A046315 -id:A046315 - cp's OEIS frontend

A338901 Position of the first appearance of prime(n) as a factor in the list of squarefree semiprimes.

1, 1, 2, 3, 6, 7, 9, 11, 13, 17, 18, 21, 23, 25, 29, 31, 34, 36, 40, 42, 45, 47, 50, 52, 56, 58, 61, 64, 67, 70, 76, 78, 81, 82, 86, 89, 93, 97, 100, 104, 106, 107, 112, 113, 116, 118, 125, 129, 133, 134, 135, 139, 141, 147, 150, 154, 159, 160, 165, 167, 169

Offset: 1

Gus Wiseman, Nov 16 2020

nonn

The a(n)-th squarefree semiprime is the first divisible by prime(n).

After a(1) = 1, these are the positions of even terms in the list of all squarefree semiprimes A006881.

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

A001358 lists semiprimes, with odds A046315 and evens A100484.
A004526 counts 2-part partitions, with strict case A140106 (shifted left).
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odds A046388 and evens A100484.
A115392 is the not necessarily squarefree version.
A166237 gives the first differences of squarefree semiprimes.
A270650 and A270652 give the prime indices of squarefree semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A338898 gives prime indices of semiprimes, with differences A176506.
A338899 gives prime indices of squarefree semiprimes, differences A338900.
A338912 and A338913 give the prime indices of semiprimes.
Cf. A001221, A001222, A002100, A056239, A065516, A112798, A167171, A320891, A320911, A338903, A338905.

rs=First/@FactorInteger[#]&/@Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&];
Table[Position[rs,i][[1,1]],{i,Union@@rs}]

A006881(a(n)) = A100484(n).

A358777 Dirichlet inverse of A353557, the characteristic function of odd numbers with an even number of prime factors (counted with multiplicity).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, -1, 0, -1, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 1

Offset: 1

Antti Karttunen, Dec 20 2022

As A353557 is not multiplicative, neither is this sequence.
Absolute values differ from A353557 for the first time at n=81, where a(81) = 0.
Absolute values differ from A353480 for the first time at n=1, and then at n=135.
The first value greater than 1 occurs as a(225) = 2. The first value less than -1 occurs as a(2835) = -2.
From Antti Karttunen, Jan 12 2023: (Start)
Few properties concerning this sequence:
(1) For all even numbers n, a(n) = 0. Proof: In the convolution formula, at least the other of the divisors (n/d) and d is always even, for any such divisor pair of an even n. As A353557 is zero for all even numbers, it is easy to show by induction that also a(n) is zero for all even n.
(2) For all numbers n with an odd number of prime factors (with multiplicity), a(n) = 0. Proof: In the convolution formula, either the divisor (n/d) or d (but not both) has an odd number of prime factors for any divisor pair d and (n/d) of any n in A026424. As A353557 is zero for all A026424, it is easy to show by induction that also a(n) is zero for all such numbers.
(3) Therefore, nonzero values occur only on indices that are a subset of A046337. (See A359607 for exceptions).
(4) For any two odd numbers x and y with the same prime signature (A046523(x) = A046523(y)), a(x) = a(y).
(5) a(A046315(n)) = -1.
(6) Apparently it also holds that for any n that is a square that is the 4th, 6th, 8th, ..., 2k-th power (k>=2) of some natural number > 1, a(n) is even.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..100000
Jon Maiga, Computer-generated formulas for A358777, Sequence Machine.

Cf. A046315, A046337, A065043, A353557, A358778 (positions of positive terms), A359595 (parity of terms), A359596 (positions of odd terms), A359599 (terms with record absolute values), A359598 (their positions in this sequence), A359607, A359609 (distinct values in the order of their appearance), A359608 (their positions in this sequence).
Agrees paritywise with A359589 and A366265.
Cf. also A323239 (Dirichlet inverse of A166698(n) = A353557(n) - A353558(n)).
Cf. A359763, A359773, A359780, A359814, A359815 for similar sequences.

A353557(n) = ((n%2)&&(!(bigomega(n)%2)));
memoA358777 = Map();
A358777(n) = if(1==n,1,my(v); if(mapisdefined(memoA358777,n,&v), v, v = -sumdiv(n,d,if(dA353557(n/d)*A358777(d),0)); mapput(memoA358777,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA353557(n/d) * a(d).
From Antti Karttunen, Nov 22 2023: (Start)
Following identities (among others) are listed by Sequence Machine:
a(n) = o(n)*A359763(n) = o(n)*A359773(n) = o(n)*A359780(n) = o(n)*A359814(n) = o(n)*A359815(n), where o(n) = A000035(n), parity of n.
a(n) = A353557(n) * A359763(n) = A353557(n) * A359814(n).
a(n) = A065043(n) * A359773(n).
(End)

A096173 Numbers k such that k^3+1 is an odd semiprime.

Original entry on oeis.org

2, 4, 6, 16, 18, 22, 28, 42, 58, 60, 70, 72, 78, 100, 102, 106, 112, 148, 156, 162, 190, 210, 232, 280, 310, 330, 352, 358, 382, 396, 448, 456, 490, 568, 606, 672, 756, 786, 820, 826, 828, 856, 858, 876, 928, 970, 982, 1008, 1012, 1030, 1068, 1092, 1108, 1150

Offset: 1

Hugo Pfoertner, Jun 20 2004

nonn

Numbers n such that n^3 + 1 is a semiprime, because then n^3 + 1 must be odd, since n^3 + 1 = (n+1)*(n^2 - n + 1) is a semiprime only if n+1 is odd. - Jonathan Sondow, Feb 02 2014

Obviously, n + 1 is always a prime number. Sequence is intersection of A006093 and A055494. - Altug Alkan, Dec 20 2015

			a(1)=2 because 2^3+1=9=3*3, a(13)=100: 100^3+1=1000001=101*9901.

R. J. Mathar, Table of n, a(n) for n = 1..2086

Cf. A001358; A081256: largest prime factor of k^3+1; A096174: (k^3+1)/(k+1) is prime; A046315, A237037, A237038, A237039, A237040.

Cf. A006093, A055494.

[n: n in [1..2*10^3] | IsPrime(n+1) and IsPrime(n^2-n+1)]; // Vincenzo Librandi, Dec 21 2015

select(n -> isprime(n+1) and isprime(n^2-n+1), [seq(2*i,i=1..1000)]); # Robert Israel, Dec 20 2015

Select[Range[1200], PrimeQ[#^2 - # + 1] && PrimeQ[# + 1] &] (* Jonathan Sondow, Feb 02 2014 *)

for(n=1, 1e5, if(bigomega(n^3+1)==2, print1(n, ", "))); \\ Altug Alkan, Dec 20 2015

a(n) = 2*A237037(n) = (A237040(n)-1)^(1/3). - Jonathan Sondow, Feb 02 2014

Corrected by Zak Seidov, Mar 08 2006

A339113 Products of primes of squarefree semiprime index (A322551).

Original entry on oeis.org

1, 13, 29, 43, 47, 73, 79, 101, 137, 139, 149, 163, 167, 169, 199, 233, 257, 269, 271, 293, 313, 347, 373, 377, 389, 421, 439, 443, 449, 467, 487, 491, 499, 559, 577, 607, 611, 631, 647, 653, 673, 677, 727, 751, 757, 811, 821, 823, 829, 839, 841, 907, 929, 937

Offset: 1

Gus Wiseman, Mar 12 2021

nonn

A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

Also MM-numbers of labeled multigraphs (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:
      1: {}               233: {{2,7}}          487: {{2,11}}
     13: {{1,2}}          257: {{3,5}}          491: {{1,15}}
     29: {{1,3}}          269: {{2,8}}          499: {{3,8}}
     43: {{1,4}}          271: {{1,10}}         559: {{1,2},{1,4}}
     47: {{2,3}}          293: {{1,11}}         577: {{1,16}}
     73: {{2,4}}          313: {{3,6}}          607: {{2,12}}
     79: {{1,5}}          347: {{2,9}}          611: {{1,2},{2,3}}
    101: {{1,6}}          373: {{1,12}}         631: {{3,9}}
    137: {{2,5}}          377: {{1,2},{1,3}}    647: {{1,17}}
    139: {{1,7}}          389: {{4,5}}          653: {{4,7}}
    149: {{3,4}}          421: {{1,13}}         673: {{1,18}}
    163: {{1,8}}          439: {{3,7}}          677: {{2,13}}
    167: {{2,6}}          443: {{1,14}}         727: {{2,14}}
    169: {{1,2},{1,2}}    449: {{2,10}}         751: {{4,8}}
    199: {{1,9}}          467: {{4,6}}          757: {{1,19}}

These primes (of squarefree semiprime index) are listed by A322551.
The strict (squarefree) case is A309356.
The prime instead of squarefree semiprime version:
  primes: A006450
  products: A076610
  strict: A302590
The nonprime instead of squarefree semiprime version:
  primes: A007821
  products: A320628
  odd: A320629
  strict: A340104
  odd strict: A340105
The semiprime instead of squarefree semiprime version:
  primes: A106349
  products: A339112
  strict: A340020
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A002100 counts partitions into squarefree semiprimes.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A056239 gives the sum of prime indices, which are listed by A112798.
A302242 is the weight of the multiset of multisets with MM-number n.
A305079 is the number of connected components for MM-number n.
A320911 lists products of squarefree semiprimes (Heinz numbers of A338914).
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
A339561 lists products of distinct squarefree semiprimes (ranking: A339560).
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, A289509, A320461.

sqfsemiQ[n_]:=SquareFreeQ[n]&&PrimeOmega[n]==2;
Select[Range[1000],FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;!sqfsemiQ[PrimePi[p]]]&]

A339114 Least semiprime whose prime indices sum to n.

Original entry on oeis.org

4, 6, 9, 14, 21, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514, 526

Offset: 2

Gus Wiseman, Nov 28 2020

nonn

Converges to A100484.
After a(4) = 9, also the least squarefree semiprime whose prime indices sum to n.
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}     106: {1,16}    254: {1,31}
      6: {1,2}     118: {1,17}    262: {1,32}
      9: {2,2}     122: {1,18}    274: {1,33}
     14: {1,4}     134: {1,19}    278: {1,34}
     21: {2,4}     142: {1,20}    298: {1,35}
     26: {1,6}     146: {1,21}    302: {1,36}
     34: {1,7}     158: {1,22}    314: {1,37}
     38: {1,8}     166: {1,23}    326: {1,38}
     46: {1,9}     178: {1,24}    334: {1,39}
     58: {1,10}    194: {1,25}    346: {1,40}
     62: {1,11}    202: {1,26}    358: {1,41}
     74: {1,12}    206: {1,27}    362: {1,42}
     82: {1,13}    214: {1,28}    382: {1,43}
     86: {1,14}    218: {1,29}    386: {1,44}
     94: {1,15}    226: {1,30}    394: {1,45}

A024697 is the sum of the same semiprimes.
A098350 has this sequence as antidiagonal minima.
A338904 has this sequence as row minima.
A339114 (this sequence) is the squarefree case for n > 4.
A339115 is the greatest 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.
A056239 gives the sum of prime indices of n.
A084126 and A084127 give the prime factors of semiprimes.
A087112 groups semiprimes by greater factor.
A320655 counts factorizations into semiprimes.
A332765/A332877 is the greatest squarefree semiprime of weight n.
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, A062198, A112798, A338905, A339116.

Table[Min@@Table[Prime[k]*Prime[n-k],{k,n-1}],{n,2,30}]
Take[DeleteDuplicates[SortBy[{Times@@#,Total[PrimePi[#]]}&/@Tuples[ Prime[ Range[ 200]],2],{Last,First}],GreaterEqual[#1[[2]],#2[[2]]]&][[All,1]],60] (* Harvey P. Dale, Sep 06 2022 *)

a(n) = vecmin(vector(n-1, k, prime(k)*prime(n-k))); \\ Michel Marcus, Dec 03 2020

A339657 Heinz numbers of non-loop-graphical partitions of even numbers.

Original entry on oeis.org

7, 13, 19, 21, 22, 29, 34, 37, 39, 43, 46, 49, 52, 53, 55, 57, 61, 62, 66, 71, 76, 79, 82, 85, 87, 89, 91, 94, 101, 102, 107, 111, 113, 115, 116, 117, 118, 121, 129, 130, 131, 133, 134, 136, 138, 139, 146, 148, 151, 154, 155, 156, 159, 163, 165, 166, 169, 171

Offset: 1

Gus Wiseman, Dec 18 2020

nonn

Equals the image of A181819 applied to the set of terms of A320892.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
An integer partition is loop-graphical if it comprises the multiset of vertex-degrees of some graph with loops, where a loop is an edge with two equal vertices. Loop-graphical partitions are counted by A339656, with Heinz numbers A339658.
The following are equivalent characteristics for any positive integer n:
(1) the prime factors of n can be partitioned into distinct pairs, i.e., into a set of edges and loops;
(2) n can be factored into distinct semiprimes;
(3) the prime signature of n is loop-graphical.

			The sequence of terms together with their prime indices begins:
      7: {4}         57: {2,8}      107: {28}
     13: {6}         61: {18}       111: {2,12}
     19: {8}         62: {1,11}     113: {30}
     21: {2,4}       66: {1,2,5}    115: {3,9}
     22: {1,5}       71: {20}       116: {1,1,10}
     29: {10}        76: {1,1,8}    117: {2,2,6}
     34: {1,7}       79: {22}       118: {1,17}
     37: {12}        82: {1,13}     121: {5,5}
     39: {2,6}       85: {3,7}      129: {2,14}
     43: {14}        87: {2,10}     130: {1,3,6}
     46: {1,9}       89: {24}       131: {32}
     49: {4,4}       91: {4,6}      133: {4,8}
     52: {1,1,6}     94: {1,15}     134: {1,19}
     53: {16}       101: {26}       136: {1,1,1,7}
     55: {3,5}      102: {1,2,7}    138: {1,2,9}
For example, the three loop-multigraphs with degrees y = (5,2,1) are:
  {{1,1},{1,1},{1,2},{2,3}}
  {{1,1},{1,1},{1,3},{2,2}}
  {{1,1},{1,2},{1,2},{1,3}},
but since none of these is a loop-graph (they have multiple edges), the Heinz number 66 is in the sequence.

Eric Weisstein's World of Mathematics, Graphical partition.

A320892 has these prime shadows (see A181819).
A321728 is conjectured to be the version for half-loops {x} instead of loops {x,x}.
A339655 counts these partitions.
A339658 ranks the complement, counted by A339656.
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A006881 lists squarefree semiprimes, with odd and even terms A046388 and A100484.
A101048 counts partitions into semiprimes.
A320655 counts factorizations into semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A339844 counts loop-graphical partitions by length.
factorizations of n into distinct primes or squarefree semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A058696 counts partitions of 2n (A300061).
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A209816 counts multigraphical partitions (A320924).
- A339655 counts non-loop-graphical partitions of 2n (A339657 [this sequence]).
- A339656 counts loop-graphical partitions (A339658).
- A339617 counts non-graphical partitions of 2n (A339618).
- A000569 counts graphical partitions (A320922).
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A001055, A001221, A001222, A007717, A056239, A112798, A320732, A338898, A338912, A338913, A339742, A339839.

spsbin[{}]:={{}};spsbin[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsbin[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpsbin[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@spsbin[Range[Length[set]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[50],EvenQ[Length[nrmptn[#]]]&&Select[mpsbin[nrmptn[#]],UnsameQ@@#&]=={}&]

A300061 = A339657 \/ A339658.

A046328 Palindromes with exactly 2 prime factors (counted with multiplicity).

Original entry on oeis.org

4, 6, 9, 22, 33, 55, 77, 111, 121, 141, 161, 202, 262, 303, 323, 393, 454, 505, 515, 535, 545, 565, 626, 707, 717, 737, 767, 818, 838, 878, 898, 939, 949, 959, 979, 989, 1111, 1441, 1661, 1991, 3113, 3223, 3443, 3883, 7117, 7447, 7997, 9119, 9229, 9449, 10001

Offset: 1

Patrick De Geest, Jun 15 1998

			111 is a palindrome and 111 = 3*37. 3 and 37 are primes.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 2000 terms from Zak Seidov, terms a(2001)-a(2816) from Michael De Vlieger)

Cf. A046315, A046408, A108505.

Subsequence of A001358 and A046338.

fQ[n_] := Block[{id = IntegerDigits[n]}, Plus @@ Last /@ FactorInteger[n] == 2 && id == Reverse[id]]; Select[ Range[ 10000], fQ[ # ] &] (* Robert G. Wilson v, Jun 06 2005 *)
Select[Range[10002], Reverse[x = IntegerDigits[#]] == x && PrimeOmega[#] == 2 &] (* Jayanta Basu, Jun 23 2013 *)
Select[Range[11000],PalindromeQ[#]&&PrimeOmega[#]==2&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 30 2018 *)

ispal(n) = my(d=digits(n));d == Vecrev(d) \\ A002113
for(k=1,1e4,if(ispal(k)&&bigomega(k)==2, print1(k, ", "))) \\ Alexandru Petrescu, Jul 07 2022

from sympy import factorint
from itertools import product
def ispal(n): s = str(n); return s == s[::-1]
def pals(d, base=10): # all d-digit palindromes
    digits = "".join(str(i) for i in range(base))
    for p in product(digits, repeat=d//2):
        if d > 1 and p[0] == "0": continue
        left = "".join(p); right = left[::-1]
        for mid in [[""], digits][d%2]: yield int(left + mid + right)
def ok(pal): return sum(factorint(pal).values()) == 2
print(list(filter(ok, (p for d in range(1, 6) for p in pals(d) if ok(p))))) # Michael S. Branicky, Aug 14 2022

A338906 Semiprimes whose prime indices sum to an even number.

Original entry on oeis.org

4, 9, 10, 21, 22, 25, 34, 39, 46, 49, 55, 57, 62, 82, 85, 87, 91, 94, 111, 115, 118, 121, 129, 133, 134, 146, 155, 159, 166, 169, 183, 187, 194, 203, 205, 206, 213, 218, 235, 237, 247, 253, 254, 259, 267, 274, 289, 295, 298, 301, 303, 314, 321, 334, 335, 339

Offset: 1

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}      87: {2,10}    183: {2,18}    274: {1,33}
      9: {2,2}      91: {4,6}     187: {5,7}     289: {7,7}
     10: {1,3}      94: {1,15}    194: {1,25}    295: {3,17}
     21: {2,4}     111: {2,12}    203: {4,10}    298: {1,35}
     22: {1,5}     115: {3,9}     205: {3,13}    301: {4,14}
     25: {3,3}     118: {1,17}    206: {1,27}    303: {2,26}
     34: {1,7}     121: {5,5}     213: {2,20}    314: {1,37}
     39: {2,6}     129: {2,14}    218: {1,29}    321: {2,28}
     46: {1,9}     133: {4,8}     235: {3,15}    334: {1,39}
     49: {4,4}     134: {1,19}    237: {2,22}    335: {3,19}
     55: {3,5}     146: {1,21}    247: {6,8}     339: {2,30}
     57: {2,8}     155: {3,11}    253: {5,9}     341: {5,11}
     62: {1,11}    159: {2,16}    254: {1,31}    358: {1,41}
     82: {1,13}    166: {1,23}    259: {4,12}    361: {8,8}
     85: {3,7}     169: {6,6}     267: {2,24}    365: {3,21}

A031215 looks at primes instead of semiprimes.
A098350 has this as union of even-indexed antidiagonals.
A300061 looks at all numbers (not just semiprimes).
A338904 has this as union of even-indexed rows.
A338907 is the odd version.
A338908 is the squarefree case.
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A056239 gives the sum of prime indices (Heinz weight).
A084126 and A084127 give the prime factors of semiprimes.
A087112 groups semiprimes by greater factor.
A289182/A115392 list the positions of odd/even terms in A001358.
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.
A338911 lists products of pairs of primes both of even index.
A339114/A339115 give the least/greatest semiprime of weight n.
Cf. A000040, A001222, A024697, A037143, A112798, A300063, A319242, A320655, A332765, A338910, A339004.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],PrimeOmega[#]==2&&EvenQ[Total[primeMS[#]]]&]

from math import isqrt
from sympy import primepi, primerange
def A338906(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))
    return bisection(f,n,n) # Chai Wah Wu, Apr 03 2025

A338906 \/ A338907 = A001358.

A338910 Numbers of the form prime(x) * prime(y) where x and y are both odd.

Original entry on oeis.org

4, 10, 22, 25, 34, 46, 55, 62, 82, 85, 94, 115, 118, 121, 134, 146, 155, 166, 187, 194, 205, 206, 218, 235, 253, 254, 274, 289, 295, 298, 314, 334, 335, 341, 358, 365, 382, 391, 394, 415, 422, 451, 454, 466, 482, 485, 514, 515, 517, 527, 529, 538, 545, 554

Offset: 1

Gus Wiseman, Nov 20 2020

nonn

			The sequence of terms together with their prime indices begins:
      4: {1,1}     146: {1,21}    314: {1,37}
     10: {1,3}     155: {3,11}    334: {1,39}
     22: {1,5}     166: {1,23}    335: {3,19}
     25: {3,3}     187: {5,7}     341: {5,11}
     34: {1,7}     194: {1,25}    358: {1,41}
     46: {1,9}     205: {3,13}    365: {3,21}
     55: {3,5}     206: {1,27}    382: {1,43}
     62: {1,11}    218: {1,29}    391: {7,9}
     82: {1,13}    235: {3,15}    394: {1,45}
     85: {3,7}     253: {5,9}     415: {3,23}
     94: {1,15}    254: {1,31}    422: {1,47}
    115: {3,9}     274: {1,33}    451: {5,13}
    118: {1,17}    289: {7,7}     454: {1,49}
    121: {5,5}     295: {3,17}    466: {1,51}
    134: {1,19}    298: {1,35}    482: {1,53}

A338911 is the even instead of odd version.
A339003 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.
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 are semiprimes of even/odd weight.
A338898, A338912, and A338913 give the prime indices of semiprimes, with product A087794, sum A176504, and difference A176506.
A338899, A270650, and A270652 give prime indices of squarefree semiprimes.
A338909 lists semiprimes with non-relatively prime indices.
Cf. A005117, A037143, A055684, A056239, A065516, A112798, A195017, A320655, A320732, A320892, A339004.

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

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

from math import isqrt
from sympy import primepi, primerange
def A338910(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)
    return bisection(f,n,n) # Chai Wah Wu, Apr 03 2025

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

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

Original entry on oeis.org

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

cp's OEIS Frontend

A338901 Position of the first appearance of prime(n) as a factor in the list of squarefree semiprimes.

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A358777 Dirichlet inverse of A353557, the characteristic function of odd numbers with an even number of prime factors (counted with multiplicity).

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A096173 Numbers k such that k^3+1 is an odd semiprime.

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A339113 Products of primes of squarefree semiprime index (A322551).

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A339114 Least semiprime whose prime indices sum to n.

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A339657 Heinz numbers of non-loop-graphical partitions of even numbers.

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A046328 Palindromes with exactly 2 prime factors (counted with multiplicity).

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A338906 Semiprimes whose prime indices sum to an even number.

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