A036467 -id:A036467 - cp's OEIS frontend

A073738 Sum of every other prime <= n-th prime down to 2 or 1; equals the partial sums of A036467 (in which sums of two consecutive terms form the primes).

1, 2, 4, 7, 11, 18, 24, 35, 43, 58, 72, 89, 109, 130, 152, 177, 205, 236, 266, 303, 337, 376, 416, 459, 505, 556, 606, 659, 713, 768, 826, 895, 957, 1032, 1096, 1181, 1247, 1338, 1410, 1505, 1583, 1684, 1764, 1875, 1957, 2072, 2156, 2283, 2379, 2510, 2608

Offset: 0

Paul D. Hanna, Aug 07 2002

			a(10) = p_10 + p_8 + p_6 + p_4 + p_2 + p_0 = 29 + 19 + 13 + 7 + 3 + 1 = 72.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Harvey P. Dale)

Cf. A000040, A036467, A073736.

Cf. A006005.

a073738 n = a073738_list !! n
a073738_list = tail zs where
   zs = 1 : 1 : zipWith (+) a006005_list zs
-- Reinhard Zumkeller, Apr 28 2013

a:= proc(n) a(n):= `if`(n<1, n+1, ithprime(n) + a(n-2)) end:
seq(a(n), n=0..50);  # Alois P. Heinz, Jun 04 2021

nn=60;Join[{1},Sort[Join[Accumulate[Prime[Range[1,nn+1,2]]], 1+#&/@ Accumulate[Prime[Range[2,nn,2]]]]]] (* Harvey P. Dale, May 04 2011 *)

a(n) = Sum_{m<=n, m=n (mod 2)} p_m, where p_m is the m-th prime; that is, a(2n+k) = p_(2n+k) + p_(2(n-1)+k) + p_(2(n-2)+k) +... +p_k, for 0<=k<2, where a(0)=1 and the 0th prime is taken to be 1.

a(n) = prime(n) + a(n-2) for n >= 2. - Alois P. Heinz, Jun 04 2021

A135528 1, then repeat 1,0.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0

Offset: 1

N. J. A. Sloane, based on a message from Guy Steele and Don Knuth, Mar 01 2008

This is Guy Steele's sequence GS(2, 1) (see A135416).
2-adic expansion of 1/3 (right to left): 1/3 = ...01010101010101011. - Philippe Deléham, Mar 24 2009
Also, with offset 0, parity of A036467(n-1). - Omar E. Pol, Mar 17 2015
Appears to be the Gilbreath transform of 1,2,3,5,7,11,13,... (A008578). (This is essentially the same as the Gilbreath conjecture, see A036262.) - N. J. A. Sloane, May 08 2023

			G.f. = x + x^2 + x^4 + x^6 + x^8 + x^10 + x^12 + x^14 + x^16 + x^18 + x^20 + ...

Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A008578, A036467, A049711, A135416.

a135528 n = a135528_list !! (n-1)
a135528_list = concat $ iterate ([1,0] *) [1]
instance Num a => Num [a] where
fromInteger k = [fromInteger k]
   (p:ps) + (q:qs) = p + q : ps + qs
   ps + qs         = ps ++ qs
   (0:ps) * qs         = 0 : ps * qs
   (p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
    *                = []
-- Reinhard Zumkeller, Apr 02 2011

Maple
```
GS(2,1,200); [see A135416].
```

Prepend[Table[Mod[n + 1, 2], {n, 2, 60}], 1] (* Michael De Vlieger, Mar 17 2015 *)
PadRight[{1},120,{0,1}] (* Harvey P. Dale, Apr 23 2024 *)

G.f.: x*(1+x-x^2)/(1-x^2). - Philippe Deléham, Feb 08 2012
G.f.: x / (1 - x / (1 + x / (1 + x / (1 - x)))). - Michael Somos, Apr 02 2012
a(n) = A049711(n+2) mod 2. - Ctibor O. Zizka, Jan 28 2019

A147541 Result of using the primes as coefficients in an infinite polynomial series in x and then expressing this series as (1+x)(1+a(1)x)(1+a(2)x^2) ... .

Original entry on oeis.org

1, 2, 1, 3, 2, -4, 2, 5, 4, -6, 4, 4, 10, -36, 18, 45, 34, -72, 64, -24, 124, -358, 258, 170, 458, -1260, 916, 148, 1888, -4296, 3690, 887, 7272, -17616, 14718, -5096, 29610, -67164, 58722, -26036, 119602, -244496, 242256, -104754, 487352, -1029384

Offset: 1

Neil Fernandez, Nov 06 2008

sign

This is the PPE (power product expansion) of A036467. - R. J. Mathar, Feb 01 2010

			From the primes, construct the series 1+2x+3x^2+5x^3+7x^4+... Divide this by (1+x) to get the quotient (1+a(1)x+...), which here gives a(1)=1. Then divide this quotient by (1+a(1)x), i.e. here (1+x), to get (1+a(2)x^2+...), giving a(2)=2.

H. Gingold, A note on reduction of operations via power product approximations, Utilitas Math. 37 (1990), 79-89. [From _R. J. Mathar_, Nov 10 2008]
H. Gingold and A. Knopfmacher, Analytic properties of power product expansions, Canad. J. Math. 47 (1995), 1219-1239. [From _R. J. Mathar_, Nov 10 2008]
H. Gingold, A. Knopfmacher and D. Lubinsky, The zero distribution of the partial products of power product expansions, Analysis 13 (1993), 133-157. [From _R. J. Mathar_, Nov 10 2008]

Cf. A000040, A147542.

From R. J. Mathar, Feb 01 2010: (Start)
# Partition n into a set of distinct positive integers, the maximum one
# being m.
# Example: partitionsQ(7,5) returns [[2,5],[3,4],[1,2,4]] ;
# Richard J. Mathar, 2008-11-10
partitionsQ := proc(n,m)
local p,t,rec,q;
p := [] ;
# take 't' of the n and recursively determine the partitions of
# what has been left over.
for t from min(m,n) to 1 by -1 do
# Since we are only considering partitions into distinct parts,
# the triangular numbers set a lower bound on the t.
if t*(t+1)/2 >= n then
rec := partitionsQ(n-t,t-1) ;
if nops(rec) = 0 then
p := [op(p),[t]] ;
else
for q in rec do
p := [op(p),[op(q),t]] ;
end do:
end if;
end if;
end do:
RETURN(p) ;
end proc:
# Power product expansion of L.
# L is a list starting with 1, which is considered L[0].
# Returns the list [a(1),a(2),..] such that
# product_(i=1,2,..) (1+a(i)x^i) = sum_(j=0,1,2,...) L[j]x^j.
# Richard J. Mathar, 2008-11-10
ppe := proc(L)
local pro,i,par,swithi,snoti,m,p,k ;
pro := [] ;
for i from 1 to nops(L)-1 do
par := partitionsQ(i,i) ;
swithi := 0 ;
snoti := 0 ;
for p in par do
if i in p then
m := 1 ;
for k from 1 to nops(p)-1 do
m := m*op(op(k,p),pro) ;
end do;
swithi := swithi+m ;
else
snoti := snoti+mul( op(k,pro),k=p) ;
end if;
end do:
pro := [op(pro), (op(i+1,L)-snoti)/swithi] ;
end do:
RETURN(pro) ;
end proc:
read("transforms") ;
A147541 := proc(nmax)
local L,L1,L2 ;
L := [1,seq(ithprime(n),n=1..nmax)] ;
L1 := [seq((-1)^n,n=0..nmax+10)] ;
A036467 := CONV(L,L1) ;
ppe(A036467) ;
end:
A147541(47) ;
(End)

Extended by R. J. Mathar, Feb 01 2010

A073737 Sums of three successive terms form the odd primes.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 5, 7, 7, 9, 13, 9, 15, 17, 11, 19, 23, 17, 21, 29, 21, 23, 35, 25, 29, 43, 29, 31, 47, 31, 35, 61, 35, 41, 63, 45, 43, 69, 51, 47, 75, 57, 49, 85, 59, 53, 87, 71, 65, 91, 73, 69, 97, 75, 79, 103, 81, 85, 105, 87, 89, 107, 97, 103, 111, 99, 107, 125, 105, 117

Offset: 1

Paul D. Hanna, Aug 07 2002

			At n=10, a(10) + a(9) + a(8) = 13 + 9 + 7 = 29 = p_10.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A036467, A001223, A073736.

Cf. A065091, A000040.

a073737 n = a073737_list !! (n-1)
a073737_list =
   1 : 1 : zipWith (-) a065091_list
                       (zipWith (+) a073737_list $ tail a073737_list)
-- Reinhard Zumkeller, Aug 14 2011

a[0] = 1; a[-1] = 0; a[-2] = 0; p[0] = 1; p[n_?Positive] := Prime[n]; a[n_] := a[n] = p[n] - a[n-1] - a[n-2]; Table[a[n], {n, 0, 69}] (* Jean-François Alcover, Sep 30 2011 *)
nxt[{a_,b_,c_}]:={b,c,NextPrime[a+b+c]-(b+c)}; Transpose[NestList[nxt,{1,1,1},70]][[1]] (* Harvey P. Dale, Mar 15 2015 *)

a(n) + a(n-1) + a(n-2) = n-th prime, where a(0)=1, a(-1)=0, a(-2)=0 and the 0th prime is taken to be 1.

A073739 Least positive integers whose convolution forms a sequence whose odd-indexed terms are twice the odd primes (see: A073740).

Original entry on oeis.org

1, 1, 1, 2, 0, 3, 0, 4, 0, 7, 0, 6, 0, 11, 0, 8, 0, 15, 0, 14, 0, 17, 0, 20, 0, 21, 0, 22, 0, 25, 0, 28, 0, 31, 0, 30, 0, 37, 0, 34, 0, 39, 0, 40, 0, 43, 0, 46, 0, 51, 0, 50, 0, 53, 0, 54, 0, 55, 0, 58, 0, 69, 0, 62, 0, 75, 0, 64, 0, 85, 0, 66

Offset: 0

Paul D. Hanna, Aug 07 2002

The odd-indexed bisection forms A036467, in which the pairwise sums yield the primes.

			a(10) = p_10 +p_8 +p_6 +p_4 +p_2 +p_0 = 29 + 19 + 13 +7 +3 + 1 = 72.

Cf. A036467, A073740.

import Data.List (transpose)
a073739 n = a073739_list !! n
a073739_list = concat $ transpose [1 : 1 : repeat 0, tail a036467_list]
-- Reinhard Zumkeller, Aug 09 2015

a[n_ /; n <= 2] = 1; a[?EvenQ] = 0; a[n] := a[n] = Prime[(n + 1)/2] - a[n - 2]; Table[a[n], {n, 0, 71}] (* Jean-François Alcover, Aug 01 2013 *)

a(n) = p_n - p_{n-1} when n>1, where a(0)=a(1)=1.

A073740 Self-convolution of A073739; odd-indexed terms are twice the odd primes.

Original entry on oeis.org

1, 2, 3, 6, 5, 10, 10, 14, 20, 22, 39, 26, 64, 34, 104, 38, 152, 46, 225, 58, 308, 62, 434, 74, 556, 82, 763, 86, 936, 94, 1224, 106, 1488, 118, 1857, 122, 2244, 134, 2706, 142, 3232, 146, 3827, 158, 4472, 166, 5240, 178, 6020, 194, 6997, 202, 7936, 206, 9122

Offset: 0

Paul D. Hanna, Aug 07 2002

The first odd prime is here considered to be p_0 = 1.

			a(9) = 2*p_4 = 22; a(8) = 1*0+1*4+1*0+2*3+0*0+3*2+0*1+4*1+0*1 = 20.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A036467, A073739.

a073740 n = a073740_list !! n
a073740_list = tail $ f a073739_list [] where
   f (x:xs) ys = (sum $ zipWith (*) ys a073739_list) : f xs (x : ys)
-- Reinhard Zumkeller, Aug 09 2015

(* b = A073739 *) b[n_ /; n <= 2] = 1; b[?EvenQ] = 0; b[n] := b[n] = Prime[(n+1)/2] - b[n-2]; a[0] = 1; a[1] = 2; a[n_?OddQ] := 2*Prime[(n-1)/2+1]; a[n_?EvenQ] := Sum[b[k]*b[n-k], {k, 0, n}]; Table[a[n], {n, 0, 54}] (* Jean-François Alcover, Aug 01 2013 *)

a(0) = 1, a(2n+1) = 2*p_n, a(2n) = Sum_{k=0..2n} A073739(k)*A073739(2n-k).

A137462 a(n) + a(n-1) = n-th semiprime.

Original entry on oeis.org

1, 3, 3, 6, 4, 10, 5, 16, 6, 19, 7, 26, 8, 27, 11, 28, 18, 31, 20, 35, 22, 36, 26, 39, 30, 44, 33, 49, 36, 50, 37, 54, 39, 55, 40, 66, 45, 70, 48, 71, 50, 72, 51, 78, 55, 79, 62, 80, 63, 82, 64, 91, 67, 92, 69, 97, 72, 105, 73, 110, 75, 112, 82, 119, 83, 120, 85, 121, 88, 125

Offset: 0

Jonathan Vos Post, Apr 19 2008

This is to A001358 as A036467 is to A000040.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A001358, A036467, A065516.

A001358 := proc(n) option remember ; if n =1 then 4; else for a from A001358(n-1)+1 do if numtheory[bigomega](a) = 2 then RETURN(a) ; fi ; od: fi ; end: A137462 := proc(n) option remember; if n =0 then 1; else A001358(n)-A137462(n-1) ; fi ; end: seq(A137462(n),n=0..100) ; # R. J. Mathar, Apr 23 2008

Module[{nn=300,sp,k=1},sp=Select[Range[nn],PrimeOmega[#]==2&];Join[{1}, Table[k=sp[[n]]-k,{n,Length[sp]}]]] (* Harvey P. Dale, Apr 30 2015 *)

a(n) + a(n-1) = A001358(n).

More terms from R. J. Mathar, Apr 23 2008

A338467 a(n+1) = prime(n) + 2*n - a(n). a(1) = 1.

Original entry on oeis.org

1, 3, 4, 7, 8, 13, 12, 19, 16, 25, 24, 29, 32, 35, 36, 41, 44, 49, 48, 57, 54, 61, 62, 67, 70, 77, 76, 81, 82, 85, 88, 101, 94, 109, 98, 121, 102, 129, 110, 135, 118, 143, 122, 155, 126, 161, 130, 175, 144, 181, 148, 187, 156, 191, 168, 199, 176, 207, 180, 215

Offset: 1

Carole Dubois, Jan 31 2021

			a(1) + a(2) - 2*1 = 1st prime; 1 + 3 - 2*1 = 2.
a(13) + a(14) - 2*13 = 13th prime; 32 + 35 - 2*13 = 41.

Cf. A000040, A001223, A036467, A078916.

a:= proc(n) option remember; `if`(n=1, 1,
      ithprime(n-1)-a(n-1)+2*n-2)
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 31 2021

a[1] = 1; a[n_] := a[n] = Prime[n - 1] + 2*(n - 1) - a[n - 1]; Array[a, 60] (* Amiram Eldar, Feb 01 2021 *)

a(n) = if (n==1, 1, prime(n-1) + 2*(n-1) - a(n-1)); \\ Michel Marcus, Jan 31 2021

from sympy import prime
S=[1]
nomb=100
for n in range(1,nomb):
    derterm=S[-1]
    terme= prime(n)-derterm+2*(len(S))
    S.append(terme)
print(S)

a(n+1) = A078916(n) - a(n). - Michel Marcus, Jan 31 2021

cp's OEIS Frontend

A073738 Sum of every other prime <= n-th prime down to 2 or 1; equals the partial sums of A036467 (in which sums of two consecutive terms form the primes).

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A135528 1, then repeat 1,0.

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A147541 Result of using the primes as coefficients in an infinite polynomial series in x and then expressing this series as (1+x)(1+a(1)*x)(1+a(2)*x^2) ... .

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A073737 Sums of three successive terms form the odd primes.

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A073739 Least positive integers whose convolution forms a sequence whose odd-indexed terms are twice the odd primes (see: A073740).

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A073740 Self-convolution of A073739; odd-indexed terms are twice the odd primes.

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A137462 a(n) + a(n-1) = n-th semiprime.

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A147541 Result of using the primes as coefficients in an infinite polynomial series in x and then expressing this series as (1+x)(1+a(1)x)(1+a(2)x^2) ... .