A365067 -id:A365067 - cp's OEIS frontend

A366528 Sum of odd prime indices of n.

0, 1, 0, 2, 3, 1, 0, 3, 0, 4, 5, 2, 0, 1, 3, 4, 7, 1, 0, 5, 0, 6, 9, 3, 6, 1, 0, 2, 0, 4, 11, 5, 5, 8, 3, 2, 0, 1, 0, 6, 13, 1, 0, 7, 3, 10, 15, 4, 0, 7, 7, 2, 0, 1, 8, 3, 0, 1, 17, 5, 0, 12, 0, 6, 3, 6, 19, 9, 9, 4, 0, 3, 21, 1, 6, 2, 5, 1, 0, 7, 0, 14, 23, 2

Offset: 1

Gus Wiseman, Oct 22 2023

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, sum A056239(n).

			The prime indices of 198 are {1,2,2,5}, so a(198) = 1+5 = 6.

Zeros are A066207, counted by A035363.
The triangle for this rank statistic is A113685, without zeros A365067.
For count instead of sum we have A257991, even A257992.
Nonzeros are A366322, counted by A086543.
The even version is A366531, halved A366533, triangle A113686.
A000009 counts partitions into odd parts, ranks A066208.
A053253 = partitions with all odd parts and conjugate parts, ranks A352143.
A066967 adds up sums of odd parts over all partitions.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A162641 counts even prime exponents, odd A162642.
A352142 = odd indices with odd exponents, counted by A117958.
Cf. A000720, A055396, A130780, A171966, A174713, A325698, A325700, A366748.

Table[Total[Cases[FactorInteger[n], {p_?(OddQ@*PrimePi),k_}:>PrimePi[p]*k]],{n,100}]

a(n) = A056239(n) - A366531(n).

A239261 Number of partitions of n having (sum of odd parts) = (sum of even parts).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 12, 0, 0, 0, 30, 0, 0, 0, 70, 0, 0, 0, 165, 0, 0, 0, 330, 0, 0, 0, 704, 0, 0, 0, 1380, 0, 0, 0, 2688, 0, 0, 0, 4984, 0, 0, 0, 9394, 0, 0, 0, 16665, 0, 0, 0, 29970, 0, 0, 0, 52096, 0, 0, 0, 90090, 0, 0, 0, 152064, 0, 0, 0

Offset: 0

Clark Kimberling, Mar 13 2014

			a(8) counts these 4 partitions:  431, 41111, 3221, 221111.
From _Gus Wiseman_, Oct 24 2023: (Start)
The a(0) = 1 through a(12) = 12 partitions:
  ()  .  .  .  (211)  .  .  .  (431)     .  .  .  (633)
                               (3221)             (651)
                               (41111)            (4332)
                               (221111)           (5421)
                                                  (33222)
                                                  (52221)
                                                  (63111)
                                                  (432111)
                                                  (3222111)
                                                  (6111111)
                                                  (42111111)
                                                  (222111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A239259, A239260, A239262, A239263, A000041.
The LHS (sum of odd parts) is counted by A113685.
The RHS (sum of even parts) is counted by A113686.
Without all the zeros we have a(4n) = A249914(n).
The strict case (without zeros) is A255001.
The Heinz numbers of these partitions are A366748, see also A019507.
A000009 counts partitions into odd parts, ranks A066208.
A035363 counts partitions into even parts, ranks A066207.
Cf. A028260, A045931, A106529, A239241, A241638, A325698, A365067.

z = 40; p[n_] := p[n] = IntegerPartitions[n]; f[t_] := f[t] = Length[t]
t1 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] < n &]], {n, z}] (* A239259 *)
t2 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] <= n &]], {n, z}] (* A239260 *)
t3 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] == n &]], {n, z}] (* A239261 *)
t4 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] > n &]], {n, z}] (* A239262 *)
t5 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] >= n &]], {n, z}] (* A239263 *)
(* Peter J. C. Moses, Mar 12 2014 *)

A239260(n) + a(n) + A239262(n) = A000041(n).
From David A. Corneth, Oct 25 2023: (Start)
a(4*n) = A000009(2*n) * A000041(n) for n >= 0.
a(4*n + r) = 0 for n >= 0 and r in {1, 2, 3}. (End)

More terms from Alois P. Heinz, Mar 15 2014

A174713 Triangle read by rows, A173305 (A000009 shifted down twice) * A174712 (diagonalized variant of A000041).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 2, 4, 2, 2, 3, 5, 3, 4, 3, 6, 4, 4, 3, 5, 8, 5, 6, 6, 5, 10, 6, 8, 6, 5, 7, 12, 8, 10, 9, 10, 7, 15, 10, 12, 12, 10, 7, 11, 18, 12, 16, 15, 15, 14, 11, 22, 15, 20, 18, 20, 14, 11, 15

Offset: 0

Gary W. Adamson, Mar 27 2010

Row sums = A000041, the partition numbers.
The current triangle is the 2nd in an infinite set, followed by A174714 (k=3), and A174715, (k=4); in which row sums of each triangle = A000041.
k-th triangle in the infinite set can be defined as having the sequence:
"Euler transform of ones: (1,1,1,...) interleaved with (k-1) zeros"; shifted down k times (except column 0) in successive columns, then multiplied * triangle A174712, the diagonalized variant of A000041, A174713 begins with A000009 shifted down twice (triangle A173305); where A000009 = the Euler transform of period 2 sequence: [1,0,1,0,...].
Similarly, triangle A174714 begins with A000716 shifted down thrice; where A000716 = the Euler transform of period 3 series: [1,1,0,1,1,0,...]. Then multiply the latter as an infinite lower triangular matrix * A174712, the diagonalized variant of A000041, obtaining triangle A174714 with row sums = A000041.
Case k=4 = triangle A174715 which begins with the Euler transform of period 4 series: [1,1,1,0,1,1,1,0,...], shifted down 4 times in successive columns then multiplied * A174712, the diagonalized variant of A000041.
All triangles in the infinite set have row sums = A000041.
The sequences: "Euler transform of ones interleaved with (k-1) zeros" have the following properties, beginning with k=2:
...
k=2, A000009: = Euler transform of [1,0,1,0,1,0,...] and satisfies
.....A000009. = p(x)/p(x^2), where p(x) = polcoeff A000041; and A000041 =
.....A000009(x) = r(x), then p(x) = r(x) * r(x^2) * r(x^4) * r(x^8) * ...
...
k=3, A000726: = Euler transform of [1,1,0,1,1,0,...] and satisfies
.....A000726(x): = p(x)/p(x^3), and given s(x) = polcoeff A000726, we get
.....A000041(x) = p(x) = s(x) * s(x^3) * s(x^9) * s(x^27) * ...
...
k=4, A001935: = Euler transform of [1,1,1,0,1,1,1,0,...] and satisfies
.....A001935(x) = p(x)/p(x^4) and given t(x) = polcoeff A001935, we get
.....A000041(x) = p(x) = t(x) * t(x^4) * t(x^16) * t(x^64) * ...
...
Also the number of integer partitions of n whose even parts sum to k, for k an even number from zero to n. The version including odd k is A113686. - Gus Wiseman, Oct 23 2023

			First few rows of the triangle =
1;
1;
1, 1;
2, 1;
2, 1, 2;
3, 2, 2;
4, 2, 2, 3;
5, 3, 4, 3;
6, 4, 4, 3, 5;
8, 5, 6, 6, 5;
10, 6, 8, 6, 5, 7;
12, 8, 10, 9, 10, 7;
15, 10, 12, 12, 10, 7, 11;
18, 12, 16, 15, 15, 14, 11;
22, 15, 20, 18, 20, 14, 11, 15;
...
From _Gus Wiseman_, Oct 23 2023: (Start)
Row n = 9 counts the following partitions:
  (9)          (72)        (54)       (63)      (81)
  (711)        (5211)      (522)      (6111)    (621)
  (531)        (3321)      (4311)     (432)     (441)
  (51111)      (321111)    (411111)   (42111)   (4221)
  (333)        (21111111)  (32211)    (3222)    (22221)
  (33111)                  (2211111)  (222111)
  (3111111)
  (111111111)
(End)

Cf. A000009, A173305, A174712, A174714, A174715.
Row sums are A000041.
The odd version is A365067.
The corresponding rank statistic is A366531, odd version A366528.
A000009 counts partitions into odd parts, ranks A066208.
A113685 counts partitions by sum of odd parts, even version A113686.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
Cf. A035363, A066967, A130780, A171966, A182616, A366527, A366533.

Table[Length[Select[IntegerPartitions[n],Total[Select[#,EvenQ]]==k&]],{n,0,15},{k,0,n,2}] (* Gus Wiseman, Oct 23 2023 *)

As infinite lower triangular matrices, A173305 * A174712.

T(n,k) = A000009(n-2k) * A000041(k). - Gus Wiseman, Oct 23 2023

A366531 Sum of even prime indices of n.

Original entry on oeis.org

0, 0, 2, 0, 0, 2, 4, 0, 4, 0, 0, 2, 6, 4, 2, 0, 0, 4, 8, 0, 6, 0, 0, 2, 0, 6, 6, 4, 10, 2, 0, 0, 2, 0, 4, 4, 12, 8, 8, 0, 0, 6, 14, 0, 4, 0, 0, 2, 8, 0, 2, 6, 16, 6, 0, 4, 10, 10, 0, 2, 18, 0, 8, 0, 6, 2, 0, 0, 2, 4, 20, 4, 0, 12, 2, 8, 4, 8, 22, 0, 8, 0, 0, 6

Offset: 1

Gus Wiseman, Oct 22 2023

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 198 are {1,2,2,5}, so a(198) = 2+2 = 4.

Zeros are A066208, counted by A000009.
The triangle for the odd version is A113685, without zeros A365067.
The triangle for this statistic is A113686, without zeros A174713.
The odd version is A366528.
The halved version is A366533.
A066207 lists numbers with all even prime indices, counted by A035363.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A239261 counts partitions with sum of odd parts = sum of even parts.
A257991 counts odd prime indices, even A257992.
A346697 adds up odd-indexed prime indices, even-indexed A346698.
A366322 lists numbers with not all prime indices even, counted by A086543.
Cf. A000720, A055396, A055922, A061395, A162641, A171966, A258117, A325698, A325700, A352140, A352141.

Table[Total[Cases[FactorInteger[n], {p_?(EvenQ@*PrimePi),k_}:>PrimePi[p]*k]],{n,100}]

a(n) = A056239(n) - A366528(n).

A366533 Sum of even prime indices of n divided by 2.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 2, 0, 2, 0, 0, 1, 3, 2, 1, 0, 0, 2, 4, 0, 3, 0, 0, 1, 0, 3, 3, 2, 5, 1, 0, 0, 1, 0, 2, 2, 6, 4, 4, 0, 0, 3, 7, 0, 2, 0, 0, 1, 4, 0, 1, 3, 8, 3, 0, 2, 5, 5, 0, 1, 9, 0, 4, 0, 3, 1, 0, 0, 1, 2, 10, 2, 0, 6, 1, 4, 2, 4, 11, 0, 4, 0, 0, 3, 0, 7

Offset: 1

Gus Wiseman, Oct 23 2023

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 198 are {1,2,2,5}, so a(198) = (2+2)/2 = 2.

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

Zeros are A066208, counted by A000009.
The triangle for this statistic (without zeros) is A174713.
The un-halved odd version is A366528.
The un-halved version is A366531.
A066207 lists numbers with all even prime indices, counted by A035363.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A113685 counts partitions by sum of odd parts, even version A113686.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
A257991 counts odd prime indices, even A257992.
A346697 adds up odd-indexed prime indices, even-indexed A346698.
A365067 counts partitions by sum of odd parts (without zeros).
A366322 lists numbers with not all prime indices even, counted by A086543.
Cf. A000720, A055396, A055922, A061395, A162641, A171966, A258117, A325698, A325700, A352140, A352141.

f:= proc(n) local F,t;
  F:= map(t -> [numtheory:-Pi(t[1]),t[2]], ifactors(n)[2]);
  add(`if`(t[1]::even, t[1]*t[2]/2, 0), t=F)
end proc:
map(f, [$1..100]); # Robert Israel, Nov 22 2023

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[Select[prix[n],EvenQ]]/2,{n,100}]

a(n) = A366531(n)/2.

A366842 Number of integer partitions of n whose odd parts have a common divisor > 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 1, 4, 1, 8, 3, 13, 6, 21, 10, 36, 15, 53, 28, 80, 41, 122, 63, 174, 97, 250, 140, 359, 201, 496, 299, 685, 410, 949, 575, 1284, 804, 1726, 1093, 2327, 1482, 3076, 2023, 4060, 2684, 5358, 3572, 6970, 4745, 9050, 6221, 11734, 8115, 15060, 10609

Offset: 0

Gus Wiseman, Oct 28 2023

nonn

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

This is the odd case of A018783, complement A000837.
The even version is A047967.
The complement is counted by A366850, ranks A366846.
A000041 counts integer partitions, strict A000009.
A000740 counts relatively prime compositions.
A113685 counts partitions by sum of odds, stat A366528, w/o zeros A365067.
A168532 counts partitions by gcd.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
A289508 gives gcd of prime indices, positions of ones A289509.
Cf. A007359, A051424, A055922, A066208, A078374, A087436, A116598, A337485, A366843, A366844, A366845.

Table[Length[Select[IntegerPartitions[n], GCD@@Select[#,OddQ]>1&]], {n,0,30}]

from math import gcd
from sympy.utilities.iterables import partitions
def A366842(n): return sum(1 for p in partitions(n) if gcd(*(q for q in p if q&1))>1) # Chai Wah Wu, Oct 28 2023

A366844 Number of strict integer partitions of n into odd relatively prime parts.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 0, 2, 1, 2, 1, 2, 2, 3, 3, 5, 4, 4, 5, 6, 7, 8, 8, 9, 11, 12, 12, 15, 16, 15, 19, 23, 23, 26, 28, 30, 34, 37, 38, 44, 48, 48, 56, 62, 63, 72, 77, 82, 92, 96, 102, 116, 124, 128, 142, 155, 162, 178, 191, 200, 222, 236, 246, 276, 291, 303, 334

Offset: 0

Gus Wiseman, Oct 29 2023

nonn

			The a(n) partitions for n = 1, 8, 14, 17, 16, 20, 21:
  (1)  (5,3)  (9,5)   (9,5,3)   (9,7)      (11,9)      (9,7,5)
       (7,1)  (11,3)  (9,7,1)   (11,5)     (13,7)      (11,7,3)
              (13,1)  (11,5,1)  (13,3)     (17,3)      (11,9,1)
                      (13,3,1)  (15,1)     (19,1)      (13,5,3)
                                (7,5,3,1)  (9,7,3,1)   (13,7,1)
                                           (11,5,3,1)  (15,5,1)
                                                       (17,3,1)

This is the relatively prime case of A000700.
The pairwise coprime version is the odd-part case of A007360.
Allowing even parts gives A078374.
The halved even version is A078374 aerated.
The non-strict version is A366843, with evens A000837.
The complement is counted by the strict case of A366852, with evens A018783.
A000041 counts integer partitions, strict A000009 (also into odds).
A051424 counts pairwise coprime partitions, for odd parts A366853.
A113685 counts partitions by sum of odd parts, rank statistic A366528.
A168532 counts partitions by gcd.
A366842 counts partitions whose odd parts have a common divisor > 1.
Cf. A007359, A047967, A055922, A066208, A116598, A239261, A302697, A337485, A365067, A366845, A366848.

Table[Length[Select[IntegerPartitions[n], And@@OddQ/@#&&UnsameQ@@#&&GCD@@#==1&]],{n,0,30}]

from math import gcd
from sympy.utilities.iterables import partitions
def A366844(n): return sum(1 for p in partitions(n) if all(d==1 for d in p.values()) and all(d&1 for d in p) and gcd(*p)==1) # Chai Wah Wu, Oct 30 2023

More terms from Chai Wah Wu, Oct 30 2023

A366848 Odd numbers whose odd prime indices are relatively prime.

Original entry on oeis.org

55, 85, 155, 165, 187, 205, 253, 255, 275, 295, 335, 341, 385, 391, 415, 425, 451, 465, 485, 495, 527, 545, 561, 595, 605, 615, 635, 649, 697, 713, 715, 737, 745, 759, 765, 775, 785, 799, 803, 825, 885, 895, 913, 935, 943, 955, 1003, 1005, 1023, 1025, 1045

Offset: 1

Gus Wiseman, Nov 01 2023

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 odd prime indices of 345 are {3,9}, which are not relatively prime, so 345 is not in the sequence.
The odd prime indices of 825 are {3,3,5}, which are relatively prime, so 825 is in the sequence
The terms together with their prime indices begin:
    55: {3,5}
    85: {3,7}
   155: {3,11}
   165: {2,3,5}
   187: {5,7}
   205: {3,13}
   253: {5,9}
   255: {2,3,7}
   275: {3,3,5}
   295: {3,17}
   335: {3,19}
   341: {5,11}
   385: {3,4,5}
   391: {7,9}
   415: {3,23}
   425: {3,3,7}
   451: {5,13}
   465: {2,3,11}
   485: {3,25}
   495: {2,2,3,5}

Including even terms and prime indices gives A289509, ones of A289508, counted by A000837.
Including even prime indices gives A302697, counted by A302698.
Including even terms gives A366846, counted by A366850.
For halved even instead of odd prime indices we have A366849.
A000041 counts integer partitions, strict A000009 (also into odds).
A066208 lists numbers with all odd prime indices, even A066207.
A112798 lists prime indices, length A001222, sum A056239.
A257991 counts odd prime indices, even A257992.
A366528 adds up odd prime indices, partition triangle A113685.
A366531 = 2*A366533 adds up even prime indices, triangle A113686/A174713.
Cf. A000720, A018783, A055396, A061395, A087436, A325698, A365067, A366842, A366843, A366844, A366847.

Select[Range[1000], OddQ[#]&&GCD@@Select[PrimePi/@First/@FactorInteger[#], OddQ]==1&]

A366850 Number of integer partitions of n whose odd parts are relatively prime.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 7, 11, 16, 22, 32, 43, 60, 80, 110, 140, 194, 244, 327, 410, 544, 670, 883, 1081, 1401, 1708, 2195, 2651, 3382, 4069, 5129, 6157, 7708, 9194, 11438, 13599, 16788, 19911, 24432, 28858, 35229, 41507, 50359, 59201, 71489, 83776, 100731, 117784

Offset: 0

Gus Wiseman, Oct 28 2023

nonn

			The a(1) = 1 through a(8) = 16 partitions:
  (1)  (11)  (21)   (31)    (41)     (51)      (61)       (53)
             (111)  (211)   (221)    (321)     (331)      (71)
                    (1111)  (311)    (411)     (421)      (431)
                            (2111)   (2211)    (511)      (521)
                            (11111)  (3111)    (2221)     (611)
                                     (21111)   (3211)     (3221)
                                     (111111)  (4111)     (3311)
                                               (22111)    (4211)
                                               (31111)    (5111)
                                               (211111)   (22211)
                                               (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

For all parts (not just odd) we have A000837, complement A018783.
The complement is counted by A366842.
These partitions have ranks A366846.
A000041 counts integer partitions, strict A000009 (also into odds).
A000740 counts relatively prime compositions.
A078374 counts relatively prime strict partitions.
A113685 counts partitions by sum of odd parts, rank statistic A366528.
A168532 counts partitions by gcd.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
Cf. A007359, A047967, A051424, A066208, A116598, A365067, A366843, A366844, A366845, A366848.

Table[Length[Select[IntegerPartitions[n],GCD@@Select[#,OddQ]==1&]],{n,0,30}]

A366852 Number of integer partitions of n into odd parts with a common divisor > 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 2, 1, 1, 4, 0, 1, 4, 1, 2, 6, 1, 1, 6, 3, 1, 8, 2, 1, 13, 1, 0, 13, 1, 7, 15, 1, 1, 19, 6, 1, 25, 1, 2, 33, 1, 1, 32, 5, 10, 39, 2, 1, 46, 14, 6, 55, 1, 1, 77, 1, 1, 82, 0, 20, 92, 1, 2, 105, 31, 1, 122, 1, 1, 166, 2, 16, 168

Offset: 0

Gus Wiseman, Nov 01 2023

nonn

			The a(n) partitions for n = 3, 9, 15, 21, 25, 27:
(3)  (9)      (15)         (21)             (25)         (27)
     (3,3,3)  (5,5,5)      (7,7,7)          (15,5,5)     (9,9,9)
              (9,3,3)      (9,9,3)          (5,5,5,5,5)  (15,9,3)
              (3,3,3,3,3)  (15,3,3)                      (21,3,3)
                           (9,3,3,3,3)                   (9,9,3,3,3)
                           (3,3,3,3,3,3,3)               (15,3,3,3,3)
                                                         (9,3,3,3,3,3,3)
                                                         (3,3,3,3,3,3,3,3,3)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Allowing even parts gives A018783, complement A000837.
For parts > 1 instead of gcd > 1 we have A087897.
For gcd = 1 instead of gcd > 1 we have A366843.
The strict case is A366750, with evens A303280.
The strict complement is A366844, with evens A078374.
A000041 counts integer partitions, strict A000009 (also into odd parts).
A000700 counts strict partitions into odd parts.
A113685 counts partitions by sum of odd parts, rank statistic A366528.
A168532 counts partitions by gcd.
A366842 counts partitions whose odd parts have a common divisor > 1.
Cf. A007359, A047967, A051424, A055922, A066208, A302698, A337485, A365067, A366845, A366848.

Table[Length[Select[IntegerPartitions[n],And@@OddQ/@#&&GCD@@#>1&]],{n,15}]

from math import gcd
from sympy.utilities.iterables import partitions
def A366852(n): return sum(1 for p in partitions(n) if all(d&1 for d in p) and gcd(*p)>1) # Chai Wah Wu, Nov 02 2023

More terms from Chai Wah Wu, Nov 02 2023

a(0)=0 prepended by Alois P. Heinz, Jan 11 2024

cp's OEIS Frontend

A366528 Sum of odd prime indices of n.

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A239261 Number of partitions of n having (sum of odd parts) = (sum of even parts).

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A174713 Triangle read by rows, A173305 (A000009 shifted down twice) * A174712 (diagonalized variant of A000041).

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A366531 Sum of even prime indices of n.

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A366533 Sum of even prime indices of n divided by 2.

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A366842 Number of integer partitions of n whose odd parts have a common divisor > 1.

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A366844 Number of strict integer partitions of n into odd relatively prime parts.

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A366848 Odd numbers whose odd prime indices are relatively prime.

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