
"At position 1, there is a prime in S" [indeed, this is 2]
"At position 7, there is a prime in S" [indeed, this is 11]
"At position 8, there is a prime in S" [indeed, this is 13]
"At position 11, there is a prime in S" [indeed, this is 19]
"At position 13, there is a prime in S" [indeed, this is 23]
"At position 10, there is a prime in S" [indeed, this is 17], etc.
S is built with this rule: when you are about to write a term of S, always use the smallest integer not yet present in S and not leading to a contradiction.
Thus one cannot start with 1; this would read: "At position 1, there is a prime number in S" [no, 1 is not a prime]
So start S with 2 and the rest follows smoothly.
S contains all the primes and they appear in their natural order.
Does the ratio primes/composites in S tend to a limit?
The definition and the comments above are Eric Angelini's original submission. A more formal definition would be "Lexicographically earliest sequence of distinct positive numbers such that k is a term of the sequence iff a(k) is a prime". However, to honor Eric Angelini's memory, we will retain his enigmatic definition. - N. J. A. Sloane, Dec 20 2024
Comments from N. J. A. Sloane, Nov 14 2024 (Start)
Theorem. Let p(k) = k-th prime, c(k) = k-th composite number. For n >= 5, if n is a prime or n = c(2*t+1) for some t, then a(n) = p(k) where k = floor((n+PrimePi(n))/2); otherwise, n = c(2*t) for some t and a(n) = c(2*t+1).
The proof will be added later (see reference).
The theorem implies that the sequence consists of the primes and the odd-subscripted composite numbers.
All of Dean Hickerson's comments below follow from this theorem. (End)
Comments from Dean Hickerson, Aug 11 2006: (Start)
In the limit, exactly half of the terms are primes. Here's a formula, found empirically, for a(n) for n >= 5:
Let pi(n) be the number of primes <= n and p(n) be the n-th prime. Then for n >= 5:
- if n is prime or (n is composite and n+pi(n) is even) then a(n) = p(floor((n+pi(n))/2));
- if n is composite and n+pi(n) is odd and n+1 is composite then a(n) = n+1;
- if n is composite and n+pi(n) is odd and n+1 is prime then a(n) = n+2.
Also, for n >= 5, a(n) is in the sequence iff either n is prime or n+pi(n) is even.
(This could all be proved by induction on n.)
It follows from this that, for n >= 4, the number of primes among a(1), ..., a(n) is exactly floor((n+pi(n))/2). Since pi(n)/n -> 0 as n -> infinity, this is asymptotic to n/2. (End)

N. J. A. Sloane, The Remarkable Sequences of Éric Angelini, MS in preparation, December 2024.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 [First 1000 terms from Kerry Mitchell]
Eric Angelini, A sequence describing the position of its prime terms, Blog Post, August 2006.
E. Angelini, A sequence describing the position of its prime terms [Cached copy, with permission]
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 9.
N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]

Cf. A000720, A002808, A065855, A073783, A099861, A105753, A377899.

See A377901 for the analogous sequence if 1 is regarded as a prime.

chi := proc(n) if n <= 3 then 0 else n - numtheory:-pi(n) - 1; fi; end; # A065855, number of composites <= n
A002808 := proc(n) option remember ; local a ; if n = 1 then 4; else for a from procname(n-1)+1 do if not isprime(a) then return a; end if; end do ; end if; end proc;
A121053 := proc(n) local init,t1;
init := [2,3,5,1,7];
if n<=5 then return(init[n]); fi;
if isprime(n) or (not isprime(n) and ((chi(n) mod 2) = 1))
   then ithprime(floor((n+numtheory:-pi(n))/2));
else t1 := chi(n); A002808(t1+1);
fi; end;
[seq(A121053(n),n=1..120)]; # N. J. A. Sloane, Nov 14 2024

a[1]=2; a[2]=3; a[3]=5; a[4]=1; a[n_ /; PrimeQ[n] || !PrimeQ[n] && EvenQ[n+PrimePi[n]]] := Prime[Floor[(n+PrimePi[n])/2]]; a[n_ /; !PrimeQ[n] && OddQ[n+PrimePi[n]]] := If[!PrimeQ[n+1], n+1, n+2]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 21 2011, based on Dean Hickerson's formulas *)

from sympy import isprime, prime, primepi, composite, compositepi
def a(n): return [2, 3, 5, 1, 7][n-1] if n < 6 else prime(n+primepi(n)>>1) if isprime(n) or (c:=compositepi(n))&1 else composite(c+1)
print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Nov 29 2024

# faster for initial segment of sequence
from sympy import isprime, sieve
from itertools import count, islice
def nextcomposite(n): return next(k for k in count(n+1) if k not in sieve)
def agen(): # generator of terms
    alst, chin, pin, nextc = [2, 3, 5, 1, 7], 1, 3, 6
    yield from alst
    for n in count(6):
        if isprimen:=n < nextc: pin += 1
        else: chin, nextc = chin + 1, nextcomposite(nextc)
        yield sieve[(n+pin)>>1] if isprimen or chin&1 else nextc
print(list(islice(agen(), 80))) # Michael S. Branicky, Nov 29 2024

A379315 Number of strict integer partitions of n with a unique 1 or prime part.

Original entry on oeis.org

0, 1, 1, 1, 0, 2, 1, 3, 1, 3, 2, 7, 3, 7, 4, 10, 7, 15, 7, 17, 13, 23, 16, 31, 20, 37, 31, 48, 38, 62, 48, 76, 68, 93, 80, 119, 105, 147, 137, 175, 166, 226, 208, 267, 263, 326, 322, 407, 391, 481, 492, 586, 591, 714, 714, 849, 884, 1020, 1050, 1232, 1263

Offset: 0

Gus Wiseman, Dec 28 2024

nonn

The "old" primes are listed by A008578.

			The a(10) = 2 through a(15) = 10 partitions:
  (8,2)  (11)     (9,3)    (13)     (9,5)    (8,7)
  (9,1)  (6,5)    (10,2)   (7,6)    (12,2)   (10,5)
         (7,4)    (6,4,2)  (8,5)    (8,4,2)  (11,4)
         (8,3)             (10,3)   (9,4,1)  (12,3)
         (9,2)             (12,1)            (14,1)
         (10,1)            (6,4,3)           (6,5,4)
         (6,4,1)           (8,4,1)           (8,4,3)
                                             (8,6,1)
                                             (9,4,2)
                                             (10,4,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

For all prime parts we have A000586, non-strict A000607 (ranks A076610).
For no prime parts we have A096258, non-strict A002095 (ranks A320628).
For a unique composite part we have A379303, non-strict A379302 (ranks A379301).
Considering 1 nonprime gives A379305, non-strict A379304 (ranks A331915).
For squarefree instead of old prime we have A379309, non-strict A379308 (ranks A379316).
Ranked by A379312 /\ A005117 = squarefree positions of 1 in A379311.
The non-strict version is A379314.
A000040 lists the prime numbers, differences A001223.
A000041 counts integer partitions, strict A000009.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A376682 gives k-th differences of old primes.
Cf. A000070, A023895, A034891, A036497, A038348, A095195, A175804, A204389, A257994, A302540.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,_?(#==1||PrimeQ[#]&)]==1&]],{n,0,30}]

seq(n)={Vec(sum(k=1, n, if(isprime(k) || k==1, x^k)) * prod(k=4, n, 1 + if(!isprime(k), x^k), 1 + O(x^n)), -n-1)} \\ Andrew Howroyd, Dec 28 2024

A373400 Numbers k such that the k-th maximal run of composite numbers has length different from all prior maximal runs. Sorted positions of first appearances in A176246 (or A046933 shifted).

Original entry on oeis.org

1, 3, 8, 23, 29, 33, 45, 98, 153, 188, 216, 262, 281, 366, 428, 589, 737, 1182, 1830, 1878, 2190, 2224, 3076, 3301, 3384, 3426, 3643, 3792, 4521, 4611, 7969, 8027, 8687, 12541, 14356, 14861, 15782, 17005, 19025, 23282, 30801, 31544, 33607, 34201, 34214, 38589

Offset: 1

Gus Wiseman, Jun 10 2024

nonn

The unsorted version is A073051.

A run of a sequence (in this case A002808) is an interval of positions at which consecutive terms differ by one.

			The maximal runs of composite numbers begin:
   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  45  46
  48  49  50  51  52
  54  55  56  57  58
  60
  62  63  64  65  66
  68  69  70
  72
  74  75  76  77  78
  80  81  82
  84  85  86  87  88
  90  91  92  93  94  95  96
  98  99 100
The a(n)-th rows are:
   4
   8   9  10
  24  25  26  27  28
  90  91  92  93  94  95  96
 114 115 116 117 118 119 120 121 122 123 124 125 126
 140 141 142 143 144 145 146 147 148
 200 201 202 203 204 205 206 207 208 209 210

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers

The unsorted version is A073051, firsts of A176246.
For squarefree runs we have the triple (1,3,5), firsts of A120992.
For prime runs we have the triple (1,2,3), firsts of A175632.
For squarefree antiruns we have A373128, firsts of A373127.
For nonsquarefree runs we have A373199 (assuming sorted), firsts of A053797.
For prime antiruns we have A373402, unsorted A373401, firsts of A027833.
For composite runs we have the triple (1,2,7), firsts of A373403.
A000040 lists the primes, differences A001223.
A002808 lists the composite numbers, differences A073783.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
Cf. A006512, A007674, A049093, A068781, A072284, A077641, A174965, A251092, A373198, A373408, A373411.

t=Length/@Split[Select[Range[10000],CompositeQ],#1+1==#2&]//Most;
Select[Range[Length[t]],FreeQ[Take[t,#-1],t[[#]]]&]

A376602 Inflection and undulation points in the sequence of composite numbers (A002808).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 14, 15, 16, 18, 20, 21, 22, 25, 27, 29, 32, 33, 34, 37, 38, 39, 41, 43, 44, 45, 48, 50, 52, 53, 54, 57, 60, 61, 62, 65, 66, 67, 68, 69, 72, 74, 76, 78, 80, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 96, 99, 100, 101, 103, 105, 106, 107, 108

Offset: 1

Gus Wiseman, Oct 05 2024

nonn

These are points at which the second differences (A073445) are zero.

			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 zeros at (A376602):
  1, 3, 5, 7, 9, 11, 14, 15, 16, 18, 20, 21, 22, 25, 27, 29, 32, 33, 34, 37, 38, ...

Wikipedia, Inflection point
Gus Wiseman, Inflection and undulation points in the sequence of composite numbers.

Partitions into composite numbers are counted by A023895, factorizations A050370.
For prime instead of composite we have A064113.
These are the positions of zeros in A073445.
For first differences we had A073783, ones A375929, complement A065890.
For concavity in primes we have A258025/A258026, weak A333230/A333231.
For upward concavity (instead of zero) we have A376651, downward A376652.
The complement is A376603.
For composite numbers: A002808 (terms), A073783 (first differences), A073445 (second differences), A376603 (nonzero curvature), A376651 (concave-up), A376652 (concave-down).
For inflection and undulation points: A064113 (prime), A376588 (non-perfect-power), A376591 (squarefree), A376594 (nonsquarefree), A376597 (prime-power), A376600 (non-prime-power).
Cf. A000040, A036263, A251092, A333214, A333216, A333254.

Join@@Position[Differences[Select[Range[100],CompositeQ],2],0]

A379301 Positive integers whose prime indices include a unique composite number.

Original entry on oeis.org

7, 13, 14, 19, 21, 23, 26, 28, 29, 35, 37, 38, 39, 42, 43, 46, 47, 52, 53, 56, 57, 58, 61, 63, 65, 69, 70, 71, 73, 74, 76, 77, 78, 79, 84, 86, 87, 89, 92, 94, 95, 97, 101, 103, 104, 105, 106, 107, 111, 112, 113, 114, 115, 116, 117, 119, 122, 126, 129, 130, 131

Offset: 1

Gus Wiseman, Dec 25 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 70 are {1,3,4}, so 70 is in the sequence.
The prime indices of 98 are {1,4,4}, so 98 is not in the sequence.

For no composite parts we have A302540, counted by A034891 (strict A036497).
For all composite parts we have A320629, counted by A023895 (strict A204389).
For a unique prime part we have A331915, counted by A379304 (strict A379305).
Positions of one in A379300.
Partitions of this type are counted by A379302 (strict A379303).
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A066247 is the characteristic function for the composite numbers.
A377033 gives k-th differences of composite numbers, see A073445, A377034-A377037.
Other counts of prime indices:
- A087436 postpositive, see A038550.
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
- A379311 old prime, see A379312-A379315.
Cf. A113646, A376680.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Length[Select[prix[#],CompositeQ]]==1&]

A379304 Number of integer partitions of n with a unique prime part.

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 4, 6, 7, 9, 11, 17, 20, 26, 31, 41, 47, 62, 72, 93, 108, 136, 156, 199, 226, 279, 321, 398, 452, 555, 630, 767, 873, 1051, 1188, 1433, 1618, 1930, 2185, 2595, 2921, 3458, 3891, 4580, 5155, 6036, 6776, 7926, 8883, 10324, 11577, 13421, 15014

Offset: 0

Gus Wiseman, Dec 27 2024

nonn

			The a(2) = 1 through a(9) = 9 partitions:
  (2)  (3)   (31)   (5)     (42)     (7)       (62)       (54)
       (21)  (211)  (311)   (51)     (43)      (71)       (63)
                    (2111)  (3111)   (421)     (431)      (621)
                            (21111)  (511)     (4211)     (711)
                                     (31111)   (5111)     (4311)
                                     (211111)  (311111)   (42111)
                                               (2111111)  (51111)
                                                          (3111111)
                                                          (21111111)

For all prime parts we have A000607 (strict A000586), ranks A076610.
For no prime parts we have A002095 (strict A096258), ranks A320628.
Ranked by A331915 = positions of one in A257994.
For a unique composite part we have A379302 (strict A379303), ranks A379301.
The strict case is A379305.
For squarefree instead of prime we have A379308 (strict A379309), ranks A379316.
Considering 1 prime gives A379314 (strict A379315), ranks A379312.
A000040 lists the prime numbers, differences A001223.
A000041 counts integer partitions, strict A000009.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A095195 gives k-th differences of prime numbers.
Cf. A000070, A023895, A034891, A036497, A204389, A302540, A330944.

Table[Length[Select[IntegerPartitions[n],Count[#,_?PrimeQ]==1&]],{n,0,30}]

A379305 Number of strict integer partitions of n with a unique prime part.

Original entry on oeis.org

0, 0, 1, 2, 1, 1, 2, 3, 3, 3, 3, 6, 8, 8, 8, 10, 12, 17, 18, 18, 22, 28, 30, 36, 40, 44, 52, 62, 67, 78, 87, 97, 113, 129, 137, 156, 177, 200, 227, 251, 271, 312, 350, 382, 425, 475, 521, 588, 648, 705, 785, 876, 957, 1061, 1164, 1272, 1411, 1558, 1693, 1866

Offset: 0

Gus Wiseman, Dec 27 2024

nonn

			The a(2) = 1 through a(12) = 8 partitions (A=10, B=11):
  (2)  (3)   (31)  (5)  (42)  (7)    (62)   (54)   (82)   (B)    (93)
       (21)             (51)  (43)   (71)   (63)   (541)  (65)   (A2)
                              (421)  (431)  (621)  (631)  (74)   (B1)
                                                          (83)   (642)
                                                          (92)   (651)
                                                          (821)  (741)
                                                                 (831)
                                                                 (921)

For all prime parts we have A000586, non-strict A000607 (ranks A076610).
For no prime parts we have A096258, non-strict A002095 (ranks A320628).
Ranked by A331915 /\ A005117 = squarefree positions of one in A257994.
For a composite instead of prime we have A379303, non-strict A379302 (ranks A379301).
The non-strict version is A379304.
For squarefree instead of prime we have A379309, non-strict A379308 (ranks A379316).
Considering 1 prime gives A379315, non-strict A379314 (ranks A379312).
A000040 lists the prime numbers, differences A001223.
A000041 counts integer partitions, strict A000009.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A095195 gives k-th differences of prime numbers.
Cf. A000070, A023895, A034891, A036497, A038348, A204389, A302540, A320629, A330944.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,_?PrimeQ]==1&]],{n,0,30}]

A373401 Least k such that the k-th maximal antirun of prime numbers > 3 has length n. Position of first appearance of n in A027833. The sequence ends if no such antirun exists.

Original entry on oeis.org

1, 2, 4, 6, 10, 8, 69, 40, 24, 46, 41, 21, 140, 82, 131, 210, 50, 199, 35, 30, 248, 192, 277, 185, 458, 1053, 251, 325, 271, 645, 748, 815, 811, 1629, 987, 826, 1967, 423, 1456, 2946, 1109, 406, 1870, 1590, 3681, 2920, 3564, 6423, 1426, 5953, 8345, 12687, 6846

Offset: 1

Gus Wiseman, Jun 09 2024

nonn

The sorted version is A373402.

For this sequence, we define an antirun to be an interval of positions at which consecutive primes differ by at least 3.

			The maximal antiruns of prime numbers > 3 begin:
    5
    7  11
   13  17
   19  23  29
   31  37  41
   43  47  53  59
   61  67  71
   73  79  83  89  97 101
  103 107
  109 113 127 131 137
  139 149
  151 157 163 167 173 179
The a(n)-th rows are:
     5
     7   11
    19   23   29
    43   47   53   59
   109  113  127  131  137
    73   79   83   89   97  101
  2269 2273 2281 2287 2293 2297 2309
  1093 1097 1103 1109 1117 1123 1129 1151
   463  467  479  487  491  499  503  509  521
For example, (19, 23, 29) is the first maximal antirun of length 3, so a(3) = 4.

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

For composite instead of prime we have A073051.
For runs instead of antiruns we have the triple (4,2,1), firsts of A251092.
For squarefree instead of prime we have A373128, firsts of A373127.
The sorted version is A373402.
A000040 lists the primes, differences A001223.
A002808 lists the composite numbers, differences A073783.
A046933 counts composite numbers between primes.
Cf. A001359, A005117, A027833, A006512, A053797, A373199, A373405.

t=Length/@Split[Select[Range[4,100000],PrimeQ],#1+2!=#2&]//Most;
spna[y_]:=Max@@Select[Range[Length[y]],SubsetQ[t,Range[#]]&];
Table[Position[t,k][[1,1]],{k,spna[t]}]

A375714 Positions of non-successions of consecutive non-perfect-powers. Numbers k such that the k-th non-perfect-power is at least two fewer than the next.

Original entry on oeis.org

2, 5, 11, 19, 20, 24, 27, 39, 53, 69, 87, 107, 110, 112, 127, 151, 177, 196, 204, 221, 233, 265, 299, 317, 334, 372, 412, 454, 481, 497, 543, 591, 641, 693, 747, 803, 861, 921, 959, 982, 1046, 1112, 1180, 1250, 1284, 1321, 1395, 1471, 1549, 1629, 1675, 1710

Offset: 1

Gus Wiseman, Sep 10 2024

nonn

Non-perfect-powers (A007916) are numbers with no proper integer roots.

			The initial non-perfect-powers are 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, which increase by more than one after term 2, term 5, term 11, etc.

First differences are A375702.
Positions of terms > 1 in A375706 (differences of A007916).
The complement for non-prime-powers is A375713, differences A373672.
The complement is A375740.
The version for non-prime-powers is A375928, differences A110969.
Prime-powers inclusive:
- terms: A000961
- differences: A057820
- runs: A373675, A373673, A373674, A174965
- antiruns: A373576, A120430, A006549, A373671
Non-prime-powers inclusive:
- terms: A361102
- differences: A375708
- runs: A373678, A373676, A373677, A110969
- antiruns: A373679, A373575, A255346, A373672
Cf. A000040, A001223, A046933, A053289, A073783, A246655, A251092, A375734.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
ce=Select[Range[100],radQ];
Select[Range[Length[ce]-1],!ce[[#+1]]==ce[[#]]+1&]

from itertools import count, islice
from sympy import perfect_power
def A375714_gen(): # generator of terms
    a, b = -1, 0
    for n in count(1):
        c = not perfect_power(n)
        if c:
            a += 1
        if b&(c^1):
            yield a
        b = c
A375714_list = list(islice(A375714_gen(),52)) # Chai Wah Wu, Sep 11 2024

A007916(a(n)+1) - A007916(a(n)) > 1.

A377034 Antidiagonal-sums of the array A377033(n,k) = n-th term of the k-th differences of the composite numbers (A002808).

Original entry on oeis.org

4, 8, 10, 8, 14, 14, 11, 24, 10, 20, 37, -10, 56, 26, -52, 260, -659, 2393, -8128, 25703, -72318, 184486, -430901, 933125, -1888651, 3597261, -6479654, 11086964, -18096083, 28307672, -42644743, 62031050, -86466235, 110902085, -110907437, 52379, 483682985

Offset: 1

Gus Wiseman, Oct 17 2024

sign

Row-sums of the triangle version of A377033.

			The fourth antidiagonal of A377033 is (9, 1, -1, -1), so a(4) = 8.

The version for prime instead of composite is A140119, noncomposite A376683.
This is the antidiagonal-sums of the array A377033, absolute version A377035.
For squarefree instead of composite we have A377039, absolute version A377040.
For nonsquarefree instead of composite we have A377047, absolute version A377048.
For prime-power instead of composite we have A377052, absolute version A377053.
Other arrays of differences: A095195 (prime), A376682 (noncomposite), A377033 (composite), A377038 (squarefree), A377046 (nonsquarefree), A377051 (prime-power).
A000040 lists the primes, differences A001223, second A036263.
A002808 lists the composite numbers, differences A073783, second A073445.
A008578 lists the noncomposites, differences A075526.
Cf. A018252, A065310, A065890, A333254, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680, A377036.

q=Select[Range[100],CompositeQ];
t=Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[i+k]],{k,0,j}],{j,0,Length[q]/2},{i,Length[q]/2}];
Total/@Table[t[[j,i-j+1]],{i,Length[q]/2},{j,i}]

cp's OEIS Frontend

A121053 A sequence S describing the position of its prime terms.

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A379315 Number of strict integer partitions of n with a unique 1 or prime part.

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A373400 Numbers k such that the k-th maximal run of composite numbers has length different from all prior maximal runs. Sorted positions of first appearances in A176246 (or A046933 shifted).

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A376602 Inflection and undulation points in the sequence of composite numbers (A002808).

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A379301 Positive integers whose prime indices include a unique composite number.

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A379304 Number of integer partitions of n with a unique prime part.

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A379305 Number of strict integer partitions of n with a unique prime part.

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A373401 Least k such that the k-th maximal antirun of prime numbers > 3 has length n. Position of first appearance of n in A027833. The sequence ends if no such antirun exists.

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