A006005 -id:A006005 - cp's OEIS frontend

A298604 Number of partitions of n into distinct odd prime parts (including 1).

1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 6, 5, 5, 6, 6, 7, 7, 8, 9, 8, 9, 10, 11, 12, 11, 12, 14, 14, 15, 16, 17, 17, 17, 20, 22, 21, 22, 24, 25, 27, 28, 30, 31, 31, 33, 36, 39, 40, 40, 42, 46, 47, 49, 53, 54, 55, 58, 63, 67, 68, 70, 73, 77, 81, 84

Offset: 0

Ilya Gutkovskiy, Jan 22 2018

nonn

			a(16) = 3 because we have [13, 3], [11, 5] and [7, 5, 3, 1].

Index entries for related partition-counting sequences

Cf. A000586, A006005, A008578, A024939, A036497, A298603.

nmax = 78; CoefficientList[Series[(1 + x) Product[(1 + x^Prime[k]), {k, 2, nmax}], {x, 0, nmax}], x]

G.f.: (1 + x)*Product_{k>=2} (1 + x^prime(k)).

A299766 Greatest odd noncomposite divisor of n.

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 7, 1, 3, 5, 11, 3, 13, 7, 5, 1, 17, 3, 19, 5, 7, 11, 23, 3, 5, 13, 3, 7, 29, 5, 31, 1, 11, 17, 7, 3, 37, 19, 13, 5, 41, 7, 43, 11, 5, 23, 47, 3, 7, 5, 17, 13, 53, 3, 11, 7, 19, 29, 59, 5, 61, 31, 7, 1, 13, 11, 67, 17, 23, 7, 71, 3, 73, 37, 5, 19, 11, 13, 79, 5, 3, 41, 83, 7, 17, 43

Offset: 1

Omar E. Pol, Mar 12 2018

The sequence can be constructed replacing the 2's with 1's in A006530.

Cf. A000265 (odd part), A008578 (noncomposite numbers), A006005 (odd noncomposite numbers), A006530 (greatest noncomposite divisor).

[#f eq 1 select 1 else f[#f][1] where f is Factorization(2*n): n in [1..90]]; // Vincenzo Librandi, Mar 20 2018

A299766 := proc(n)
    local dvs ;
    dvs := sort(convert(numtheory[factorset](n) minus {2},list)) ;
    if nops(dvs) >= 1 then
        op(-1,dvs) ;
    else
        1;
    end if;
end proc: # R. J. Mathar, May 05 2018

Array[FactorInteger[#][[-1, 1]] /. 2 -> 1 &, 86] (* Michael De Vlieger, Mar 16 2018 *)
Table[Max[Select[Divisors[n],OddQ[#]&&!CompositeQ[#]&]],{n,100}] (* Harvey P. Dale, May 17 2024 *)

a(n) = my(m = n >> valuation(n, 2)); if (m==1, 1, vecmax(factor(m)[,1])); \\ Michel Marcus, Mar 15 2018

a(n) = A006530(A000265(n)). - Michel Marcus, Mar 15 2018

a(n) = 1 for n in A000079 (powers of 2). - Michel Marcus, Apr 14 2018

A300893 L.g.f.: log(Product_{k>=1} (1 + x^k)/(1 + x^prime(k))) = Sum_{n>=1} a(n)*x^n/n.

Original entry on oeis.org

1, -1, 1, 3, 1, 5, 1, 3, 10, 9, 1, 9, 1, 13, 16, 3, 1, 14, 1, 13, 22, 21, 1, 9, 26, 25, 37, 17, 1, 30, 1, 3, 34, 33, 36, 18, 1, 37, 40, 13, 1, 40, 1, 25, 70, 45, 1, 9, 50, 34, 52, 29, 1, 41, 56, 17, 58, 57, 1, 34, 1, 61, 94, 3, 66, 60, 1, 37, 70, 58, 1, 18, 1, 73, 116, 41, 78, 70, 1, 13, 118, 81, 1, 44, 86

Offset: 1

Ilya Gutkovskiy, Mar 14 2018

sign

			L.g.f.: L(x) = x - x^2/2 + x^3/3 + 3*x^4/4 + x^5/5 + 5*x^6/6 + x^7/7 + 3*x^8/8 + 10*x^9/9 + 9*x^10/10 + ...
exp(L(x)) = 1 + x + x^4 + x^5 + x^6 + x^7 + x^8 + 2*x^9 + 3*x^10 + ... + A096258(n)*x^n + ...

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A006005, A018252, A023890, A096258, A300852.

nmax = 85; Rest[CoefficientList[Series[Log[Product[(1 + x^k)/(1 + x^Prime[k]), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 85; Rest[CoefficientList[Series[Sum[Boole[!PrimeQ[k]] k x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[DivisorSum[n, (-1)^(n/# + 1) # &, !PrimeQ[#] &], {n, 85}]

G.f.: Sum_{k>=1} A018252(k)*x^A018252(k)/(1 + x^A018252(k)).

a(n) = 1 if n is an odd prime or 1 (A006005).

A328456 LCM of the prime indices of 2n + 1, all minus 1; a(0) = 0.

Original entry on oeis.org

0, 1, 2, 3, 1, 4, 5, 2, 6, 7, 3, 8, 2, 1, 9, 10, 4, 6, 11, 5, 12, 13, 2, 14, 3, 6, 15, 4, 7, 16, 17, 3, 10, 18, 8, 19, 20, 2, 12, 21, 1, 22, 6, 9, 23, 15, 10, 14, 24, 4, 25, 26, 6, 27, 28, 11, 29, 8, 5, 6, 4, 12, 2, 30, 13, 31, 21, 2, 32, 33, 14, 20, 18, 3, 34

Offset: 0

Gus Wiseman, Oct 17 2019

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 2 * 17 + 1 = 35, all minus 1, are {2,3}, with LCM 6, so a(17) = 6.

Positions of records (first appearances) are A006005.
The GCD of the prime indices of n, all minus 1, is A328167(n).
The LCM of the prime indices of n, all plus 1, is A328219(n).
Partitions whose parts minus 1 are relatively prime are A328170.
Numbers whose prime indices minus 1 are relatively prime are A328168.
Cf. A056239, A112798, A258409, A289508, A289509, A290103, A318981, A328169, A328451.

Table[If[n==1,0,LCM@@(PrimePi/@First/@FactorInteger[n]-1)],{n,1,100,2}]

A353160 Product_{n>=1} (1 + x^n)^a(n) = 1 + Sum_{n>=1} prime(n+1) * x^n.

Original entry on oeis.org

3, 2, 0, 4, -4, 6, -8, 11, 0, -18, 40, -67, 88, -78, -4, 205, -524, 926, -1234, 1060, 140, -2998, 7900, -14132, 18858, -16280, -2710, 48783, -127826, 228784, -306494, 263582, 55144, -829538, 2160120, -3864533, 5172792, -4406277, -1118324, 14531101, -37606792

Offset: 1

Ilya Gutkovskiy, Apr 28 2022

sign

Inverse weigh transform of odd primes.

Cf. A006005, A065091, A305871, A352609, A353161.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= proc(n) option remember; ithprime(n+1)-b(n, n-1) end:
seq(a(n), n=1..50);  # Alois P. Heinz, Apr 28 2022

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; a[n_] := a[n] = Prime[n + 1] - b[n, n - 1]; Table[a[n], {n, 1, 41}]

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).

Original entry on oeis.org

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

A126793 a(1) = 1; a(n+1) = Sum_{k|n} floor(a(k)/a(n/k)).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 5, 7, 8, 11, 11, 16, 16, 21, 22, 28, 28, 36, 36, 45, 47, 58, 58, 72, 73, 89, 92, 110, 110, 137, 137, 161, 166, 194, 195, 232, 232, 268, 276, 317, 317, 371, 371, 423, 435, 493, 493, 568, 569, 643, 657, 738, 738, 843, 846, 948, 966, 1076, 1076, 1219, 1219

Offset: 1

Leroy Quet, Feb 20 2007

nonn

a(n+1) = a(n) if and only if n is 1 or an odd prime (A006005). - Robert Israel, Dec 22 2016

			a(13) = sum{k|12} [a(k)/a(12/k)] = [a(1)/a(12)] + [a(2)/a(6)] + [a(3)/a(4)] + [a(4)/a(3)] + [a(6)/a(2)] + [a(12)/a(1)] = [1/11] + [1/3] + [2/2] + [2/2] + [3/1] + [11/1] = 0 +0 +1 +1 +3 +11 = 16.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006005.

A[1]:= 1:
for n from 1 to 100 do
  A[n+1] := add(floor(A[k]/A[n/k]),k=numtheory:-divisors(n))
od:
seq(A[i],i=1..100); # Robert Israel, Dec 22 2016

f[l_List] := Block[{n = Length[l], d = Divisors[n]},Append[l, Sum[ Floor[l[[d[[k]]]]/l[[n/d[[k]]]]], {k, Length[d]}]]];Nest[f, {1}, 61] (* Ray Chandler, Mar 03 2007 *)
a[1] = 1; a[n_] := a[n] = Sum[Floor[a[k]/a[(n - 1)/k]], {k, Divisors[n - 1]}]; Array[a, 62] (* Michael De Vlieger, Dec 22 2016 *)

Extended by Ray Chandler, Mar 03 2007

A129768 Number of odd nonprime numbers less than the n-th prime.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 3, 4, 6, 6, 8, 9, 9, 10, 12, 14, 14, 16, 17, 17, 19, 20, 22, 25, 26, 26, 27, 27, 28, 34, 35, 37, 37, 41, 41, 43, 45, 46, 48, 50, 50, 54, 54, 55, 55, 60, 65, 66, 66, 67, 69, 69, 73, 75, 77, 79, 79, 81, 82, 82, 86, 92, 93, 93, 94, 100, 102, 106, 106, 107, 109

Offset: 1

Giovanni Teofilatto, May 16 2007

nonn

Also, a(n)=sum_{k=n-th even number..n-th odd prime together with 1}-k*(-1)^k. - Juri-Stepan Gerasimov, Jul 30 2009

A129768 := proc(n) local res,p,i ; res := 0 ; p := ithprime(n) ; for i from 1 to p-1 by 2 do if not isprime(i) then res := res+1 ; fi ; od ; RETURN(res) ; end: for n from 1 to 80 do printf("%d, ",A129768(n)) ; od ; # R. J. Mathar, May 17 2007

f[n_] := Ceiling[ Prime@n / 2] - n + 1; Array[f, 72] (* Robert G. Wilson v *)
nn=401;With[{np=Select[Range[1,nn,2],!PrimeQ[#]&]},Table[Count[np,?(#<n&)],{n,Prime[Range[PrimePi[nn]]]}]] (* _Harvey P. Dale, Mar 01 2015 *)

a(n)=A008507(n-1)+1. - R. J. Mathar, Jul 06 2009

a(n)=sum_{k=A005843(n)..A006005(n)}-k*(-1)^k. - Juri-Stepan Gerasimov, Jul 30 2009

Edited, corrected and extended by Robert G. Wilson v and R. J. Mathar, May 16 2007

A130891 a(n) = n if n is not an odd prime number. Otherwise, a(n) = k*floor(n/10), where k is the smallest integer such that n < 10^k.

Original entry on oeis.org

0, 1, 2, 0, 4, 0, 6, 0, 8, 9, 10, 2, 12, 2, 14, 15, 16, 2, 18, 2, 20, 21, 22, 4, 24, 25, 26, 27, 28, 4, 30, 6, 32, 33, 34, 35, 36, 6, 38, 39, 40, 8, 42, 8, 44, 45, 46, 8, 48, 49, 50, 51, 52, 10, 54, 55, 56, 57, 58, 10, 60, 12, 62, 63, 64, 65, 66, 12, 68

Offset: 0

Mohammad K. Azarian, Aug 21 2007

nonn

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

Cf. A131516, A130758, A000027, A001477, A006005.

f:= proc(n) local k;
  if n::odd and isprime(n) then
    k:= ilog10(n)+1;
    k*floor(n/10)
  else n
  fi
end proc:
map(f, [$0..100]); # Robert Israel, Aug 04 2019

Table[If[n>2&&PrimeQ[n],IntegerLength[n]Floor[n/10],n],{n,0,70}] (* Harvey P. Dale, Mar 31 2025 *)

Offset changed by Mohammad K. Azarian, Nov 19 2008

Offset corrected by Robert Israel, Aug 04 2019

A160656 The odd prime numbers together with 0: p - (-1)^p - 1 where p = n-th prime.

Original entry on oeis.org

0, 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, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277

Offset: 1

Juri-Stepan Gerasimov, May 23 2009

Cf. A000040, A006005, A158611.

seq(ithprime(n) mod n^2, n=1..59) # Gary Detlefs, Jan 14 2012

a(n)=if(n>1,prime(n)) \\ Charles R Greathouse IV, Aug 26 2011

a(n) = p - (-1)^p - 1 where p is the n-th prime.

a(n) = p mod n^2, where p is the n-th prime. - Gary Detlefs, Jan 14 2012

cp's OEIS Frontend

A298604 Number of partitions of n into distinct odd prime parts (including 1).

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A299766 Greatest odd noncomposite divisor of n.

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A300893 L.g.f.: log(Product_{k>=1} (1 + x^k)/(1 + x^prime(k))) = Sum_{n>=1} a(n)*x^n/n.

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A328456 LCM of the prime indices of 2n + 1, all minus 1; a(0) = 0.

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A353160 Product_{n>=1} (1 + x^n)^a(n) = 1 + Sum_{n>=1} prime(n+1) * x^n.

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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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A126793 a(1) = 1; a(n+1) = Sum_{k|n} floor(a(k)/a(n/k)).

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A129768 Number of odd nonprime numbers less than the n-th prime.

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