A220264 -id:A220264 - cp's OEIS frontend

A350424 Numbers for which the number of their semiprime divisors sets a new record.

4, 12, 30, 60, 180, 210, 420, 1260, 2310, 4620, 13860, 30030, 60060, 180180, 510510, 1021020, 3063060, 9699690, 19399380, 58198140, 223092870, 446185740, 1338557220, 6469693230, 12939386460, 38818159380, 194090796900, 200560490130, 401120980260, 1203362940780, 6016814703900

Offset: 1

Hugo Pfoertner, Dec 30 2021

nonn

Aside from the first term a(1)=4, the sequence appears to be a subset of A129912. - Bill McEachen, Dec 31 2021

A350425 gives the corresponding number of semiprime divisors.

Cf. A129912, A220264.

nspd[n_]:=Count[Divisors[n],?(PrimeOmega[#]==2&)]; DeleteDuplicates[Table[{n,nspd[n]},{n,194*10^5}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* The program generates the first 19 terms of the sequence. *) (* _Harvey P. Dale, Dec 03 2024 *)

A367099 Least positive integer such that the number of divisors having two distinct prime factors is n.

Original entry on oeis.org

1, 6, 12, 24, 36, 60, 72, 120, 144, 216, 288, 360, 432, 960, 720, 864, 1296, 1440, 1728, 2160, 2592, 3456, 7560, 4320, 5184, 7776, 10800, 8640, 10368, 12960, 15552, 17280, 20736, 40320, 25920, 31104, 41472, 60480, 64800, 51840, 62208, 77760, 93312

Offset: 0

Gus Wiseman, Nov 09 2023

nonn

Does this contain every power of six, namely 1, 6, 36, 216, 1296, 7776, ...?

Yes, every power of six is a term, since 6^k = 2^k * 3^k is the least positive integer having n = tau(6^k) - (2k+1) divisors with two distinct prime factors. - Ivan N. Ianakiev, Nov 11 2023

			The divisors of 60 having two distinct prime factors are: 6, 10, 12, 15, 20. Since 60 is the first number having five such divisors, we have a(5) = 60.
The terms together with their prime indices begin:
     1: {}
     6: {1,2}
    12: {1,1,2}
    24: {1,1,1,2}
    36: {1,1,2,2}
    60: {1,1,2,3}
    72: {1,1,1,2,2}
   120: {1,1,1,2,3}
   144: {1,1,1,1,2,2}
   216: {1,1,1,2,2,2}
   288: {1,1,1,1,1,2,2}
   360: {1,1,1,2,2,3}
   432: {1,1,1,1,2,2,2}
   960: {1,1,1,1,1,1,2,3}
   720: {1,1,1,1,2,2,3}
   864: {1,1,1,1,1,2,2,2}

Amiram Eldar, Table of n, a(n) for n = 0..4469

The version for all divisors is A005179 (firsts of A000005).
For all prime factors (A001222) we have A220264, firsts of A086971.
Positions of first appearances in A367098 (counts divisors in A007774).
A000961 lists prime powers, complement A024619.
A001221 counts distinct prime factors.
A001358 lists semiprimes, squarefree A006881, complement A100959.
A367096 lists semiprime divisors, sum A076290.
Cf. A001248, A054753, A056170, A079275, A146289, A366740, A367093.

nn=1000;
w=Table[Length[Select[Divisors[n],PrimeNu[#]==2&]],{n,nn}];
spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
Table[Position[w,k][[1,1]],{k,0,spnm[w]}]

a(n) = my(k=1); while (sumdiv(k, d, omega(d)==2) != n, k++); k; \\ Michel Marcus, Nov 11 2023

A350416 Numbers with exactly 9 semiprime divisors.

Original entry on oeis.org

6300, 8820, 9900, 11700, 12600, 14700, 15300, 17100, 17640, 18900, 19404, 19800, 20700, 21780, 22050, 22932, 23400, 25200, 26100, 26460, 27900, 29400, 29700, 29988, 30420, 30492, 30600, 31500, 33300, 33516, 34200, 35100, 35280, 36300, 36900, 37800, 38700, 38808

Offset: 1

Wesley Ivan Hurt, Dec 29 2021

Numbers with exactly four distinct prime divisors (cf. A033993), one of which has multiplicity 1 and the others at least 2. - David A. Corneth, Jun 10 2022

			6300 is in the sequence as 4, 6, 9, 10, 14, 15, 21, 25, 35 are the exactly 9 of its semiprime divisors. - _David A. Corneth_, Jun 10 2022

David A. Corneth, Table of n, a(n) for n = 1..10000

Numbers with exactly k semiprime divisors: A346041 (k=1), A345381 (k=2), A345382 (k=3), A350371 (k=4), A350372 (k=5), A350373 (k=6), A350374 (k=7), A350375 (k=8), this sequence (k=9).

Cf. A033993, A220264.

q[n_] := DivisorSum[n, 1 &, PrimeOmega[#] == 2 &] == 9; Select[Range[40000], q] (* Amiram Eldar, Dec 30 2021 *)
spd9Q[n_]:=Count[Divisors[n],?(PrimeOmega[#]==2&)]==9; Select[Range[ 40000],spd9Q] (* _Harvey P. Dale, Jun 09 2022 *)

isok(k) = sumdiv(k, d, bigomega(d)==2) == 9; \\ Michel Marcus, Dec 30 2021

is(n)= if(n==1, return(0)); my(f = vecsort(factor(n)[,2])); #f == 4 && f[1] == 1 && f[2]>=2 \\ David A. Corneth, Jun 10 2022

A350425 Records of the number of semiprime divisors corresponding to A350424.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 15, 16, 17, 21, 22, 23, 28, 29, 30, 36, 37, 38, 45, 46, 47, 48, 55, 56, 57, 58, 66, 67, 68, 69, 78, 79, 80, 81, 91, 92, 93, 94, 105, 106, 107, 108, 120, 121, 122, 123, 136, 137, 138, 139, 153, 154, 155, 156, 171, 172, 173, 174, 190, 191, 192, 193, 210

Offset: 1

Hugo Pfoertner, Dec 30 2021

nonn

A350424 gives the numbers setting the records.

Cf. A220264.

A367105 Least positive integer with n more divisors than distinct subset-sums of prime indices.

Original entry on oeis.org

1, 12, 24, 48, 60, 192, 144, 120, 180, 336, 240, 630, 420, 360, 900, 1344, 960, 1008, 720, 840, 2340, 1980, 1260, 1440, 3120, 2640, 1680, 4032, 2880, 6840, 3600, 4620, 3780, 2520, 6480, 11700, 8820, 6300, 7200, 10560, 6720, 12240, 9360, 7920, 5040, 10920, 9240

Offset: 1

Gus Wiseman, Nov 09 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.An integer n is a subset-sum (A299701, A304792) of a multiset y if there exists a submultiset of y with sum n.

			The divisors of 60 are {1,2,3,4,5,6,10,12,15,20,30,60}, and the distinct subset-sums of its prime indices {1,1,2,3} are {0,1,2,3,4,5,6,7}, so the difference is 12 - 8 = 4. Since 60 is the first number with this difference, we have a(4) = 60.
The terms together with their prime indices begin:
     1: {}
    12: {1,1,2}
    24: {1,1,1,2}
    48: {1,1,1,1,2}
    60: {1,1,2,3}
   120: {1,1,1,2,3}
   144: {1,1,1,1,2,2}
   180: {1,1,2,2,3}
   192: {1,1,1,1,1,1,2}
   240: {1,1,1,1,2,3}
   336: {1,1,1,1,2,4}
   360: {1,1,1,2,2,3}
   420: {1,1,2,3,4}
   630: {1,2,2,3,4}
   720: {1,1,1,1,2,2,3}
   840: {1,1,1,2,3,4}
   900: {1,1,2,2,3,3}
   960: {1,1,1,1,1,1,2,3}

The first part (divisors) is A000005.
The second part (subset-sums of prime indices) is A299701, positive A304793.
These are the positions of first appearances in the difference A325801.
The binary version is A367093, firsts of  A086971 - A366739.
A001222 counts prime factors (or prime indices), distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A000720, A220264, A304792, A365920, A366740, A367097.

nn=1000;
w=Table[DivisorSigma[0,n]-Length[Union[Total/@Subsets[prix[n]]]],{n,nn}];
spnm[y_]:=Max@@Select[Union[y],Function[i,Union[Select[y,#<=i&]]==Range[0,i]]];
Table[Position[w,k][[1,1]],{k,0,spnm[w]}]

A000005(a(n)) - A299701(a(n)) = n.

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A350424 Numbers for which the number of their semiprime divisors sets a new record.

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A367099 Least positive integer such that the number of divisors having two distinct prime factors is n.

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A350416 Numbers with exactly 9 semiprime divisors.

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A350425 Records of the number of semiprime divisors corresponding to A350424.

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A367105 Least positive integer with n more divisors than distinct subset-sums of prime indices.

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