A087436 -id:A087436 - cp's OEIS frontend

A106799 Number of prime factors of n apart from 2 or 3, counted with multiplicity.

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

Offset: 1

Henry Bottomley, May 17 2005

nonn

Self-similar in every second and in every third term, i.e., a(n) = a(2n) = a(3n).
Logarithmic since a(b*c) = a(b) + a(c).
Coincidentally, a(n) = A101040(n+78) for 1 < n < 20.

			a(24) = 0 since 24 = 2*2*2*3.
a(25) = 2 since 25 = 5*5.
a(26) = 1 since 26 = 2*13.

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

Cf. A001222, A007814, A007949, A027746, A065330, A087436, A101040, A169611.

a106799 = a001222 . a065330  -- Reinhard Zumkeller, Nov 19 2015

a[n_] := PrimeOmega[n] - IntegerExponent[n, 2] - IntegerExponent[n, 3]; Array[a, 100] (* Amiram Eldar, Jan 16 2022 *)

a(n) = bigomega(n) - valuation(n, 2) - valuation(n, 3); \\ Michel Marcus, Jan 16 2022

a(n) = A001222(n) - A007814(n) - A007949(n) = A087436(n) - A007949(n).

a(n) = A001222(A065330(n)). - Reinhard Zumkeller, May 19 2005

A359640 a(n) is the least odd prime not in A001359 such that all subsequent composites in the gap up to the next prime have exactly n odd prime factors, counted with multiplicity.

Original entry on oeis.org

307, 1999, 101527, 7146697, 272572999, 4809363523

Offset: 2

Hugo Pfoertner, Jan 16 2023

			a(2) = 307: 308 = 2^2*7*11, 309 = 3*103, 310 = 2*5*31, all have exactly 2 odd prime factors.

Cf. A001222, A001359, A087436, A359637, A359639.

a087436(n) = bigomega (n >> valuation (n, 2));
a359640(maxp) = {my(k=2, pp=5); forprime (p=7, maxp, my(mi=oo, ma=0); if (p-pp>2, for (j=pp+1, p-1, my(mo=a087436(j)); if (mo

A366839 Sum of even prime factors of 2n, counted with multiplicity.

Original entry on oeis.org

2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 10, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 12, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 10, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 14, 2, 4, 2, 6, 2, 4, 2, 8, 2, 4, 2, 6, 2, 4, 2, 10, 2, 4, 2, 6, 2

Offset: 1

Gus Wiseman, Oct 25 2023

nonn

			The prime factors of 2*60 are {2,2,2,3,5}, of which the even factors are {2,2,2}, so a(60) = 6.

A compound version is A001414, triangle A331416.
Dividing by 2 gives A001511.
Positions of 2's are A005408.
For count instead of sum we have A007814, odd version A087436.
The partition triangle for this statistic is A116598 aerated.
For indices we have A366531, halved A366533, triangle A113686/A174713.
The odd version is A366840.
A019507 lists numbers with (even factor sum) = (odd factor sum).
A066207 lists numbers with all even prime indices, counted by A035363.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A162641 counts even prime exponents, odd A162642.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
A257992 counts even prime indices, odd A257991.
A366528 adds up odd prime indices, triangle A113685 (without zeros A365067).
Cf. A000720, A066208, A258117, A325698.

Table[2*Length[NestWhileList[#/2&,n,EvenQ]],{n,100}]

a(n) = 2 * valuation(n, 2) + 2; \\ Amiram Eldar, Sep 13 2024

a(n) = 2*A001511(n).
a(n) = A100006(n) - A366840(2n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4. - Amiram Eldar, Sep 13 2024

A366840 Sum of odd prime factors of n, counted with multiplicity.

Original entry on oeis.org

0, 0, 3, 0, 5, 3, 7, 0, 6, 5, 11, 3, 13, 7, 8, 0, 17, 6, 19, 5, 10, 11, 23, 3, 10, 13, 9, 7, 29, 8, 31, 0, 14, 17, 12, 6, 37, 19, 16, 5, 41, 10, 43, 11, 11, 23, 47, 3, 14, 10, 20, 13, 53, 9, 16, 7, 22, 29, 59, 8, 61, 31, 13, 0, 18, 14, 67, 17, 26, 12, 71, 6

Offset: 1

Gus Wiseman, Oct 27 2023

Contains all positive integers except 1, 2, 4.

			The prime factors of 60 are {2,2,2,3,5}, of which the odd factors are {3,5}, so a(60) = 8.

The compound version is A001414, triangle A331416.
For count instead of sum we have A087436, even version A007814.
Odd-indexed terms are A100005.
Positions of odd terms are A335657, even A036349.
For prime indices we have A366528, triangle A113685 (without zeros A365067)
The even version is A366839 = 2*A001511.
The partition triangle for this statistic is A366851, even version A116598.
A019507 lists numbers with (even factor sum) = (odd factor sum).
A066207 lists numbers with all even prime indices, counted by A035363.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A162641 counts even prime exponents, odd A162642.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
A257992 counts even prime indices, odd A257991.
Cf. A000009, A066208, A113686, A174713, A258117, A325698, A366531, A366850.

Table[Total[Times@@@DeleteCases[If[n==1,{}, FactorInteger[n]],{2,_}]],{n,100}]

a(n) = my(f=factor(n), j=if(n%2, 1, 2)); sum(i=j, #f~, f[i,1]*f[i,2]); \\ Michel Marcus, Oct 30 2023

a(n) = A100006(n) - A366839(n).
a(2n) = a(n).
a(2n-1) = A001414(2n-1) = A100005(n).
Completely additive with a(2^e) = 0 and a(p^e) = e*p for an odd prime p. - Amiram Eldar, Nov 03 2023

A322824 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = 0 if n is an odd prime, and f(n) = A242424(n) for all other numbers.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 7, 8, 3, 9, 3, 10, 11, 12, 3, 13, 3, 10, 14, 15, 3, 16, 17, 18, 19, 20, 3, 14, 3, 21, 22, 23, 24, 25, 3, 26, 27, 15, 3, 28, 3, 29, 30, 31, 3, 32, 33, 24, 34, 35, 3, 36, 37, 29, 38, 39, 3, 22, 3, 40, 41, 42, 43, 44, 3, 45, 46, 33, 3, 47, 3, 48, 49, 50, 51, 52, 3, 18, 53, 54, 3, 44, 55, 56, 57, 58, 3, 59, 60, 61, 62, 63, 64, 65, 3, 66, 67, 37, 3

Offset: 1

Antti Karttunen, Dec 29 2018

nonn

For all i, j: a(i) = a(j) => A087436(i) = A087436(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A087436, A242424, A322814.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A242424(n) = if(1==n,n,prime(bigomega(n))*A064989(n));
A322824aux(n) = if((n>2)&&isprime(n),0,A242424(n));
v322824 = rgs_transform(vector(up_to,n,A322824aux(n)));
A322824(n) = v322824[n];

A359639 a(n) is the least odd prime not in A001359 such that all subsequent composites in the gap up to the next prime have at least n odd prime factors, counted with multiplicity.

Original entry on oeis.org

97, 1999, 101527, 6666547, 272572999, 3819770107, 410274361249

Offset: 2

Hugo Pfoertner, Jan 16 2023

			a(2) = 97: 98 = 2*7^2, 99 = 3^2*11, 100 = 2^2*5^2 have 2 or 3 odd prime factors, so the minimum 2 is achieved.
a(3) = 1999: 2000 has the 3 odd prime factors 5^3, 2001 = 3*23*29, 2002 = 2*7*11*13.

Cf. A001222, A001359, A087436, A359637, A359640.

a087436(n) = bigomega (n >> valuation (n, 2));
a359639(maxp) = {my(k=2,pp=5); forprime (p=7, maxp, my(mi=oo); if (p-pp>2, for (j=pp+1, p-1, my(mo=a087436(j)); if (mo=k, print1(pp,", "); k++)); pp=p)};
a359639(3*10^8)

A379313 Positive integers whose prime indices are not all composite.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82

Offset: 1

Gus Wiseman, Dec 28 2024

nonn

Or, positive integers whose prime indices include at least one 1 or prime number.

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 terms together with their prime indices begin:
     2: {1}
     3: {2}
     4: {1,1}
     5: {3}
     6: {1,2}
     8: {1,1,1}
     9: {2,2}
    10: {1,3}
    11: {5}
    12: {1,1,2}
    14: {1,4}
    15: {2,3}
    16: {1,1,1,1}
    17: {7}
    18: {1,2,2}
    20: {1,1,3}
    21: {2,4}
    22: {1,5}
    24: {1,1,1,2}

Partitions of this type are counted by A000041 - A023895.
The "old" primes are listed by A008578.
For no composite parts we have A302540, counted by A034891 (strict A036497).
The complement is A320629, counted by A023895 (strict A204389).
For a unique prime we have A331915, counted by A379304 (strict A379305).
Positions of nonzeros in A379311.
For a unique 1 or prime we have A379312, counted by A379314 (strict A379315).
A000040 lists the prime numbers, differences A001223.
A002808 lists the composite numbers, nonprimes A018252, differences A073783 or A065310.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A080339 is the characteristic function for the old prime numbers.
A376682 gives k-th differences of old prime numbers, see A030016, A075526.
A377033 gives k-th differences of composite numbers, see A073445, A377034.
Other counts of prime indices:
- A257991 odd, see A000041, A000070, A066207, A349158.
- A257992 even, see A000009, A038348, A066208, A379317.
- A257994 prime, see A000586, A000607, A076610, A331386.
- A330944 nonprime, see A002095, A096258, A320628, A330945.
- A379306 squarefree, see A302478, A379308, A379309, A379316.
- A379310 nonsquarefree, see A114374, A256012, A379307.
`Cf. A038550, A087436, A173390, A379300, A379301.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!And@@CompositeQ/@prix[#]&]

A105442 Numbers with at least two odd prime factors (not necessarily distinct) such that in binary representation all divisors of n are contained in n.

Original entry on oeis.org

55, 215, 407, 1403, 1681, 3223, 3362, 3415, 6724, 13448, 13655, 15487, 25751, 80089, 146621, 160178, 218455, 237169, 320356, 445663, 464711, 474338, 873815, 948676, 1662743, 1897352, 1932377, 1975531, 2484187, 3223001, 3410639, 3872639

Offset: 1

Reinhard Zumkeller, Apr 09 2005

nonn

Intersection of A093641 and A105441;

A087436(a(n)) > 1.

Cf. A093642, A093641.

A284233 Sum of odd prime power divisors of n (not including 1).

Original entry on oeis.org

0, 0, 3, 0, 5, 3, 7, 0, 12, 5, 11, 3, 13, 7, 8, 0, 17, 12, 19, 5, 10, 11, 23, 3, 30, 13, 39, 7, 29, 8, 31, 0, 14, 17, 12, 12, 37, 19, 16, 5, 41, 10, 43, 11, 17, 23, 47, 3, 56, 30, 20, 13, 53, 39, 16, 7, 22, 29, 59, 8, 61, 31, 19, 0, 18, 14, 67, 17, 26, 12, 71, 12, 73, 37, 33, 19, 18, 16, 79, 5

Offset: 1

Ilya Gutkovskiy, Mar 23 2017

			a(15) = 8 because 15 has 4 divisors {1, 3, 5, 15} among which 2 are odd prime powers {3, 5} therefore 3 + 5 = 8.

Eric Weisstein's World of Mathematics, Prime Power.
Index entries for sequences related to sums of divisors.

Cf. A000961, A005069, A023888, A023889, A038712, A061345, A065091 (fixed points), A087436 (number of odd prime power divisors of n), A206787, A246655, A284117.

nmax = 80; Rest[CoefficientList[Series[Sum[Boole[PrimePowerQ[k] && Mod[k, 2] == 1] k x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Total[Select[Divisors[n], PrimePowerQ[#] && Mod[#, 2] == 1 &]], {n, 80}]
f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; f[2, e_] := 0; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 24 2024 *)

G.f.: Sum_{k>=1} A061345(k)*x^A061345(k)/(1 - x^A061345(k)).
a(n) = Sum_{d|n, d = p^k, p prime, p > 2, k > 0} d.
a(p^k) = p*(p^k - 1)/(p - 1) for p is a prime > 2.
a(2^k*p) = p for p is a prime > 2.
a(2^k) = 0.
Additive with a(2^e) = 0, and a(p^e) = (p^(e+1)-1)/(p-1) - 1 for an odd prime p. - Amiram Eldar, Jul 24 2024

A359587 Fully multiplicative with a(p) = A008578(1+A329697(p)).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 3, 2, 3, 3, 4, 1, 2, 4, 5, 2, 6, 3, 5, 2, 4, 3, 8, 3, 5, 4, 5, 1, 6, 2, 6, 4, 5, 5, 6, 2, 3, 6, 7, 3, 8, 5, 7, 2, 9, 4, 4, 3, 5, 8, 6, 3, 10, 5, 7, 4, 5, 5, 12, 1, 6, 6, 7, 2, 10, 6, 7, 4, 5, 5, 8, 5, 9, 6, 7, 2, 16, 3, 5, 6, 4, 7, 10, 3, 5, 8, 9, 5, 10, 7, 10, 2, 3, 9, 12, 4, 5, 4, 5, 3, 12

Offset: 1

Antti Karttunen, Jan 08 2023

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000265, A001222, A008578, A056239, A087436, A318995, A329697, A336396, A336466.

A008578(n) = if(1==n,1,prime(n-1));
A329697(n) = if(!bitand(n, n-1), 0, 1+A329697(n-(n/vecmax(factor(n)[, 1]))));
A359587(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 1] = A008578(1+A329697(f[i, 1]))); factorback(f); };

For n >= 1: (Start)
a(A000265(n)) = a(2*n) = a(n).
A001222(a(n)) = A087436(n),
A056239(a(n)) = A329697(n),
A318995(a(n)) = A336396(n) = A329697(A336466(n)).
(End)

cp's OEIS Frontend

A106799 Number of prime factors of n apart from 2 or 3, counted with multiplicity.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A359640 a(n) is the least odd prime not in A001359 such that all subsequent composites in the gap up to the next prime have exactly n odd prime factors, counted with multiplicity.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

A366839 Sum of even prime factors of 2n, counted with multiplicity.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A366840 Sum of odd prime factors of n, counted with multiplicity.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A322824 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = 0 if n is an odd prime, and f(n) = A242424(n) for all other numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A359639 a(n) is the least odd prime not in A001359 such that all subsequent composites in the gap up to the next prime have at least n odd prime factors, counted with multiplicity.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

A379313 Positive integers whose prime indices are not all composite.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A105442 Numbers with at least two odd prime factors (not necessarily distinct) such that in binary representation all divisors of n are contained in n.

Views

Author

Keywords

Comments

Crossrefs

A284233 Sum of odd prime power divisors of n (not including 1).

Views

Author

Keywords

Examples