A037143 -id:A037143 - cp's OEIS frontend

A338911 Numbers of the form prime(x) * prime(y) where x and y are both even.

9, 21, 39, 49, 57, 87, 91, 111, 129, 133, 159, 169, 183, 203, 213, 237, 247, 259, 267, 301, 303, 321, 339, 361, 371, 377, 393, 417, 427, 453, 481, 489, 497, 519, 543, 551, 553, 559, 579, 597, 623, 669, 687, 689, 703, 707, 717, 749, 753, 789, 791, 793, 813, 817

Offset: 1

Gus Wiseman, Nov 20 2020

nonn

			The sequence of terms together with their prime indices begins:
      9: {2,2}     237: {2,22}    481: {6,12}
     21: {2,4}     247: {6,8}     489: {2,38}
     39: {2,6}     259: {4,12}    497: {4,20}
     49: {4,4}     267: {2,24}    519: {2,40}
     57: {2,8}     301: {4,14}    543: {2,42}
     87: {2,10}    303: {2,26}    551: {8,10}
     91: {4,6}     321: {2,28}    553: {4,22}
    111: {2,12}    339: {2,30}    559: {6,14}
    129: {2,14}    361: {8,8}     579: {2,44}
    133: {4,8}     371: {4,16}    597: {2,46}
    159: {2,16}    377: {6,10}    623: {4,24}
    169: {6,6}     393: {2,32}    669: {2,48}
    183: {2,18}    417: {2,34}    687: {2,50}
    203: {4,10}    427: {4,18}    689: {6,16}
    213: {2,20}    453: {2,36}    703: {8,12}

A338910 is the odd instead of even version.
A339004 is the squarefree case.
A001221 counts distinct prime indices.
A001222 counts prime indices.
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A338899, A270650, A270652 list prime indices of squarefree semiprimes.
A289182/A115392 list the positions of odd/even terms of A001358.
A300912 lists semiprimes with relatively prime indices.
A318990 lists semiprimes with divisible indices.
A338904 groups semiprimes by weight.
A338906/A338907 list semiprimes of even/odd weight.
A338909 lists semiprimes with non-relatively prime indices.
A338912 and A338913 list prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
Cf. A005117, A037143, A055684, A056239, A065516, A112798, A128301, A195017, A320655, A320732, A320892, A338898, A339002, A339003.

q:= n-> (l-> add(i[2], i=l)=2 and andmap(i->
    numtheory[pi](i[1])::even, l))(ifactors(n)[2]):
select(q, [$1..1000])[];  # Alois P. Heinz, Nov 23 2020

Select[Range[100],PrimeOmega[#]==2&&OddQ[Times@@(1+PrimePi/@First/@FactorInteger[#])]&]

from math import isqrt
from sympy import primerange, primepi
def A338911(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(x//p)-a>>1 for a,p in enumerate(primerange(isqrt(x)+1),-1) if a&1^1)
    return bisection(f,n,n) # Chai Wah Wu, Apr 03 2025

Numbers m such that A001222(m) = 2 and A195017(m) = -2. - Peter Munn, Jan 17 2021

A339112 Products of primes of semiprime index (A106349).

Original entry on oeis.org

1, 7, 13, 23, 29, 43, 47, 49, 73, 79, 91, 97, 101, 137, 139, 149, 161, 163, 167, 169, 199, 203, 227, 233, 257, 269, 271, 293, 299, 301, 313, 329, 343, 347, 373, 377, 389, 421, 439, 443, 449, 467, 487, 491, 499, 511, 529, 553, 559, 577, 607, 611, 631, 637, 647

Offset: 1

Gus Wiseman, Mar 12 2021

nonn

A semiprime (A001358) is a product of any two prime numbers.

Also MM-numbers of labeled multigraphs with loops (without uncovered vertices). 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 multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of terms together with the corresponding multigraphs begins (A..F = 10..15):
     1:            149:   (34)     313:     (36)
     7:   (11)     161: (11)(22)   329:   (11)(23)
    13:   (12)     163:   (18)     343: (11)(11)(11)
    23:   (22)     167:   (26)     347:     (29)
    29:   (13)     169: (12)(12)   373:     (1C)
    43:   (14)     199:   (19)     377:   (12)(13)
    47:   (23)     203: (11)(13)   389:     (45)
    49: (11)(11)   227:   (44)     421:     (1D)
    73:   (24)     233:   (27)     439:     (37)
    79:   (15)     257:   (35)     443:     (1E)
    91: (11)(12)   269:   (28)     449:     (2A)
    97:   (33)     271:   (1A)     467:     (46)
   101:   (16)     293:   (1B)     487:     (2B)
   137:   (25)     299: (12)(22)   491:     (1F)
   139:   (17)     301: (11)(14)   499:     (38)

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

These primes (of semiprime index) are listed by A106349.
The strict (squarefree) case is A340020.
The prime instead of semiprime version:
  primes: A006450
  products: A076610
  strict: A302590
The nonprime instead of semiprime version:
  primes: A007821
  products: A320628
  odd: A320629
  strict: A340104
  odd strict: A340105
The squarefree semiprime instead of semiprime version:
  strict: A309356
  primes: A322551
  products: A339113
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A006881 lists squarefree semiprimes.
A037143 lists primes and semiprimes (and 1).
A056239 gives the sum of prime indices, which are listed by A112798.
A084126 and A084127 give the prime factors of semiprimes.
A101048 counts partitions into semiprimes.
A302242 is the weight of the multiset of multisets with MM-number n.
A305079 is the number of connected components for MM-number n.
A320892 lists even-omega non-products of distinct semiprimes.
A320911 lists products of squarefree semiprimes (Heinz numbers of A338914).
A320912 lists products of distinct semiprimes (Heinz numbers of A338916).
A338898, A338912, and A338913 give the prime indices of semiprimes.
MM-numbers: A255397 (normal), A302478 (set multisystems), A320630 (set multipartitions), A302494 (sets of sets), A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A328514 (connected sets of sets), A329559 (clutters), A340019 (half-loop graphs).
Cf. A000040, A000720, A001055, A001222, A003963, A005117, A007097, A289509, A320461.

N:= 1000: # for terms up to N
SP:= {}: p:= 1:
for i from 1 do
  p:= nextprime(p);
  if 2*p > N then break fi;
  Q:= map(t -> p*t, select(isprime, {2,seq(i,i=3..min(p,N/p),2)}));
  SP:= SP union Q;
od:
SP:= sort(convert(SP,list)):
PSP:= map(ithprime,SP):
R:= {1}:
for p in PSP do
  Rp:= {}:
  for k from 1 while p^k <= N do
    Rpk:= select(`<=`,R, N/p^k);
    Rp:= Rp union map(`*`,Rpk, p^k);
  od;
  R:= R union Rp;
od:
sort(convert(R,list)); # Robert Israel, Nov 03 2024

semiQ[n_]:=PrimeOmega[n]==2;
Select[Range[100],FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;!semiQ[PrimePi[p]]]&]

A346703 Product of primes at odd positions in the weakly increasing list (with multiplicity) of prime factors of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 4, 3, 2, 11, 6, 13, 2, 3, 4, 17, 6, 19, 10, 3, 2, 23, 4, 5, 2, 9, 14, 29, 10, 31, 8, 3, 2, 5, 6, 37, 2, 3, 4, 41, 14, 43, 22, 15, 2, 47, 12, 7, 10, 3, 26, 53, 6, 5, 4, 3, 2, 59, 6, 61, 2, 21, 8, 5, 22, 67, 34, 3, 14, 71, 12, 73, 2, 15, 38

Offset: 1

Gus Wiseman, Aug 08 2021

nonn

			The prime factors of 108 are (2,2,3,3,3), with odd bisection (2,3,3), with product 18, so a(108) = 18.
The prime factors of 720 are (2,2,2,2,3,3,5), with odd bisection (2,2,3,5), with product 60, so a(720) = 60.

Positions of 2's are A001747.
Positions of primes are A037143 (complement: A033942).
The even reverse version appears to be A329888.
Positions of first appearances are A342768.
The sum of prime indices of a(n) is A346697(n), reverse: A346699.
The reverse version is A346701.
The even version is A346704.
A001221 counts distinct prime factors.
A001222 counts all prime factors.
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A209281 (shifted) adds up the odd bisection of standard compositions.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A335433/A335448 rank separable/inseparable partitions.
A344606 counts alternating permutations of prime indices.
A344617 gives the sign of the alternating sum of prime indices.
A346633 adds up the even bisection of standard compositions.
A346698 gives the sum of the even bisection of prime indices.
A346700 gives the sum of the even bisection of reversed prime indices.
Cf. A025047, A027187, A027193, A053738, A097805, A106356, A341446, A344653, A345957, A345958, A345959.

Table[Times@@First/@Partition[Append[Flatten[Apply[ConstantArray,FactorInteger[n],{1}]],0],2],{n,100}]

a(n) * A346704(n) = n.

A056239(a(n)) = A346697(n).

A339115 Greatest semiprime whose prime indices sum to n.

Original entry on oeis.org

4, 6, 10, 15, 25, 35, 55, 77, 121, 143, 187, 221, 289, 323, 391, 493, 551, 667, 841, 899, 1073, 1189, 1369, 1517, 1681, 1763, 1961, 2183, 2419, 2537, 2809, 3127, 3481, 3599, 3953, 4189, 4489, 4757, 5041, 5293, 5723, 5963, 6499, 6887, 7171, 7663, 8051, 8633

Offset: 2

Gus Wiseman, Nov 28 2020

nonn

A semiprime is a product of any two prime numbers. A prime index of n is a number m such that the m-th prime number divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices begins:
        4: {1,1}      493: {7,10}      2809: {16,16}
        6: {1,2}      551: {8,10}      3127: {16,17}
       10: {1,3}      667: {9,10}      3481: {17,17}
       15: {2,3}      841: {10,10}     3599: {17,18}
       25: {3,3}      899: {10,11}     3953: {17,19}
       35: {3,4}     1073: {10,12}     4189: {17,20}
       55: {3,5}     1189: {10,13}     4489: {19,19}
       77: {4,5}     1369: {12,12}     4757: {19,20}
      121: {5,5}     1517: {12,13}     5041: {20,20}
      143: {5,6}     1681: {13,13}     5293: {19,22}
      187: {5,7}     1763: {13,14}     5723: {17,25}
      221: {6,7}     1961: {12,16}     5963: {19,24}
      289: {7,7}     2183: {12,17}     6499: {19,25}
      323: {7,8}     2419: {13,17}     6887: {20,25}
      391: {7,9}     2537: {14,17}     7171: {20,26}

Robert Israel, Table of n, a(n) for n = 2..10000

A024697 is the sum of the same semiprimes.
A332765/A332877 is the squarefree case.
A338904 has this sequence as row maxima.
A339114 is the least among the same semiprimes.
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A037143 lists primes and semiprimes.
A084126 and A084127 give the prime factors of semiprimes.
A087112 groups semiprimes by greater factor.
A320655 counts factorizations into semiprimes.
A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
A338899, A270650, and A270652 give the prime indices of squarefree semiprimes, with difference A338900.
A338907/A338906 list semiprimes of odd/even weight.
A338907/A338908 list squarefree semiprimes of odd/even weight.
Cf. A000040, A001221, A001222, A014342, A025129, A056239, A062198, A098350, A112798, A338905, A339116.

P:= [seq(ithprime(i),i=1..200)]:
[seq(max(seq(P[i]*P[j-i],i=1..j-1)),j=2..200)]; # Robert Israel, Dec 06 2020

Table[Max@@Table[Prime[k]*Prime[n-k],{k,n-1}],{n,2,30}]

A054576 Largest proper factor of the largest proper factor of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 3, 1, 5, 1, 1, 1, 6, 1, 1, 3, 7, 1, 5, 1, 8, 1, 1, 1, 9, 1, 1, 1, 10, 1, 7, 1, 11, 5, 1, 1, 12, 1, 5, 1, 13, 1, 9, 1, 14, 1, 1, 1, 15, 1, 1, 7, 16, 1, 11, 1, 17, 1, 7, 1, 18, 1, 1, 5, 19, 1, 13, 1, 20, 9, 1, 1, 21, 1, 1, 1, 22, 1, 15, 1, 23, 1, 1, 1, 24

Offset: 1

Henry Bottomley, Apr 11 2000

Here a "proper factor of n" means 1 if n = 1, and otherwise any d that divides n with 1 <= d < n. - N. J. A. Sloane, Dec 26 2022

			The largest proper factor of 8 is 4, the largest proper factor of 4 is 2, so a(8) = 2. - _N. J. A. Sloane_, Dec 26 2022

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

Cf. A053598.

Cf. A020639, A032742, A033942, A037143, A117357, A117358.

a054576 = a032742 . a032742  -- Reinhard Zumkeller, Jan 31 2015

A054576(n)=A032742(A032742(n)) \\ M. F. Hasler, Apr 03 2017

a(n) = A053598(A053598(n))

a(n) = A032742(A032742(n)); A117357(n) = A020639(a(n)); A117358(n) = A032742(a(n)) = a(n) / A117357(n); a(A037143(n)) = 1, a(A033942(n)) > 1. - Reinhard Zumkeller, Mar 10 2006

Deleted an incorrect comment and link. - N. J. A. Sloane, Dec 26 2022

A122179 Number of ways to write n as n = xyz with 1

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 1, 0, 2, 0, 0, 0, 3, 0, 0, 0, 2, 0, 1, 0, 1, 1, 0, 0, 4, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 4, 0, 0, 1, 3, 0, 1, 0, 1, 0, 1, 0, 6, 0, 0, 1, 1, 0, 1, 0, 4, 1, 0, 0, 4, 0, 0, 0, 2, 0, 4, 0, 1, 0, 0, 0, 6, 0, 1, 1, 3, 0, 1, 0, 2, 1

Offset: 1

Rick L. Shepherd, Aug 23 2006

nonn

x,y,z are proper factors of n. a(n) > 0 iff n is a term of A033942; a(n) = 0 iff n is a term of A037143.

			a(24) = 2 because 24 = 2*2*6 = 2*3*4, two products of three proper factors of 24.

Seiichi Manyama, Table of n, a(n) for n = 1..8192 (terms 1..2048 from Antti Karttunen)

Cf. A034836, A033942, A037143, A088432, A088433, A088434, A122180.

for(n=1,105, t=0; for(x=2,n-1, for(y=x,n-1, for(z=y,n-1, if(x*y*z==n, t++)))); print1(t,", "))

A122179(n) = { my(s=0); fordiv(n, x, if((x>1)&&(xAntti Karttunen, Aug 24 2017

A339841 Numbers that can be factored into distinct primes or semiprimes in exactly one way.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 48, 49, 53, 59, 61, 67, 71, 73, 79, 80, 83, 89, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 131, 137, 139, 144, 149, 151, 157, 162, 163, 167, 169, 173, 176, 179, 181, 191, 193

Offset: 1

Gus Wiseman, Dec 25 2020

nonn

A semiprime (A001358) is a product of any two prime numbers.

			The sequence of terms together with their one factorization begins:
     1 =        29 = 29        80 = 2*4*10
     2 = 2      31 = 31        83 = 83
     3 = 3      37 = 37        89 = 89
     4 = 4      41 = 41        97 = 97
     5 = 5      43 = 43       101 = 101
     7 = 7      47 = 47       103 = 103
     8 = 2*4    48 = 2*4*6    107 = 107
     9 = 9      49 = 49       109 = 109
    11 = 11     53 = 53       112 = 2*4*14
    13 = 13     59 = 59       113 = 113
    17 = 17     61 = 61       121 = 121
    19 = 19     67 = 67       125 = 5*25
    23 = 23     71 = 71       127 = 127
    25 = 25     73 = 73       131 = 131
    27 = 3*9    79 = 79       137 = 137
For example, we have 360 = 2*3*6*10, so 360 is in the sequence. But 360 is absent from A293511, because we also have 360 = 2*6*30.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Degree Sequence.
Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.

See link for additional cross-references.
These are the positions of ones in A339839.
The version for no factorizations is A339840.
The version for at least one factorization is A339889.
A001055 counts factorizations.
A001358 lists semiprimes, with squarefree case A006881.
A037143 lists primes and semiprimes.
A293511 are a product of distinct squarefree numbers in exactly one way.
A320663 counts non-isomorphic multiset partitions into singletons or pairs.
A338915 counts partitions that cannot be partitioned into distinct pairs.
Cf. A002494, A013929, A028260, A320893, A320922, A339618, A339740, A339846.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Length[Select[facs[#],UnsameQ@@#&&SubsetQ[{1,2},PrimeOmega/@#]&]]==1&]

A029862 Expansion of q^(5/24) / (eta(q) * eta(q^2)^2) in powers of q.

Original entry on oeis.org

1, 1, 4, 5, 14, 18, 41, 54, 109, 145, 267, 357, 618, 826, 1359, 1815, 2872, 3824, 5859, 7774, 11600, 15329, 22362, 29425, 42113, 55167, 77648, 101267, 140479, 182395, 249789, 322906, 437199, 562755, 754171, 966713, 1283630, 1638716, 2157763

Offset: 0

N. J. A. Sloane

nonn

Number of partitions of n where there are 3 kinds of even parts. - Ilya Gutkovskiy, Jan 17 2018

Also the number of non-isomorphic multiset partitions of weight n using singletons or pairs where no vertex appears more than twice. - Gus Wiseman, Oct 18 2018 (Proved by Andrew Howroyd, Oct 26 2018)

			G.f. = 1 + x + 4*x^2 + 5*x^3 + 14*x^4 + 18*x^5 + 41*x^6 + 54*x^7 + 109*x^8 + ...
G.f. = q^-5 + q^19 + 4*q^43 + 5*q^67 + 14*q^91 + 18*q^115 + 41*q^139 + ...
From _Gus Wiseman_, Oct 27 2018: (Start)
Non-isomorphic representatives of the a(1) = 1 through a(5) = 18 multiset partitions using singletons or pairs where no vertex appears more than twice:
  {{1}}  {{1,1}}    {{1},{2,2}}    {{1,1},{2,2}}      {{1},{2,2},{3,3}}
         {{1,2}}    {{1},{2,3}}    {{1,2},{1,2}}      {{1},{2,3},{2,3}}
         {{1},{1}}  {{2},{1,2}}    {{1,2},{3,3}}      {{1},{2,3},{4,4}}
         {{1},{2}}  {{1},{2},{2}}  {{1,2},{3,4}}      {{1},{2,3},{4,5}}
                    {{1},{2},{3}}  {{1,3},{2,3}}      {{1},{2,4},{3,4}}
                                   {{1},{1},{2,2}}    {{2},{1,2},{3,3}}
                                   {{1},{1},{2,3}}    {{2},{1,3},{2,3}}
                                   {{1},{2},{1,2}}    {{4},{1,2},{3,4}}
                                   {{1},{2},{3,3}}    {{1},{1},{3},{2,3}}
                                   {{1},{2},{3,4}}    {{1},{2},{2},{3,3}}
                                   {{1},{3},{2,3}}    {{1},{2},{2},{3,4}}
                                   {{1},{1},{2},{2}}  {{1},{2},{3},{2,3}}
                                   {{1},{2},{3},{3}}  {{1},{2},{3},{4,4}}
                                   {{1},{2},{3},{4}}  {{1},{2},{3},{4,5}}
                                                      {{1},{2},{4},{3,4}}
                                                      {{1},{2},{2},{3},{3}}
                                                      {{1},{2},{3},{4},{4}}
                                                      {{1},{2},{3},{4},{5}}
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Cf. A001358, A007716, A007717, A037143, A320462, A320655, A320663, A320665.

nmax = 40; CoefficientList[Series[Product[1 / ((1 - x^(2*k))^3 * (1 - x^(2*k-1))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *)
QP = QPochhammer; s = 1/(QP[q]*QP[q^2]^2) + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( 1 / (eta(x + A) * eta(x^2 + A)^2), n))};

Euler transform of period 2 sequence [ 1, 3, ...].
G.f.: Product_{k>0} 1 / ((1 - x^(2*k))^3 * (1 - x^(2*k-1))). - Michael Somos, Mar 23 2003
a(n) ~ exp(2*Pi*sqrt(n/3))/(6*sqrt(2)*n^(3/2)). - Vaclav Kotesovec, Sep 07 2015

A101040 If n has one or two prime-factors then 1 else 0.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Nov 28 2004

nonn

a(A033942(n))=0; for n>1: a(A037143(n))=1;
a(A000040(n))=1; a(A001358(n))=1;
A101041(n) = Sum(a(k): 1<=k<=n) + 1.
Primes counted with multiplicity. - Harvey P. Dale, Feb 16 2024

Characteristic function of A037143 (without its initial term 1).

a[n_] := If[n == 1, 0, Boole[PrimeOmega[n] <= 2]];
Array[a, 105] (* Jean-François Alcover, Dec 02 2021 *)

vector(105,k,bigomega(k)<=2&&k>1) \\ Hugo Pfoertner, Dec 02 2021

(define (A101040 n) (if (= 1 n) 0 (A063524 (A032742 (A032742 n))))) ;; Antti Karttunen, Nov 23 2017

a(n) = A010051(n)+A064911(n) = 0^floor(A001222(n)/3)-0^(n-1).

a(1) = 0; for n > 1, a(n) = A063524(A032742(A032742(n))). - Antti Karttunen, Nov 23 2017

A346701 Heinz number of the odd bisection (odd-indexed parts) of the integer partition with Heinz number n.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 4, 3, 5, 11, 6, 13, 7, 5, 4, 17, 6, 19, 10, 7, 11, 23, 6, 5, 13, 9, 14, 29, 10, 31, 8, 11, 17, 7, 6, 37, 19, 13, 10, 41, 14, 43, 22, 15, 23, 47, 12, 7, 10, 17, 26, 53, 9, 11, 14, 19, 29, 59, 10, 61, 31, 21, 8, 13, 22, 67, 34, 23, 14, 71

Offset: 1

Gus Wiseman, Aug 03 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The partition (2,2,2,1,1) has Heinz number 108 and odd bisection (2,2,1) with Heinz number 18, so a(108) = 18.
The partitions (3,2,2,1,1), (3,2,2,2,1), (3,3,2,1,1) have Heinz numbers 180, 270, 300 and all have odd bisection (3,2,1) with Heinz number 30, so a(180) = a(270) = a(300) = 30.

Positions of last appearances are A000290 without the first term 0.
Positions of primes are A037143 (complement: A033942).
The even version is A329888.
Positions of first appearances are A342768.
The sum of prime indices of a(n) is A346699(n), non-reverse: A346697.
The non-reverse version is A346703.
The even non-reverse version is A346704.
A001221 counts distinct prime factors.
A001222 counts all prime factors.
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum, reverse A344612.
A209281 (shifted) adds up the odd bisection of standard compositions.
A316524 gives the alternating sum of prime indices, reverse A344616.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344606 counts alternating permutations of prime indices.
A344617 gives the sign of the alternating sum of prime indices.
A346700 gives the sum of the even bisection of reversed prime indices.
Cf. A025047, A027187, A027193, A053738, A106356, A277103, A341446, A344653, A345957, A345958, A345959, A346698, A346702.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@First/@Partition[Append[Reverse[primeMS[n]],0],2],{n,100}]

a(n) * A329888(n) = n.

A056239(a(n)) = A346699(n).

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