A100959 -id:A100959 - cp's OEIS frontend

A133447 Nonsemiprime hyperperfect numbers.

325, 2133, 10693, 16513, 19521, 51301, 159841, 176661, 214273, 306181, 1433701, 1570153, 1950625, 2469601, 2924101, 5199013, 9398593, 10445221, 15407173, 23548753, 28600321, 39147301, 60110701, 62722153, 88347781, 112803841

Offset: 1

Jonathan Vos Post, Dec 22 2007

nonn

The other 25 of the first 35 values of A007592 are all semiprimes A001358.

This sequence excludes perfect numbers (A000396), which are 1-hyperperfect numbers. - Jud McCranie, Mar 23 2025.

			a(1) = 325 = 5^2 * 13.
a(2) = 2133 = 3^3 * 79.
a(3) = 10693 = 17^2 * 37.
a(4) = 16513 = 7^2 * 337.
a(5) = 19521 = 3^4 * 341.
a(6) = 51301 = 29^2 * 61.
a(7) = 159841 = 11^2 * 1321.
a(8) = 176661 = 3^5 * 727.
a(9) = 214273 = 47^2 * 97.
a(10) = 306181 = 53^2 * 109.
a(12) = 1570153 = 13 * 269 * 449. - _Jud McCranie_, Mar 23 2025

Jud McCranie, Table of n, a(n) for n = 1..1167 (terms < 10^15). Earlier terms from Jonathan Vos Post and R. J. Mathar, and a b-file with 192 terms (to 10^12) by Donovan Johnson.

Cf. A001358, A007592, A100959.

a034897 := [] : fd := fopen("b034897.txt",READ) : bf := fscanf(fd,"%d %d") : while nops(bf) <> 0 do a034897 := [op(a034897), op(2,bf) ] ; bf := fscanf(fd,"%d %d") ; od: a007592 := [] : for n in a034897 do m := (n-1)/( numtheory[sigma](n)-n-1) ; if m > 1 then a007592 := [op(a007592),n] ; fi ; od: isA100959 := proc(n) if numtheory[bigomega](n) <> 2 then true ; else false ; end: end: for n in a007592 do if isA100959(n) then printf("%d, ",n) ; fi ; od: # R. J. Mathar, Jan 08 2008

A100959 INTERSECTION A007592.

More terms from R. J. Mathar, Jan 08 2008

A133795 a(n) = n-th semiprime + n-th non-semiprime.

Original entry on oeis.org

5, 8, 12, 15, 21, 23, 32, 34, 38, 42, 50, 52, 54, 58, 62, 70, 76, 79, 84, 87, 89, 94, 101, 106, 114, 118, 124, 128, 130, 132, 138, 141, 144, 147, 159, 165, 171, 177, 179, 182, 185, 187, 195, 200, 202, 211, 213, 215, 218, 221, 231, 236, 238, 241, 247, 252, 261

Offset: 1

Jonathan Vos Post, Jan 05 2008

Semiprime analog of A022797 n-th prime + n-th nonprime. a(n) is prime for n = 1, 6, 18, 21, 23. a(n) is itself semiprime for n = 4, 5, 22, 25, 38, 39, 57, 62, 69, 77 of which first 10 indices all but n=5 are themselves semiprimes.

			a(1) = 1st semiprime + 1st nonsemiprime = 4 + 1 = 5.
a(2) = 2nd semiprime + 2nd nonsemiprime = 6 + 2 = 8.
a(3) = 3rd semiprime + 3rd nonsemiprime = 9 + 3 = 12.

Cf. A000040, A001358, A022797, A100959.

A100959 := proc(n) option remember; local a ; if n = 1 then 1 ; else for a from A100959(n-1)+1 do if numtheory[bigomega](a) <> 2 then RETURN(a) ; fi ; od: fi ; end: A001358 := proc(n) option remember ; local a ; if n = 1 then 4 ; else for a from A001358(n-1)+1 do if numtheory[bigomega](a) = 2 then RETURN(a) ; fi ; od: fi ; end: A133795 := proc(n) A100959(n)+A001358(n) ; end: seq(A133795(n),n=1..100) ; # R. J. Mathar, Jan 09 2008

Module[{nn=200,spr,non},spr=Select[Range[nn],PrimeOmega[#]==2&];non=Take[ Complement[ Range[ nn],spr],Length[ spr]];Total/@Thread[{spr,non}]] (* Harvey P. Dale, Sep 06 2024 *)

a(n) = A001358(n) + A100959(n).

Corrected and extended by R. J. Mathar, Jan 09 2008

A140234 Sum of the semiprimes <= n.

Original entry on oeis.org

0, 0, 0, 0, 4, 4, 10, 10, 10, 19, 29, 29, 29, 29, 43, 58, 58, 58, 58, 58, 58, 79, 101, 101, 101, 126, 152, 152, 152, 152, 152, 152, 152, 185, 219, 254, 254, 254, 292, 331, 331, 331, 331, 331, 331, 331, 377, 377, 377, 426, 426, 477, 477, 477, 477, 532, 532, 589

Offset: 0

Jonathan Vos Post, May 13 2008

This is to semiprimes A001358 as A034387 is to primes A000040. From the prime number theorem A034387(n) has the asymptotic expression: a(n) ~ n^2 / (2 log n), so what is the asymptotic expression for a(n)?

Cf. A001358, A034387, A051352, A062198, A072000, A100959, A140235.

a[n_]:=Total[Select[Range[n],PrimeOmega[#]==2&]];Array[a,58,0] (* James C. McMahon, Jul 06 2025 *)

a(n) = Sum_{j such that j is in A001358 and j<=n} = A062198(A072000(n)).

A174653 Semiprimes s such that neither s-1 nor s+1 is semiprime.

Original entry on oeis.org

4, 6, 46, 49, 51, 55, 62, 65, 69, 74, 77, 82, 91, 106, 111, 115, 129, 155, 161, 166, 169, 183, 185, 187, 194, 209, 221, 226, 235, 237, 247, 249, 259, 262, 265, 267, 274, 278, 287, 289, 291, 295, 305, 309, 314, 319, 321, 323, 329, 339, 341, 346, 355, 358, 365

Offset: 1

Juri-Stepan Gerasimov, Mar 25 2010

nonn

A064911(a(n))*(1 - A064911(a(n)-1))*(1 - A064911(a(n)+1)) = 1. - Reinhard Zumkeller, Apr 03 2010

			a(1)=4 because 4-1 = 3 (not semiprime) and 4+1 = 5 (not semiprime).

R. Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A001358, A100959.

Mean/@SequencePosition[Table[If[PrimeOmega[n]==2,1,0],{n,400}],{0,1,0}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 19 2019 *)

Corrected by Reinhard Zumkeller and Ray Chandler, Apr 07 2010

A176551 Products of 2 primes of the form 3*k-+1.

Original entry on oeis.org

4, 10, 14, 22, 25, 26, 34, 35, 38, 46, 49, 55, 58, 62, 65, 74, 77, 82, 85, 86, 91, 94, 95, 106, 115, 118, 119, 121, 122, 133, 134, 142, 143, 145, 146, 155, 158, 161, 166, 169, 178, 185, 187, 194, 202, 203, 205, 206, 209, 214, 215, 217, 218, 221, 226, 235, 247

Offset: 1

Juri-Stepan Gerasimov, Apr 20 2010

nonn

Semiprimes without 3*primes (or triple the primes).

Numbers of the form A045344(i)*A045344(j), any i, j. [From R. J. Mathar, Apr 27 2010]

Cf. A001358, A001748, A100959.

Entries checked by R. J. Mathar, Apr 27 2010

A176650 Numbers k such that nonnegative non-semiprime(k)+3 = nonnegative non-semiprime(k+3).

Original entry on oeis.org

1, 11, 12, 18, 19, 20, 26, 27, 28, 47, 53, 63, 64, 65, 66, 67, 68, 69, 73, 83, 84, 91, 92, 93, 98, 99, 100, 101, 102, 109, 115, 116, 117, 118, 122, 128, 129, 130, 134, 135, 136, 148, 152, 153, 154, 155, 156, 161, 162, 163, 164, 165, 166, 174, 183, 184, 185, 192, 193

Offset: 1

Juri-Stepan Gerasimov, Apr 22 2010

nonn

Where nonnegative non-semiprime numbers are zero together with A100959 (i.e., 0, 1, 2, 3, 5, 7, 8, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 27, 28, 29, 30, ...).

			1 is a term because nonnegative non-semiprime(1)+3 = 3 = nonnegative non-semiprime(1+3).

Jinyuan Wang, Table of n, a(n) for n = 1..10000

Cf. A100959 (non-semiprimes).

A100959 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if numtheory[bigomega](a) <> 2 then return a; end if; end do end if; end proc:
nnnspr := proc(n) if n = 1 then 0; else A100959(n-1) ; end if; end proc:
isA176650 := proc(n) nnnspr(n) + 3 = nnnspr(n+3) ; end proc:
for n from 1 to 1200 do if isA176650(n) then printf("%d,",n) ; end if; end do:
# R. J. Mathar, Apr 26 2010

Join[{1},Flatten[With[{c=Select[Range[0,300],PrimeOmega[#]!=2&]},Position[ Partition [c,4,1],?(#[[1]]+3==#[[4]]&),1,Heads->False]]]+1] (* _Harvey P. Dale, Oct 15 2022 *)

Entries checked by R. J. Mathar, Apr 26 2010

A176847 The odd non-semiprime numbers.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 17, 19, 23, 27, 29, 31, 37, 41, 43, 45, 47, 53, 59, 61, 63, 67, 71, 73, 75, 79, 81, 83, 89, 97, 99, 101, 103, 105, 107, 109, 113, 117, 125, 127, 131, 135, 137, 139, 147, 149, 151, 153, 157, 163, 165, 167, 171, 173, 175, 179, 181, 189, 191, 193, 195

Offset: 1

Juri-Stepan Gerasimov, Apr 27 2010, May 01 2010

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A100959.

Select[Range[1000], ! PrimeOmega[#] == 2 && OddQ[#] &] (* G. C. Greubel, Sep 22 2018 *)

s=[]; for(n=1, 1000, if(n%2==1 && bigomega(n)!=2, s=concat(s, n))); s \\ Colin Barker, Jun 28 2014

concat( vector( 200, n, if( n%2 && bigomega(n)!=2, [n], []))) /* Michael Somos, Jun 28 2014 */

Corrected (115 removed) by R. J. Mathar, May 01 2010

A262095 Number of non-semiprime divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 4, 2, 3, 3, 4, 2, 4, 2, 4, 3, 3, 2, 6, 2, 3, 3, 4, 2, 5, 2, 5, 3, 3, 3, 6, 2, 3, 3, 6, 2, 5, 2, 4, 4, 3, 2, 8, 2, 4, 3, 4, 2, 6, 3, 6, 3, 3, 2, 8, 2, 3, 4, 6, 3, 5, 2, 4, 3, 5, 2, 9, 2, 3, 4, 4, 3, 5, 2, 8, 4, 3, 2, 8, 3, 3, 3, 6, 2, 8, 3, 4, 3, 3, 3, 10, 2

Offset: 1

Juri-Stepan Gerasimov, Sep 11 2015

			(1, 2, 3, 4, 6, 8, 12, 24) are the divisors of n = 24: 1, 2, 3, 8, 12, and 24 are non-semiprimes, therefore a(24) = 6.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000005, A083399, A086971, A100959.

Cf. A064911, A027750.

a262095 = sum . map ((1 -) . a064911) . a027750_row
-- Reinhard Zumkeller, Sep 14 2015

Table[Count[Divisors@ n, x_ /; PrimeOmega@ x != 2], {n, 97}] (* Michael De Vlieger, Sep 14 2015 *)

a(n) = sumdiv(n, d, bigomega(d)!=2); \\ Michel Marcus, Sep 11 2015

a(n)=my(f=factor(n)[,2]); prod(i=1,#f,f[i]+1) - sum(i=1,#f,f[i]>1) - #f*(#f-1)/2 \\ Charles R Greathouse IV, Sep 14 2015

a(n) = A000005(n) - A086971(n).
A083399(n) <= a(n) <= A000005(n).
a(n) = Sum_{k=1..A000005(n)} (1 - A064911(A027750(n,k))). - Reinhard Zumkeller, Sep 14 2015

A365829 Squarefree non-semiprimes.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 42, 43, 47, 53, 59, 61, 66, 67, 70, 71, 73, 78, 79, 83, 89, 97, 101, 102, 103, 105, 107, 109, 110, 113, 114, 127, 130, 131, 137, 138, 139, 149, 151, 154, 157, 163, 165, 167, 170, 173, 174, 179, 181, 182, 186

Offset: 1

Gus Wiseman, Oct 05 2023

nonn

First differs from A030059 in having 210.

			The terms together with their prime indices begin:
     1: {}          43: {14}       102: {1,2,7}
     2: {1}         47: {15}       103: {27}
     3: {2}         53: {16}       105: {2,3,4}
     5: {3}         59: {17}       107: {28}
     7: {4}         61: {18}       109: {29}
    11: {5}         66: {1,2,5}    110: {1,3,5}
    13: {6}         67: {19}       113: {30}
    17: {7}         70: {1,3,4}    114: {1,2,8}
    19: {8}         71: {20}       127: {31}
    23: {9}         73: {21}       130: {1,3,6}
    29: {10}        78: {1,2,6}    131: {32}
    30: {1,2,3}     79: {22}       137: {33}
    31: {11}        83: {23}       138: {1,2,9}
    37: {12}        89: {24}       139: {34}
    41: {13}        97: {25}       149: {35}
    42: {1,2,4}    101: {26}       151: {36}

First condition alone is A005117 (squarefree).
Second condition alone is A100959 (non-semiprime).
The nonprime case is 1 followed by A350352.
Partitions of this type are counted by A365827, non-strict A058984.
A001358 lists semiprimes, squarefree A006881.
Cf. A000009, A004526, A008967, A078408, A365659.

Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]!=2&]

isok(k) = my(f=factor(k)); issquarefree(f) && (bigomega(f) != 2); \\ Michel Marcus, Oct 07 2023

Intersection of A005117 and A100959.

Complement of A001358 in A005117.

A176548 Non-semiprimes whose reversal is a prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 16, 17, 20, 30, 31, 32, 37, 50, 70, 71, 73, 76, 79, 92, 97, 98, 101, 104, 107, 110, 112, 113, 124, 125, 128, 130, 131, 136, 140, 149, 151, 152, 157, 160, 164, 167, 170, 172, 175, 179, 181, 182, 188, 191, 196, 199, 200, 300, 310, 311, 313, 316

Offset: 1

Juri-Stepan Gerasimov, Apr 20 2010

Cf. A100959.

A100959 := proc(n) option remember ; if n <= 3 then n ; else for a from procname(n-1)+1 do if numtheory[bigomega](a) <> 2 then return a; end if; end do: end if: end proc: read("transforms") ; for n from 1 to 300 do c := A100959(n) ; if isprime(digrev(c)) then printf("%d,",c) ; end if; end do: # R. J. Mathar, Apr 27 2010

Select[Range[400],PrimeOmega[#]!=2&&PrimeQ[FromDigits[Reverse[ IntegerDigits[ #]]]]&] (* Harvey P. Dale, Jan 03 2015 *)

Corrected (20, 30, 50, 70, 76, 92, 110, 128, etc. inserted) by R. J. Mathar, Apr 27 2010

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A133447 Nonsemiprime hyperperfect numbers.

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A133795 a(n) = n-th semiprime + n-th non-semiprime.

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A140234 Sum of the semiprimes <= n.

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A174653 Semiprimes s such that neither s-1 nor s+1 is semiprime.

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A176551 Products of 2 primes of the form 3*k-+1.

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A176650 Numbers k such that nonnegative non-semiprime(k)+3 = nonnegative non-semiprime(k+3).

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A176847 The odd non-semiprime numbers.

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A262095 Number of non-semiprime divisors of n.

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