A120944 -id:A120944 - cp's OEIS frontend

A229273 Composite squarefree numbers n such that p-tau(n) divides n-sigma(n), where p are the prime factors of n, tau(n) = A000005(n) and sigma(n) = A000203(n).

6, 10, 15, 22, 78, 138, 273, 483, 3243, 3913, 104377, 477337, 1537627, 1904487, 2508961, 3326829, 3716167, 5148949, 6154017, 6686113, 11521842, 14355679, 16872583, 25165777, 28029883, 31232337, 32403342, 50725419, 57396469, 68815381, 86850249, 98242959

Offset: 1

Paolo P. Lava, Sep 19 2013

nonn

Subsequence of A120944.

			Prime factors of 273 are 3, 7, 13 and sigma(273) = 448, tau(273) = 8.
273 - 448 = -175 and (-175) / (3 - 8) = 35, (-175) / (7 - 8) = 175, (-175) / (13 - 8) = -35.

Cf. A000005, A000203.

Cf. A228299, A228300, A228301, A228302, A229274, A229275, A229276.

with (numtheory); P:=proc(q) local a, b, c, i, ok, p, n;
for n from 2 to q do  if not isprime(n) then a:=ifactors(n)[2]; ok:=1;
for i from 1 to nops(a) do if a[i][2]>1 or a[i][1]=tau(n) then ok:=0; break;
else if not type((n-sigma(n))/(a[i][1]-tau(n)), integer) then ok:=0; break; fi; fi; od; if ok=1 then print(n); fi; fi; od; end: P(2*10^6);

a(20)-a(33) from Giovanni Resta, Sep 20 2013

First term deleted by Paolo P. Lava, Sep 23 2013

A229274 Composite squarefree numbers n such that p+tau(n) divides n-sigma(n), where p are the prime factors of n, tau(n) = A000005(n) and sigma(n) = A000203(n).

Original entry on oeis.org

51, 93, 177, 219, 303, 471, 597, 681, 723, 807, 849, 933, 1059, 1101, 1227, 1437, 1563, 1689, 1731, 1857, 1941, 1983, 2319, 2361, 2487, 2571, 2823, 2949, 2991, 3117, 3327, 3369, 3453, 3579, 3747, 3831, 3873, 3957, 4083, 4377, 4461, 4629, 4713, 4839, 4881

Offset: 1

Paolo P. Lava, Sep 19 2013

nonn

Subsequence of A120944.

			Prime factors of 177 are 3, 59 and sigma(177) = 240 , tau(177) = 4.
177 - 240 = -63 and (-63) / (3 + 4) = -9, (-63) / (59 + 4) = -1.

Paolo P. Lava, Table of n, a(n) for n = 1..999

Cf. A000005, A000203.

Cf. A228299, A228300, A228301, A228302, A229273, A229275, A229276.

with (numtheory); P:=proc(q) local a, b, c, i, ok, p, n;
for n from 2 to q do  if not isprime(n) then a:=ifactors(n)[2]; ok:=1;
for i from 1 to nops(a) do if a[i][2]>1 then ok:=0; break;
else if not type((n-sigma(n))/(a[i][1]+tau(n)), integer) then ok:=0; break; fi; fi; od; if ok=1 then print(n); fi; fi; od; end: P(10^6);

First term deleted by Paolo P. Lava, Sep 23 2013

A229321 Composite squarefree numbers n such that p-tau(n) divides n+phi(n), where p are the prime factors of n, tau(n) = A000005(n) and phi(n) = A000010(n).

Original entry on oeis.org

6, 10, 15, 21, 39, 110, 170, 609, 897, 935, 1265, 1729, 2882, 2915, 12374, 15387, 161833, 411230, 444797, 558830, 842741, 881705, 1091810, 1122501, 1163990, 1342165, 1565565, 1898259, 2763901, 4157605, 4453697, 4675877, 5962835, 6241610, 6809690, 7201599

Offset: 1

Paolo P. Lava, Sep 20 2013

nonn

Subsequence of A120944.

			Prime factors of 1265 are 5, 11, 23 and tau(1265) = 8, phi(1265) = 880. 1265 + 880 = 2145 and 2145 / (5 - 8) = -715, 2145 / (11 - 8) = 715, 2145 / (23 - 8) = 143.

Cf. A000005, A000010, A228299-A228302, A229274-A229276, A229322-A229324.

with (numtheory); P:=proc(q) global a, b, c, i, ok, p, n;
for n from 2 to q do  if not isprime(n) then a:=ifactors(n)[2]; ok:=1;
for i from 1 to nops(a) do if a[i][2]>1 or a[i][1]=tau(n) then ok:=0; break;
else if not type((n+phi(n))/(a[i][1]-tau(n)), integer) then ok:=0; break; fi; fi; od; if ok=1 then print(n); fi; fi; od; end: P(6*10^9);

a(18)-a(37) from Giovanni Resta, Sep 20 2013

First term deleted by Paolo P. Lava, Sep 23 2013

A354911 Number of factorizations of n into relatively prime prime-powers.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 1, 0, 2, 2, 1, 0, 5, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 2, 0, 1, 2, 0, 1, 1, 0, 2, 1, 1, 0, 6, 0, 1, 2, 2, 1, 1, 0, 5, 0, 1, 0, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 25 2022

nonn

			The a(n) factorizations for n = 6, 12, 24, 36, 48, 72, 96:
  2*3  3*4    3*8      4*9      3*16       8*9        3*32
       2*2*3  2*3*4    2*2*9    2*3*8      2*4*9      3*4*8
              2*2*2*3  3*3*4    3*4*4      3*3*8      2*3*16
                       2*2*3*3  2*2*3*4    2*2*2*9    2*2*3*8
                                2*2*2*2*3  2*3*3*4    2*3*4*4
                                           2*2*2*3*3  2*2*2*3*4
                                                      2*2*2*2*2*3

Wikipedia, Coprime integers.

This is the relatively prime case of A000688, partitions A023894.
Positions of 0's are A246655 (A000961 includes 1).
For strict instead of relatively prime we have A050361, partitions A054685.
Positions of 1's are A000469 (A120944 excludes 1).
For pairwise coprime instead of relatively prime we have A143731.
The version for partitions instead of factorizations is A356067.
A000005 counts divisors.
A001055 counts factorizations.
A001221 counts distinct prime divisors, with sum A001414.
A001222 counts prime-power divisors.
A289509 lists numbers whose prime indices are relatively prime.
A295935 counts twice-factorizations with constant blocks (type PPR).
A355743 lists numbers with prime-power prime indices, squarefree A356065.
Cf. A000837, A023893, A076610, A085970, A106244, A279784, A318721, A355737, A355742, A356064, A356066.

ufacs[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[ufacs[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Table[Length[Select[ufacs[Select[Divisors[n],PrimePowerQ[#]&],n],GCD@@#<=1&]],{n,100}]

a(n) = A000688(n) if n is nonprime, otherwise a(n) = 0.

A355528 Minimal difference between adjacent 0-prepended prime indices of n > 1.

Original entry on oeis.org

1, 2, 0, 3, 1, 4, 0, 0, 1, 5, 0, 6, 1, 1, 0, 7, 0, 8, 0, 2, 1, 9, 0, 0, 1, 0, 0, 10, 1, 11, 0, 2, 1, 1, 0, 12, 1, 2, 0, 13, 1, 14, 0, 0, 1, 15, 0, 0, 0, 2, 0, 16, 0, 2, 0, 2, 1, 17, 0, 18, 1, 0, 0, 3, 1, 19, 0, 2, 1, 20, 0, 21, 1, 0, 0, 1, 1, 22, 0, 0, 1, 23

Offset: 2

Gus Wiseman, Jul 10 2022

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.

			The 0-prepended prime indices of 9842 are {0,1,4,8,12}, with differences (1,3,4,4), so a(9842) = 1.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
Positions of first appearances are 4 followed by A000040.
Positions of positive terms are A005117, complement A013929.
A similar statistic is counted by A238353.
The maximal version is A286469, without prepending A355526.
Without prepending we have A355524 or A355525.
Positions of ones are A355530.
A001522 counts partitions with a fixed point (unproved), ranked by A352827.
A112798 lists prime indices, with sum A056239.
A287352, A355533, A355534, A355536 list the differences of prime indices.
Cf. A064428, A066312, A091602, A120944, A238354, A286470, A325161, A352822, A355527, A355531, A355532.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Min@@Differences[Prepend[primeMS[n],0]],{n,2,100}]

A350352 Products of three or more distinct prime numbers.

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 210, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 330, 345, 354, 357, 366, 370, 374, 385, 390, 399, 402, 406, 410, 418, 426, 429, 430

Offset: 1

Gus Wiseman, Jan 11 2022

nonn

First differs from A336568 in lacking 420.

			The terms and their prime indices begin:
     30: {1,2,3}     182: {1,4,6}      285: {2,3,8}
     42: {1,2,4}     186: {1,2,11}     286: {1,5,6}
     66: {1,2,5}     190: {1,3,8}      290: {1,3,10}
     70: {1,3,4}     195: {2,3,6}      310: {1,3,11}
     78: {1,2,6}     210: {1,2,3,4}    318: {1,2,16}
    102: {1,2,7}     222: {1,2,12}     322: {1,4,9}
    105: {2,3,4}     230: {1,3,9}      330: {1,2,3,5}
    110: {1,3,5}     231: {2,4,5}      345: {2,3,9}
    114: {1,2,8}     238: {1,4,7}      354: {1,2,17}
    130: {1,3,6}     246: {1,2,13}     357: {2,4,7}
    138: {1,2,9}     255: {2,3,7}      366: {1,2,18}
    154: {1,4,5}     258: {1,2,14}     370: {1,3,12}
    165: {2,3,5}     266: {1,4,8}      374: {1,5,7}
    170: {1,3,7}     273: {2,4,6}      385: {3,4,5}
    174: {1,2,10}    282: {1,2,15}     390: {1,2,3,6}

Michael De Vlieger, Table of n, a(n) for n = 1..10000

This is the squarefree case of A033942.
Including squarefree semiprimes gives A120944.
The squarefree complement consists of 1 and A167171.
These are the Heinz numbers of the partitions counted by A347548.
A000040 lists prime numbers (exactly 1 prime factor).
A005117 lists squarefree numbers.
A006881 lists squarefree numbers with exactly 2 prime factors.
A007304 lists squarefree numbers with exactly 3 prime factors.
A046386 lists squarefree numbers with exactly 4 prime factors.
Cf. A000009, A000111, A001250, A002808, A027383, A349796.

Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]>=3&]

is(n,f=factor(n))=my(e=f[,2]); #e>2 && vecmax(e)==1 \\ Charles R Greathouse IV, Jul 08 2022

list(lim)=my(v=List()); forsquarefree(n=30,lim\1, if(#n[2][,2]>2, listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Jul 08 2022

from sympy import factorint
def ok(n): f = factorint(n, multiple=True); return len(f) == len(set(f)) > 2
print([k for k in range(431) if ok(k)]) # Michael S. Branicky, Jan 14 2022

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A350352(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x-sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,i)) for i in range(3,x.bit_length())))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f,n,n) # Chai Wah Wu, Sep 11 2024

A355527 Squarefree numbers having at least one pair of consecutive prime factors. Numbers n such that the minimal difference between adjacent prime indices of n is 1.

Original entry on oeis.org

6, 15, 30, 35, 42, 66, 70, 77, 78, 102, 105, 114, 138, 143, 154, 165, 174, 186, 195, 210, 221, 222, 231, 246, 255, 258, 282, 285, 286, 318, 323, 330, 345, 354, 366, 385, 390, 402, 426, 429, 435, 437, 438, 442, 455, 462, 465, 474, 498, 510, 534, 546, 555, 570

Offset: 1

Gus Wiseman, Jul 10 2022

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.
A number is squarefree if it is not divisible by any perfect square > 1.
A number has consecutive prime factors if it is divisible by both prime(k) and prime(k+1) for some k.

			The terms together with their prime indices begin:
    6: {1,2}
   15: {2,3}
   30: {1,2,3}
   35: {3,4}
   42: {1,2,4}
   66: {1,2,5}
   70: {1,3,4}
   77: {4,5}
   78: {1,2,6}
  102: {1,2,7}
  105: {2,3,4}
  114: {1,2,8}
  138: {1,2,9}
  143: {5,6}
  154: {1,4,5}
  165: {2,3,5}
  174: {1,2,10}
  186: {1,2,11}
  195: {2,3,6}
  210: {1,2,3,4}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
All terms are in A005117, complement A013929.
For minimal difference <= 1 we have A055932.
For maximal instead of minimal difference = 1 we have A066312.
For minimal difference > 1 we have A325160.
If zero is considered a prime index we get A355530.
A001522 counts partitions with a fixed point (unproved), ranked by A352827.
A287352, A355533, A355534, A355536 list the differences of prime indices.
A355524 or A355525 give minimal difference between prime indices.
Cf. A000005, A000040, A056239, A120944, A130091, A238353, A238354, A286470, A325161, A352822, A355526, A355531.

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

Intersection of A005117 (squarefree) and A104210 (has consecutive primes).

A378373 Number of composite numbers (A002808) between consecutive nonsquarefree numbers (A013929), exclusive.

Original entry on oeis.org

1, 0, 1, 2, 0, 0, 2, 0, 1, 0, 1, 3, 2, 1, 0, 1, 0, 0, 1, 0, 1, 2, 1, 0, 2, 2, 1, 0, 2, 0, 1, 3, 0, 1, 3, 0, 0, 0, 1, 2, 2, 2, 0, 2, 0, 2, 0, 0, 0, 2, 2, 0, 1, 3, 2, 0, 0, 0, 0, 2, 2, 1, 0, 2, 0, 1, 0, 1, 0, 2, 2, 3, 0, 1, 2, 0, 0, 3, 2, 0, 2, 3, 3, 2, 0, 1, 2

Offset: 1

Gus Wiseman, Dec 02 2024

nonn

All terms are 0, 1, 2, or 3 (cf. A078147).
The inclusive version is a(n) + 2.
The nonsquarefree numbers begin: 4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, ...

			The composite numbers counted by a(n) form the following set partition of A120944:
{6}, {}, {10}, {14,15}, {}, {}, {21,22}, {}, {26}, {}, {30}, {33,34,35}, {38,39}, ...

For prime (instead of nonsquarefree) we have A046933.
For squarefree (instead of nonsquarefree) we have A076259(n)-1.
For prime power (instead of nonsquarefree) we have A093555.
For prime instead of composite we have A236575.
For nonprime prime power (instead of nonsquarefree) we have A378456.
For perfect power (instead of nonsquarefree) we have A378614, primes A080769.
A002808 lists the composite numbers.
A005117 lists the squarefree numbers, differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147.
A073247 lists squarefree numbers with nonsquarefree neighbors.
A120944 lists squarefree composite numbers.
A377432 counts perfect-powers between primes, zeros A377436.
A378369 gives distance to the next nonsquarefree number (A120327).
Cf. A065890, A067535, A067871, A080101, A081221, A151800, A243348, A366833, A377046, A378033, A378039, A378086.

v=Select[Range[100],!SquareFreeQ[#]&];
Table[Length[Select[Range[v[[i]]+1,v[[i+1]]-1],CompositeQ]],{i,Length[v]-1}]

A228300 Composite squarefree numbers n such that p-d(n) divides n-d(n), where p are the prime factors of n and d(n) the number of divisors of n.

Original entry on oeis.org

6, 10, 15, 110, 170, 273, 638, 935, 1394, 2093, 2438, 2465, 4823, 5453, 7973, 11978, 16354, 17963, 34918, 43337, 46943, 62491, 64583, 68266, 71603, 72046, 74347, 75361, 85877, 134458, 148291, 155933, 186235, 188071, 201994, 209933, 280891, 307021, 367081

Offset: 1

Paolo P. Lava, Aug 20 2013

nonn

Subsequence of A120944.

			Prime factors of 17963 are 11, 23 and 71 while d(17963) = 8. We have that 17963 - 8 = 17955 and 17955 / (11 - 8) =  5985, 17955 / (23 - 8) = 1197 and 17955 / (71 - 8) = 285.

Donovan Johnson, Table of n, a(n) for n = 1..500

Cf. A000005, A228299, A228301, A228302.

with (numtheory); P:=proc(q) local a,b,c,i,ok,p,n;
for n from 2 to q do  if not isprime(n) then a:=ifactors(n)[2]; ok:=1;
for i from 1 to nops(a) do if a[i][2]>1 or a[i][1]=tau(n) then ok:=0; break;
else if not type((n-tau(n))/(a[i][1]-tau(n)),integer) then ok:=0; break; fi; fi; od; if ok=1 then print(n); fi; fi; od; end: P(10^6);

First term deleted by Paolo P. Lava, Sep 23 2013

A228301 Composite squarefree numbers n such that p-d(n) divides n+d(n), where p are the prime factors of n and d(n) the number of divisors of n.

Original entry on oeis.org

6, 10, 14, 15, 35, 70, 154, 190, 322, 385, 442, 595, 682, 2737, 3619, 14986, 15314, 19019, 24817, 26767, 33626, 78387, 85034, 130169, 155363, 166934, 189727, 214107, 225029, 238901, 243217, 285934, 381547, 395219, 415679, 417989, 455609, 466193, 544918

Offset: 1

Paolo P. Lava, Aug 20 2013

nonn

Subsequence of A120944.

			Prime factors of 19019 are 7, 11, 13 and 19 while d(19019) = 16. We have that 19019 + 16 = 19035 and 19035 / (7 - 16) =  -2115, 19035 / (11 - 16) =  -3807, 19035 / (13 - 16) = -6345 and 19035 / (19 - 16) = 6345.

Donovan Johnson, Table of n, a(n) for n = 1..500

Cf. A000005, A228299, A228300, A228302.

with (numtheory); P:=proc(q) local a,b,c,i,ok,p,n;
for n from 2 to q do  if not isprime(n) then a:=ifactors(n)[2]; ok:=1;
for i from 1 to nops(a) do if a[i][2]>1 or a[i][1]=tau(n) then ok:=0; break;
else if not type((n+tau(n))/(a[i][1]-tau(n)),integer) then ok:=0; break; fi; fi; od; if ok=1 then print(n); fi; fi; od; end: P(10^6);

First term deleted by Paolo P. Lava, Sep 23 2013

cp's OEIS Frontend

A229273 Composite squarefree numbers n such that p-tau(n) divides n-sigma(n), where p are the prime factors of n, tau(n) = A000005(n) and sigma(n) = A000203(n).

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A229274 Composite squarefree numbers n such that p+tau(n) divides n-sigma(n), where p are the prime factors of n, tau(n) = A000005(n) and sigma(n) = A000203(n).

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A229321 Composite squarefree numbers n such that p-tau(n) divides n+phi(n), where p are the prime factors of n, tau(n) = A000005(n) and phi(n) = A000010(n).

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A354911 Number of factorizations of n into relatively prime prime-powers.

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A355528 Minimal difference between adjacent 0-prepended prime indices of n > 1.

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A350352 Products of three or more distinct prime numbers.

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A355527 Squarefree numbers having at least one pair of consecutive prime factors. Numbers n such that the minimal difference between adjacent prime indices of n is 1.

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Mathematica

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A378373 Number of composite numbers (A002808) between consecutive nonsquarefree numbers (A013929), exclusive.

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