A031215 -id:A031215 - cp's OEIS frontend

A372591 Numbers whose binary weight (A000120) plus bigomega (A001222) is even.

2, 6, 7, 8, 9, 10, 11, 13, 15, 19, 24, 28, 31, 32, 33, 34, 36, 37, 39, 40, 41, 42, 44, 46, 47, 50, 51, 52, 54, 57, 58, 59, 60, 61, 65, 67, 70, 73, 76, 77, 79, 85, 86, 90, 95, 96, 97, 98, 103, 106, 107, 109, 110, 111, 112, 117, 119, 123, 124, 126, 127, 128, 129

Offset: 1

Gus Wiseman, May 14 2024

nonn

The odd version is A372590.

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
          {2}   2  (1)
        {2,3}   6  (2,1)
      {1,2,3}   7  (4)
          {4}   8  (1,1,1)
        {1,4}   9  (2,2)
        {2,4}  10  (3,1)
      {1,2,4}  11  (5)
      {1,3,4}  13  (6)
    {1,2,3,4}  15  (3,2)
      {1,2,5}  19  (8)
        {4,5}  24  (2,1,1,1)
      {3,4,5}  28  (4,1,1)
  {1,2,3,4,5}  31  (11)
          {6}  32  (1,1,1,1,1)
        {1,6}  33  (5,2)
        {2,6}  34  (7,1)
        {3,6}  36  (2,2,1,1)
      {1,3,6}  37  (12)
    {1,2,3,6}  39  (6,2)
        {4,6}  40  (3,1,1,1)
      {1,4,6}  41  (13)
      {2,4,6}  42  (4,2,1)

For sum (A372428, zeros A372427) we have A372587, complement A372586.
For minimum (A372437) we have A372440, complement A372439.
Positions of even terms in A372441, zeros A071814.
For maximum (A372442, zeros A372436) we have A372589, complement A372588.
The complement is A372590.
For just binary indices:
- length: A001969, complement A000069
- sum: A158704, complement A158705
- minimum:  A036554, complement A003159
- maximum: A053754, complement A053738
For just prime indices:
- length: A026424 A028260 (count A027187), complement (count A027193)
- sum: A300061 (count A058696), complement A300063 (count A058695)
- minimum: A340933 (count A026805), complement A340932 (count A026804)
- maximum: A244990 (count A027187), complement A244991 (count A027193)
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A031215 lists even-indexed primes, odd A031368.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A006141, A066207, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

Select[Range[100],EvenQ[DigitCount[#,2,1]+PrimeOmega[#]]&]

A247503 Completely multiplicative with a(prime(n)) = prime(n)^(n mod 2).

Original entry on oeis.org

1, 2, 1, 4, 5, 2, 1, 8, 1, 10, 11, 4, 1, 2, 5, 16, 17, 2, 1, 20, 1, 22, 23, 8, 25, 2, 1, 4, 1, 10, 31, 32, 11, 34, 5, 4, 1, 2, 1, 40, 41, 2, 1, 44, 5, 46, 47, 16, 1, 50, 17, 4, 1, 2, 55, 8, 1, 2, 59, 20, 1, 62, 1, 64, 5, 22, 67, 68, 23, 10, 1, 8, 73, 2, 25, 4

Offset: 1

Tom Edgar, Mar 03 2015

To compute a(n) replace even-indexed primes in the prime factorization of n by 1.
a(p) = p if p is in A031368.
a(p) = 1 if p is in A031215.

			Since 10 = 2*5, 2 = prime(1), and 5 = prime(3), a(10) = 2*5 = 10.
Since 9 = 3^2 and 3 is an even-indexed prime, 3 = prime(2), then a(9) = 1^2 = 1.
Since 30 = 2*3*5, 2 = prime(1), 3 = prime(2), and 5 = prime(3), we see that a(30) = 2*1*5 = 10.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index to divisibility sequences

Cf. A031368, A031215, A248101, A001222, A027746, A049084.

First 28 terms are the same as A343430.

a247503 = product . filter (odd . a049084) . a027746_row
-- Reinhard Zumkeller, Mar 06 2015

f[n_] := Block[{a, g, pf = FactorInteger@ n}, a = PrimePi[First /@ pf]; g[x_] := If[EvenQ@ x, 1, Prime@ x]; Times @@ Power @@@ Transpose@ {g /@ a, Last /@ pf}]; Array[f, 120] (* Michael De Vlieger, Mar 03 2015 *)
Array[Times @@ (FactorInteger[#] /. {p_, e_} /; e > 0 :> (p^Mod[PrimePi@ p, 2])^e) &, 76] (* Michael De Vlieger, Apr 05 2017 *)

a(n) = {f = factor(n); for (i=1, #f~, f[i,2] *= (primepi(f[i,1]) % 2);); factorback(f);} \\ Michel Marcus, Mar 03 2015

from math import prod
from sympy import factorint, primepi
def A247503(n): return prod(p**e for p, e in factorint(n).items() if primepi(p)&1) # Chai Wah Wu, Dec 26 2022

n=100; oddIndexPrimes=[primes_first_n(2*n+1)[2*i] for i in [0..n]]
[prod([(x[0]^(x[0] in oddIndexPrimes))^x[1] for x in factor(n)]) for n in [1..n]]

When n = Product_{k>=1} prime(k)^r_k, a(n) = Product_{k>=1} prime(k)^(r_k*(k mod 2)).
a(n) = n/A248101(n).
a(n) = Product_{k = 1..A001222(n)} A027746(n,k) and A049084(A027746(n,k)) is odd). - Reinhard Zumkeller, Mar 06 2015

A340933 Numbers whose least prime index is even. Heinz numbers of integer partitions whose last part is even.

Original entry on oeis.org

3, 7, 9, 13, 15, 19, 21, 27, 29, 33, 37, 39, 43, 45, 49, 51, 53, 57, 61, 63, 69, 71, 75, 77, 79, 81, 87, 89, 91, 93, 99, 101, 105, 107, 111, 113, 117, 119, 123, 129, 131, 133, 135, 139, 141, 147, 151, 153, 159, 161, 163, 165, 169, 171, 173, 177, 181, 183

Offset: 1

Gus Wiseman, Feb 12 2021

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. 1 has no prime indices so is not counted.

			The sequence of terms together with their prime indices begins:
      3: {2}         51: {2,7}         99: {2,2,5}
      7: {4}         53: {16}         101: {26}
      9: {2,2}       57: {2,8}        105: {2,3,4}
     13: {6}         61: {18}         107: {28}
     15: {2,3}       63: {2,2,4}      111: {2,12}
     19: {8}         69: {2,9}        113: {30}
     21: {2,4}       71: {20}         117: {2,2,6}
     27: {2,2,2}     75: {2,3,3}      119: {4,7}
     29: {10}        77: {4,5}        123: {2,13}
     33: {2,5}       79: {22}         129: {2,14}
     37: {12}        81: {2,2,2,2}    131: {32}
     39: {2,6}       87: {2,10}       133: {4,8}
     43: {14}        89: {24}         135: {2,2,2,3}
     45: {2,2,3}     91: {4,6}        139: {34}
     49: {4,4}       93: {2,11}       141: {2,15}

These partitions are counted by A026805.
Looking at length or at maximum gives A028260/A244990, counted by A027187.
If all prime indices are even we get A066207, counted by A035363.
The complement is {1} \/ A340932, counted by A026804.
A001222 counts prime factors.
A005843 lists even numbers.
A031215 lists even-indexed primes.
A055396 selects least prime index.
A056239 adds up prime indices.
A058695 counts partitions of even numbers, ranked by A300061.
A061395 selects greatest prime index.
A112798 lists the prime indices of each positive integer.
Cf. A006141, A030229, A067661, A101708, A236913,  A339846, A340601, A340602, A340605, A340854/A340855.

Select[Range[2,100],EvenQ[PrimePi[FactorInteger[#][[1,1]]]]&]

A055396(a(n)) belongs to A005843.

Closed under multiplication.

A372588 Numbers k > 1 such that (greatest binary index of k) + (greatest prime index of k) is odd.

Original entry on oeis.org

2, 6, 7, 8, 10, 11, 15, 18, 19, 21, 24, 26, 27, 28, 29, 32, 33, 34, 40, 41, 44, 45, 46, 47, 50, 51, 55, 59, 60, 62, 65, 70, 71, 72, 74, 76, 78, 79, 81, 84, 86, 87, 89, 91, 95, 96, 98, 101, 104, 105, 106, 107, 108, 111, 112, 113, 114, 116, 117, 122, 126, 128

Offset: 1

Gus Wiseman, May 14 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
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 even version is A372589.

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
        {2}   2  (1)
      {2,3}   6  (2,1)
    {1,2,3}   7  (4)
        {4}   8  (1,1,1)
      {2,4}  10  (3,1)
    {1,2,4}  11  (5)
  {1,2,3,4}  15  (3,2)
      {2,5}  18  (2,2,1)
    {1,2,5}  19  (8)
    {1,3,5}  21  (4,2)
      {4,5}  24  (2,1,1,1)
    {2,4,5}  26  (6,1)
  {1,2,4,5}  27  (2,2,2)
    {3,4,5}  28  (4,1,1)
  {1,3,4,5}  29  (10)
        {6}  32  (1,1,1,1,1)
      {1,6}  33  (5,2)
      {2,6}  34  (7,1)
      {4,6}  40  (3,1,1,1)
    {1,4,6}  41  (13)
    {3,4,6}  44  (5,1,1)
  {1,3,4,6}  45  (3,2,2)

For sum (A372428, zeros A372427) we have A372586.
For minimum (A372437) we have A372439, complement A372440.
For length (A372441, zeros A071814) we have A372590, complement A372591.
Positions of odd terms in A372442, zeros A372436.
The complement is A372589.
For just binary indices:
- length: A000069, complement A001969
- sum: A158705, complement A158704
- minimum: A003159, complement A036554
- maximum: A053738, complement A053754
For just prime indices:
- length: A026424 (count A027193), complement A028260 (count A027187)
- sum: A300063 (count A058695), complement A300061 (count A058696)
- minimum: A340932 (count A026804), complement A340933 (count A026805)
- maximum: A244991 (count A027193), complement A244990 (count A027187)
A005408 lists odd numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A031368 lists odd-indexed primes, even A031215.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A006141, A066208, A160786, A243055, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

Select[Range[2,100],OddQ[IntegerLength[#,2]+PrimePi[FactorInteger[#][[-1,1]]]]&]

Numbers k such that A070939(k) + A061395(k) is odd.

A248101 Completely multiplicative with a(prime(n)) = prime(n)^((n+1) mod 2).

Original entry on oeis.org

1, 1, 3, 1, 1, 3, 7, 1, 9, 1, 1, 3, 13, 7, 3, 1, 1, 9, 19, 1, 21, 1, 1, 3, 1, 13, 27, 7, 29, 3, 1, 1, 3, 1, 7, 9, 37, 19, 39, 1, 1, 21, 43, 1, 9, 1, 1, 3, 49, 1, 3, 13, 53, 27, 1, 7, 57, 29, 1, 3, 61, 1, 63, 1, 13, 3, 1, 1, 3, 7, 71, 9, 1, 37, 3, 19, 7, 39, 79

Offset: 1

Tom Edgar, Mar 03 2015

To compute a(n) replace odd-indexed primes in the prime factorization of n by 1.
a(p) = p if p is in A031215.
a(p) = 1 if p is in A031368.

			Since 10 = 2*5, 2 = prime(1), and 5 = prime(3), a(10) = 1*1 = 1.
Since 9 = 3^2 and 3 is an even-indexed prime, 3 = prime(2), then a(9) = 3^2 = 9.
Since 35 = 5*7, 5 = prime(3), and 7 = prime(4), we see that a(35) = 1*7 = 7.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index to divisibility sequences

Cf. A031368, A031215, A247503, A001222, A027746, A049084.

a248101 = product . filter (even . a049084) . a027746_row
-- Reinhard Zumkeller, Mar 06 2015

f[n_] := Block[{a, g, pf = FactorInteger@ n}, a = PrimePi[First /@ pf]; g[x_] := If[Or[OddQ@ x, x == 0], 1, Prime@ x]; Times @@ Power @@@ Transpose@ {g /@ a, Last /@ pf}]; Array[f, 120] (* Michael De Vlieger, Mar 03 2015 *)
Array[Times @@ (FactorInteger[#] /. {p_, e_} /; e > 0 :> (p^Mod[PrimePi@ p + 1, 2])^e) &, 79] (* Michael De Vlieger, Apr 05 2017 *)

a(n) = {f = factor(n); for (i=1, #f~, f[i,2] *= (primepi(f[i,1])+1) % 2;); factorback(f);} \\ Michel Marcus, Mar 03 2015

n=100; evenIndexPrimes=[primes_first_n(2*n+2)[2*i+1] for i in [0..n]]
[prod([(x[0]^(x[0] in evenIndexPrimes))^x[1] for x in factor(n)]) for n in [1..n]]

When n = Product_{k>=1} prime(k)^r_k, a(n) = Product_{k>=1} prime(k)^(r_k*((k+1) mod 2)).
a(n) = n / A247503(n).
a(n) = Product(A027746(n,k): k = 1 .. A001222(n) and A049084(A027746(n,k)) is even). - Reinhard Zumkeller, Mar 06 2015

A332822 One part of a 3-way classification of the positive integers. Numbers n for which A048675(n) == 2 (mod 3).

Original entry on oeis.org

3, 4, 7, 10, 13, 18, 19, 22, 24, 25, 29, 32, 34, 37, 42, 43, 45, 46, 53, 55, 56, 60, 61, 62, 71, 78, 79, 80, 81, 82, 85, 89, 94, 98, 99, 101, 104, 105, 107, 108, 113, 114, 115, 118, 121, 131, 132, 134, 139, 140, 144, 146, 150, 151, 152, 153, 155, 163, 166, 173, 174, 176, 181, 182, 187, 189, 192, 193, 194, 195, 199, 200, 204

Offset: 1

Antti Karttunen and Peter Munn, Feb 25 2020

nonn

The positive integers are partitioned between A332820, A332821 and this sequence.
For each prime p, the terms include exactly one of p and p^2. The primes alternate between this sequence and A332821. This sequence has the primes with even indexes, those in A031215.
The terms are the even numbers in A332820 halved. The terms are also the numbers m such that 5m is in A332820, and so on for alternate primes: 11, 17, 23 etc. Likewise, the terms are the numbers m such that 3m is in A332821, and so on for alternate primes: 7, 13, 19, 29 etc.
If we take each odd term of this sequence and replace each prime in its factorization by the next smaller prime, we get the same set of numbers as we get from halving the even terms of this sequence, and A332821 consists exactly of those numbers. The numbers that are one third of the terms that are multiples of 3 are in A332820, which consists exactly of those numbers. The numbers that are one fifth of the terms that are multiples of 5 constitute A332821, and for larger primes, an alternating pattern applies as described in the previous paragraph.
The product of any 2 terms of this sequence is in A332821, the product of any 3 terms is in A332820, and the product of a term of A332820 and a term of this sequence is in this sequence. So if a number k is present, k^2 is in A332821, k^3 is in A332820, and k^4 is in this sequence.
If k is an even number, exactly one of {k/2, k, 2k} is in the sequence (cf. A191257 / A067368 / A213258); and generally if k is a multiple of a prime p, exactly one of {k/p, k, k*p} is in the sequence.

Positions of terms valued -1 in A332823; equivalently, numbers in row 3k-1 of A277905 for some k >= 1.
Cf. A048675, A332820, A332821.
Comparable 2 or 3-way classifications: A000379/A000028, A001969/A000069, A003159/A036554, A005843/A005408, A028260/A026424, A191257/A067368/A213258, A325431/A325432, A329609/A329604/A332812.
Subsequences: intersection of A026478 and A066207, A031215 (prime terms), A033430\{0}, A117642\{0}, A169604, A244727\{0}, A244729\{0}, A338910 (semiprime terms).

Select[Range@ 204, Mod[Total@ #, 3] == 2 &@ Map[#[[-1]]*2^(PrimePi@ #[[1]] - 1) &, FactorInteger[#]] &] (* Michael De Vlieger, Mar 15 2020 *)

isA332822(n) =  { my(f = factor(n)); (2==((sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3)); };

{a(n) : n >= 1} = {2 * A332821(k) : k >= 1} U {A003961(A332821(k)) : k >= 1}.

{a(n) : n >= 1} = {A332821(k)^2 : k >= 1} U {A331590(2, A332821(k)) : k >= 1}.

A372586 Numbers k such that (sum of binary indices of k) + (sum of prime indices of k) is odd.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 12, 15, 16, 17, 20, 21, 29, 32, 36, 42, 43, 45, 46, 47, 48, 51, 53, 54, 55, 59, 60, 61, 63, 64, 65, 66, 67, 68, 71, 73, 78, 79, 80, 81, 84, 89, 91, 93, 94, 95, 97, 99, 101, 105, 110, 111, 113, 114, 115, 116, 118, 119, 121, 122, 125, 127

Offset: 1

Gus Wiseman, May 14 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
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 even version is A372587.

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
            {1}   1  ()
            {2}   2  (1)
          {1,2}   3  (2)
            {3}   4  (1,1)
          {1,3}   5  (3)
            {4}   8  (1,1,1)
          {1,4}   9  (2,2)
          {3,4}  12  (2,1,1)
      {1,2,3,4}  15  (3,2)
            {5}  16  (1,1,1,1)
          {1,5}  17  (7)
          {3,5}  20  (3,1,1)
        {1,3,5}  21  (4,2)
      {1,3,4,5}  29  (10)
            {6}  32  (1,1,1,1,1)
          {3,6}  36  (2,2,1,1)
        {2,4,6}  42  (4,2,1)
      {1,2,4,6}  43  (14)
      {1,3,4,6}  45  (3,2,2)
      {2,3,4,6}  46  (9,1)
    {1,2,3,4,6}  47  (15)
          {5,6}  48  (2,1,1,1,1)

Positions of odd terms in A372428, zeros A372427.
For minimum (A372437) we have A372439, complement A372440.
For length (A372441, zeros A071814) we have A372590, complement A372591.
For maximum (A372442, zeros A372436) we have A372588, complement A372589.
The complement is A372587.
For just binary indices:
- length: A000069, complement A001969
- sum: A158705, complement A158704
- minimum: A003159, complement A036554
- maximum: A053738, complement A053754
For just prime indices:
- length: A026424 (count A027193), complement A028260 (count A027187)
- sum: A300063 (count A058695), complement A300061 (count A058696)
- minimum: A340932 (count A026804), complement A340933 (count A026805)
- maximum: A244991 (count A027193), complement A244990 (count A027187)
A005408 lists odd numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A031368 lists odd-indexed primes, even A031215.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A066208, A160786, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],OddQ[Total[bix[#]]+Total[prix[#]]]&]

Numbers k such that A029931(k) + A056239(k) is odd.

A372589 Numbers k > 1 such that (greatest binary index of k) + (greatest prime index of k) is even.

Original entry on oeis.org

3, 4, 5, 9, 12, 13, 14, 16, 17, 20, 22, 23, 25, 30, 31, 35, 36, 37, 38, 39, 42, 43, 48, 49, 52, 53, 54, 56, 57, 58, 61, 63, 64, 66, 67, 68, 69, 73, 75, 77, 80, 82, 83, 85, 88, 90, 92, 93, 94, 97, 99, 100, 102, 103, 109, 110, 115, 118, 119, 120, 121, 123, 124

Offset: 1

Gus Wiseman, May 14 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
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 version is A372588.

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
        {1,2}   3  (2)
          {3}   4  (1,1)
        {1,3}   5  (3)
        {1,4}   9  (2,2)
        {3,4}  12  (2,1,1)
      {1,3,4}  13  (6)
      {2,3,4}  14  (4,1)
          {5}  16  (1,1,1,1)
        {1,5}  17  (7)
        {3,5}  20  (3,1,1)
      {2,3,5}  22  (5,1)
    {1,2,3,5}  23  (9)
      {1,4,5}  25  (3,3)
    {2,3,4,5}  30  (3,2,1)
  {1,2,3,4,5}  31  (11)
      {1,2,6}  35  (4,3)
        {3,6}  36  (2,2,1,1)
      {1,3,6}  37  (12)
      {2,3,6}  38  (8,1)
    {1,2,3,6}  39  (6,2)
      {2,4,6}  42  (4,2,1)
    {1,2,4,6}  43  (14)

For sum (A372428, zeros A372427) we have A372587, complement A372586.
For minimum (A372437) we have A372440, complement A372439.
For length (A372441, zeros A071814) we have A372591, complement A372590.
Positions of even terms in A372442, zeros A372436.
The complement is A372588.
For just binary indices:
- length: A001969, complement A000069
- sum: A158704, complement A158705
- minimum:  A036554, complement A003159
- maximum: A053754, complement A053738
For just prime indices:
- length: A026424 A028260 (count A027187), complement (count A027193)
- sum: A300061 (count A058696), complement A300063 (count A058695)
- minimum: A340933 (count A026805), complement A340932 (count A026804)
- maximum: A244990 (count A027187), complement A244991 (count A027193)
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A031215 lists even-indexed primes, odd A031368.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A006141, A066207, A243055, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

Select[Range[2,100],EvenQ[IntegerLength[#,2]+PrimePi[FactorInteger[#][[-1,1]]]]&]

Numbers k such that A070939(k) + A061395(k) is even.

A372590 Numbers whose binary weight (A000120) plus bigomega (A001222) is odd.

Original entry on oeis.org

1, 3, 4, 5, 12, 14, 16, 17, 18, 20, 21, 22, 23, 25, 26, 27, 29, 30, 35, 38, 43, 45, 48, 49, 53, 55, 56, 62, 63, 64, 66, 68, 69, 71, 72, 74, 75, 78, 80, 81, 82, 83, 84, 87, 88, 89, 91, 92, 93, 94, 99, 100, 101, 102, 104, 105, 108, 113, 114, 115, 116, 118, 120

Offset: 1

Gus Wiseman, May 14 2024

nonn

The even version is A372591.

			The terms (center), their binary indices (left), and their weakly decreasing prime indices (right) begin:
        {1}   1  ()
      {1,2}   3  (2)
        {3}   4  (1,1)
      {1,3}   5  (3)
      {3,4}  12  (2,1,1)
    {2,3,4}  14  (4,1)
        {5}  16  (1,1,1,1)
      {1,5}  17  (7)
      {2,5}  18  (2,2,1)
      {3,5}  20  (3,1,1)
    {1,3,5}  21  (4,2)
    {2,3,5}  22  (5,1)
  {1,2,3,5}  23  (9)
    {1,4,5}  25  (3,3)
    {2,4,5}  26  (6,1)
  {1,2,4,5}  27  (2,2,2)
  {1,3,4,5}  29  (10)
  {2,3,4,5}  30  (3,2,1)
    {1,2,6}  35  (4,3)
    {2,3,6}  38  (8,1)
  {1,2,4,6}  43  (14)
  {1,3,4,6}  45  (3,2,2)

For sum (A372428, zeros A372427) we have A372586, complement A372587.
For minimum (A372437) we have A372439, complement A372440.
Positions of odd terms in A372441, zeros A071814.
For maximum (A372442, zeros A372436) we have A372588, complement A372589.
The complement is A372591.
For just binary indices:
- length: A000069, complement A001969
- sum: A158705, complement A158704
- minimum: A003159, complement A036554
- maximum: A053738, complement A053754
For just prime indices:
- length: A026424 (count A027193), complement A028260 (count A027187)
- sum: A300063 (count A058695), complement A300061 (count A058696)
- minimum: A340932 (count A026804), complement A340933 (count A026805)
- maximum: A244991 (count A027193), complement A244990 (count A027187)
A005408 lists odd numbers.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A031368 lists odd-indexed primes, even A031215.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A066208, A160786, A257991, A300272, A304818, A340604, A341446, A372429-A372433, A372438.

Select[Range[100],OddQ[DigitCount[#,2,1]+PrimeOmega[#]]&]

A077126 Sum of even-indexed primes.

Original entry on oeis.org

3, 10, 23, 42, 71, 108, 151, 204, 265, 336, 415, 504, 605, 712, 825, 956, 1095, 1246, 1409, 1582, 1763, 1956, 2155, 2378, 2607, 2846, 3097, 3360, 3631, 3912, 4205, 4516, 4833, 5170, 5519, 5878, 6251, 6634, 7031, 7440, 7861, 8294, 8737, 9194, 9657, 10136, 10627

Offset: 1

Jon Perry, Nov 29 2002

nonn

Partial sums of A031215. - Michel Marcus, Oct 27 2015

			p_1=2, p_2=3, p_3=5 and p_4=7, therefore a(2) = p_2 + p_4 = 3 + 7 = 10.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A031215, A077131.

A077126list[nmax_]:=Accumulate[Prime[Range[2,2nmax,2]]];A077126list[100] (* Paolo Xausa, Aug 28 2023 *)

pc=1; ps=0; forprime (p=2,500,pc=3-pc; if (pc==1,ps=ps+p; print1(ps",")))

cp's OEIS Frontend

A372591 Numbers whose binary weight (A000120) plus bigomega (A001222) is even.

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A247503 Completely multiplicative with a(prime(n)) = prime(n)^(n mod 2).

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A340933 Numbers whose least prime index is even. Heinz numbers of integer partitions whose last part is even.

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A372588 Numbers k > 1 such that (greatest binary index of k) + (greatest prime index of k) is odd.

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A248101 Completely multiplicative with a(prime(n)) = prime(n)^((n+1) mod 2).

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A332822 One part of a 3-way classification of the positive integers. Numbers n for which A048675(n) == 2 (mod 3).

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A372586 Numbers k such that (sum of binary indices of k) + (sum of prime indices of k) is odd.

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A372589 Numbers k > 1 such that (greatest binary index of k) + (greatest prime index of k) is even.