A366528 -id:A366528 - cp's OEIS frontend

A366748 Numbers k such that (sum of odd prime indices of k) = (sum of even prime indices of k).

1, 12, 70, 90, 112, 144, 286, 325, 462, 520, 525, 594, 646, 675, 832, 840, 1045, 1080, 1326, 1334, 1344, 1666, 1672, 1728, 1900, 2142, 2145, 2294, 2465, 2622, 2695, 2754, 3040, 3432, 3465, 3509, 3526, 3900, 3944, 4186, 4255, 4312, 4455, 4845, 4864, 4900, 4982

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 terms together with their prime indices begin:
     1: {}
    12: {1,1,2}
    70: {1,3,4}
    90: {1,2,2,3}
   112: {1,1,1,1,4}
   144: {1,1,1,1,2,2}
   286: {1,5,6}
   325: {3,3,6}
   462: {1,2,4,5}
   520: {1,1,1,3,6}
   525: {2,3,3,4}
   594: {1,2,2,2,5}
   646: {1,7,8}
   675: {2,2,2,3,3}
   832: {1,1,1,1,1,1,6}
   840: {1,1,1,2,3,4}
For example, 525 has prime indices {2,3,3,4}, and 3+3 = 2+4, so 525 is in the sequence.

For prime factors instead of indices we have A019507.
Partitions of this type are counted by A239261.
For count instead of sum we have A325698, distinct A325700.
The LHS (sum of odd prime indices) is A366528, triangle A113685.
The RHS (sum of even prime indices) is A366531, triangle A113686.
These are the positions of zeros in A366749.
A000009 counts partitions into odd parts, ranked by A066208.
A035363 counts partitions into even parts, ranked by A066207.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A257991 counts odd prime indices, even A257992.
A300061 lists numbers with even sum of prime indices, odd A300063.
Cf. A028260, A045931, A106529, A239241, A241638, A365067, A366533.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000], Total[Select[prix[#],OddQ]]==Total[Select[prix[#],EvenQ]]&]

These are numbers k such that A346697(k) = A346698(k).

A366847 Numbers whose halved even prime indices are nonempty and relatively prime.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 91, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168, 171, 174

Offset: 1

Gus Wiseman, Oct 31 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.

Consists of powers of 2 times elements of the odd restriction A366849.

			The even prime indices of 91 are {4,6}, halved {2,3}, which are relatively prime, so 91 is in the sequence.
The prime indices of 665 are {3,4,8}, even {4,8}, halved {2,4}, which are not relatively prime, so 665 is not in the sequence.
The terms together with their prime indices begin:
    3: {2}
    6: {1,2}
    9: {2,2}
   12: {1,1,2}
   15: {2,3}
   18: {1,2,2}
   21: {2,4}
   24: {1,1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   33: {2,5}
   36: {1,1,2,2}
   39: {2,6}
   42: {1,2,4}
   45: {2,2,3}
   48: {1,1,1,1,2}

Including odd indices gives A289509, ones of A289508, counted by A000837.
The complement including odd indices is A318978, counted by A018783.
The partitions with these ranks are counted by A366845.
A version for odd indices A366846, counted by A366850.
The odd restriction is A366849.
A000041 counts integer partitions, strict A000009 (also into odds).
A035363 counts partitions into all even parts, ranks A066207.
A112798 lists prime indices, length A001222, sum A056239.
A162641 counts even prime exponents, odd A162642.
A257992 counts even prime indices, odd A257991.
A366528 adds up odd prime indices, partition triangle A113685.
A366531 = 2*A366533 adds up even prime indices, triangle A113686/A174713.
Cf. A000720, A055396, A061395, A066208, A168532, A302696, A302697, A325698, A366842, A366843, A366844, A366848.

Select[Range[100],GCD@@Select[PrimePi/@First/@FactorInteger[#],EvenQ]/2==1&]

A366527 Number of integer partitions of 2n containing at least one even part.

Original entry on oeis.org

0, 1, 3, 7, 16, 32, 62, 113, 199, 339, 563, 913, 1453, 2271, 3496, 5308, 7959, 11798, 17309, 25151, 36225, 51748, 73359, 103254, 144363, 200568, 277007, 380437, 519715, 706412, 955587, 1286762, 1725186, 2303388, 3063159, 4058041, 5356431, 7045454, 9235841

Offset: 0

Gus Wiseman, Oct 16 2023

nonn

Also partitions of 2n with even product.

			The a(1) = 1 through a(4) = 16 partitions:
  (2)  (4)    (6)      (8)
       (22)   (42)     (44)
       (211)  (222)    (62)
              (321)    (332)
              (411)    (422)
              (2211)   (431)
              (21111)  (521)
                       (611)
                       (2222)
                       (3221)
                       (4211)
                       (22211)
                       (32111)
                       (41111)
                       (221111)
                       (2111111)

This is the even bisection of A047967.
For odd instead of even parts we have A182616, ranks A366321 or A366528.
These partitions have ranks A366529, subset of A324929.
A000041 counts integer partitions, strict A000009.
A006477 counts partitions w/ at least one odd and even part, ranks A366532.
A086543 counts partitions of n not containing n/2, ranks A366319.
A086543 counts partitions w/o odds, ranks A366322, even bisection A182616.
Cf. A001255, A006827, A035363, A064914, A078408, A086543, A231429, A304710, A365828.

Table[Length[Select[IntegerPartitions[2n],Or@@EvenQ/@#&]],{n,0,15}]

a(n) = A000041(2n) - A000009(2n).

A366839 Sum of even prime factors of 2n, counted with multiplicity.

Original entry on oeis.org

2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 10, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 12, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 10, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 14, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 10, 2, 4, 2, 6, 2

Offset: 1

Gus Wiseman, Oct 25 2023

nonn

			The prime factors of 2*60 are {2,2,2,3,5}, of which the even factors are {2,2,2}, so a(60) = 6.

A compound version is A001414, triangle A331416.
Dividing by 2 gives A001511.
Positions of 2's are A005408.
For count instead of sum we have A007814, odd version A087436.
The partition triangle for this statistic is A116598 aerated.
For indices we have A366531, halved A366533, triangle A113686/A174713.
The odd version is A366840.
A019507 lists numbers with (even factor sum) = (odd factor sum).
A066207 lists numbers with all even prime indices, counted by A035363.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A162641 counts even prime exponents, odd A162642.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
A257992 counts even prime indices, odd A257991.
A366528 adds up odd prime indices, triangle A113685 (without zeros A365067).
Cf. A000720, A066208, A258117, A325698.

Table[2*Length[NestWhileList[#/2&,n,EvenQ]],{n,100}]

a(n) = 2 * valuation(n, 2) + 2; \\ Amiram Eldar, Sep 13 2024

a(n) = 2*A001511(n).
a(n) = A100006(n) - A366840(2n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4. - Amiram Eldar, Sep 13 2024

A366840 Sum of odd prime factors of n, counted with multiplicity.

Original entry on oeis.org

0, 0, 3, 0, 5, 3, 7, 0, 6, 5, 11, 3, 13, 7, 8, 0, 17, 6, 19, 5, 10, 11, 23, 3, 10, 13, 9, 7, 29, 8, 31, 0, 14, 17, 12, 6, 37, 19, 16, 5, 41, 10, 43, 11, 11, 23, 47, 3, 14, 10, 20, 13, 53, 9, 16, 7, 22, 29, 59, 8, 61, 31, 13, 0, 18, 14, 67, 17, 26, 12, 71, 6

Offset: 1

Gus Wiseman, Oct 27 2023

Contains all positive integers except 1, 2, 4.

			The prime factors of 60 are {2,2,2,3,5}, of which the odd factors are {3,5}, so a(60) = 8.

The compound version is A001414, triangle A331416.
For count instead of sum we have A087436, even version A007814.
Odd-indexed terms are A100005.
Positions of odd terms are A335657, even A036349.
For prime indices we have A366528, triangle A113685 (without zeros A365067)
The even version is A366839 = 2*A001511.
The partition triangle for this statistic is A366851, even version A116598.
A019507 lists numbers with (even factor sum) = (odd factor sum).
A066207 lists numbers with all even prime indices, counted by A035363.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A162641 counts even prime exponents, odd A162642.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
A257992 counts even prime indices, odd A257991.
Cf. A000009, A066208, A113686, A174713, A258117, A325698, A366531, A366850.

Table[Total[Times@@@DeleteCases[If[n==1,{}, FactorInteger[n]],{2,_}]],{n,100}]

a(n) = my(f=factor(n), j=if(n%2, 1, 2)); sum(i=j, #f~, f[i,1]*f[i,2]); \\ Michel Marcus, Oct 30 2023

a(n) = A100006(n) - A366839(n).
a(2n) = a(n).
a(2n-1) = A001414(2n-1) = A100005(n).
Completely additive with a(2^e) = 0 and a(p^e) = e*p for an odd prime p. - Amiram Eldar, Nov 03 2023

A366725 Sum of odd indices of distinct prime factors of n.

Original entry on oeis.org

0, 1, 0, 1, 3, 1, 0, 1, 0, 4, 5, 1, 0, 1, 3, 1, 7, 1, 0, 4, 0, 6, 9, 1, 3, 1, 0, 1, 0, 4, 11, 1, 5, 8, 3, 1, 0, 1, 0, 4, 13, 1, 0, 6, 3, 10, 15, 1, 0, 4, 7, 1, 0, 1, 8, 1, 0, 1, 17, 4, 0, 12, 0, 1, 3, 6, 19, 8, 9, 4, 0, 1, 21, 1, 3, 1, 5, 1, 0, 4, 0, 14, 23, 1, 10, 1, 0, 6, 0, 4, 0, 10, 11, 16, 3, 1, 25, 1, 5, 4

Offset: 1

Ilya Gutkovskiy, Oct 24 2023

nonn

			a(60) = 4 because 60 = 2^2 * 3 * 5 = prime(1)^2 * prime(2) * prime(3) and 1 + 3 = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000720 (pi), A066207 (positions of 0's), A066328, A324966, A332422, A344908, A366528, A366784.

nmax = 100; CoefficientList[Series[Sum[(2 k - 1) x^Prime[2 k - 1]/(1 - x^Prime[2 k - 1]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := Module[{i = PrimePi[p]}, If[OddQ[i], i, 0]]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 03 2025 *)

f(n) = if(n % 2, n, 0);
a(n) = vecsum(apply(x -> f(primepi(x)), factor(n)[, 1])); \\ Amiram Eldar, Jul 03 2025

G.f.: Sum_{k>=1} (2*k-1) * x^prime(2*k-1) / (1 - x^prime(2*k-1)).
From Amiram Eldar, Jul 03 2025: (Start)
Additive with a(p^e) = pi(p) if pi(p) is odd, and 0 otherwise.
a(n) = A066328(n) - 2*A366784(n). (End)

A366750 Number of strict integer partitions of n into odd parts with a common divisor > 1.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 2, 1, 0, 2, 1, 1, 3, 1, 0, 2, 0, 1, 3, 1, 0, 3, 2, 1, 4, 1, 1, 5, 0, 1, 5, 1, 2, 5, 1, 1, 5, 2, 2, 6, 0, 1, 9, 1, 0, 9, 0, 3, 9, 1, 1, 9, 5, 1, 11, 1, 0, 15, 1, 2, 13, 1, 5, 14, 0, 1, 18

Offset: 0

Gus Wiseman, Nov 01 2023

nonn

			The a(n) partitions for n = 3, 24, 30, 42, 45, 57, 60:
  (3)  (15,9)  (21,9)  (33,9)   (45)       (57)       (51,9)
       (21,3)  (25,5)  (35,7)   (33,9,3)   (45,9,3)   (55,5)
               (27,3)  (39,3)   (21,15,9)  (27,21,9)  (57,3)
                       (27,15)  (25,15,5)  (33,15,9)  (33,27)
                                (27,15,3)  (33,21,3)  (35,25)
                                           (39,15,3)  (39,21)
                                                      (45,15)
                                                      (27,21,9,3)
                                                      (33,15,9,3)

This is the case of A000700 with a common divisor.
Including evens gives A303280.
The complement is counted by A366844, non-strict version A366843.
The non-strict version is 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.
Cf. A000837, A007360, A055922, A066208, A078374, A302697, A337485, A366842, 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 A366750(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, Nov 02 2023

More terms from Chai Wah Wu, Nov 02 2023

A366851 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n such that the sum of primes indexed by all parts greater than one is k.

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 2, 0, 2, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 2, 2, 2, 2, 1, 1, 0, 0, 1

Offset: 0

Gus Wiseman, Nov 01 2023

To illustrate the definition, the sum of primes indexed by all parts greater than one of the partition (5,2,2,1) is prime(5) + prime(2) + prime(2) = 17.

			Triangle begins:
  1
  1
  1 0 0 1
  1 0 0 1 0 1
  1 0 0 1 0 1 1 1
  1 0 0 1 0 1 1 1 1 0 0 1
  1 0 0 1 0 1 1 1 1 1 2 1 0 1
  1 0 0 1 0 1 1 1 1 1 2 2 1 1 1 0 0 1
  1 0 0 1 0 1 1 1 1 1 2 2 2 3 2 0 2 1 0 1
  1 0 0 1 0 1 1 1 1 1 2 2 2 3 3 2 2 2 2 1 1 0 0 1
  1 0 0 1 0 1 1 1 1 1 2 2 2 3 3 3 4 4 2 3 2 0 3 1 0 0 0 0 0 1
  1 0 0 1 0 1 1 1 1 1 2 2 2 3 3 3 4 5 4 4 3 3 3 2 3 0 1 0 0 1 0 1
The T(8,13) = 3 partitions are: (6,1,1), (4,2,2), (3,3,2).
The T(10,17) = 4 partitions are: (7,1,1,1), (5,2,2,1), (4,4,2), (4,3,3).

Row lengths are A055670.
Columns appear to converge to A099773.
A bisected even version is A116598 (counts partitions by number of 1's).
Counting all parts (not just > 1) gives A331416, shifted A331385.
A000041 counts integer partitions, strict A000009 (also into odds).
A113685 counts partitions by sum of odd parts, rank statistic A366528.
A330953 counts partitions with Heinz number divisible by sum of primes.
A331381 counts partitions with (product)|(sum of primes), equality A331383.
Cf. A000837, A014689, A113686, A239261, A331417, A331418, A365067, A366842.

Table[Length[Select[IntegerPartitions[n], Total[Select[Prime/@#,OddQ]]==k&]], {n,0,10}, {k,0,If[n<=1,0,Prime[n]]}]

A366841 Least positive integer whose odd prime factors sum to n, starting with n = 5.

Original entry on oeis.org

5, 9, 7, 15, 27, 21, 11, 35, 13, 33, 105, 39, 17, 65, 19, 51, 195, 57, 23, 95, 171, 69, 285, 115, 29, 161, 31, 87, 483, 93, 261, 155, 37, 217, 465, 111, 41, 185, 43, 123, 555, 129, 47, 215, 387, 141, 645, 235, 53, 329, 705, 159, 987, 265, 59, 371, 61, 177

Offset: 5

Gus Wiseman, Oct 27 2023

nonn

All terms are odd.

It seems that all composite terms not divisible by 3 form a supersequence of A292081. - Ivan N. Ianakiev, Oct 30 2023

			The terms together with their prime factors (which sum to n) begin:
    5 = 5
    9 = 3*3
    7 = 7
   15 = 3*5
   27 = 3*3*3
   21 = 3*7
   11 = 11
   35 = 5*7
   13 = 13
   33 = 3*11
  105 = 3*5*7

Michel Marcus, Table of n, a(n) for n = 5..1000

This is the odd case of A056240.
Positions of first appearances in A366840 (sum of odd prime factors).
The partition triangle for this statistic is A366851, even A116598.
A001414 adds up prime factors, triangle A331416.
A019507 lists numbers with (even factor sum) = (odd factor sum).
A027746 lists prime factors, length A001222.
A087436 counts odd prime factors, even A007814.
A366528 adds up odd prime indices, triangle A113685 (without zeros A365067).
Cf. A031368, A036349, A056239, A066208, A100005, A112798, A239261, A257991, A335657, A366839.

nn=1000;
w=Table[Total[Times@@@DeleteCases[If[n==1,{},FactorInteger[n]],{2,_}]],{n,nn}];
spnm[y_]:=Max@@Select[Union[y],Function[i,Union[Select[y,#<=i&]]==Range[i]]];
Table[Position[w,k][[1,1]],{k,5,spnm[Join[{1,2,3,4},Take[w,nn]/.(0->1)]]}]

f(n) = my(f=factor(n), j=if (n%2, 1, 2)); sum(i=j, #f~, f[i,1]*f[i,2]); \\ A366840
a(n) = my(k=1); while (f(k) != n, k++); k; \\ Michel Marcus, Nov 02 2023

cp's OEIS Frontend

A366748 Numbers k such that (sum of odd prime indices of k) = (sum of even prime indices of k).

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A366847 Numbers whose halved even prime indices are nonempty and relatively prime.

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A366527 Number of integer partitions of 2n containing at least one even part.

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A366839 Sum of even prime factors of 2n, counted with multiplicity.

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A366840 Sum of odd prime factors of n, counted with multiplicity.

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A366725 Sum of odd indices of distinct prime factors of n.

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A366750 Number of strict integer partitions of n into odd parts with a common divisor > 1.

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A366851 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n such that the sum of primes indexed by all parts greater than one is k.

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