A046363 -id:A046363 - cp's OEIS frontend

A376548 a(1)=1,a(2)=2. Let p be the smallest prime factor of j=a(n-1), then for n>2 a(n) is the smallest novel multiple of Sopfr(j) if j is in A046363, or of p if it is not.

1, 2, 4, 6, 5, 10, 7, 14, 8, 12, 21, 3, 9, 15, 18, 16, 20, 22, 13, 26, 24, 28, 11, 33, 27, 30, 32, 34, 19, 38, 36, 40, 44, 42, 46, 48, 55, 25, 35, 45, 66, 50, 52, 17, 51, 39, 54, 77, 49, 56, 65, 60, 58, 31, 62, 64, 68, 70, 72, 74, 76, 23, 69, 57, 63, 78, 80, 91

Offset: 1

David James Sycamore, Sep 27 2024

nonn

If j=a(n-1) is in A046363 then a(n) = A001414(j) = prime q, and a(n+1) is the least novel multiple of q. Otherwise a(n) is the least novel multiple of the smallest prime factor of j. After a prime term a(n) the sequence produces a string of terms each divisible by the smallest prime factor of a(n+1) until arriving at a term in A046363, whereupon a new prime appears and the process repeats.

Conjectured to be a permutation of the positive integers A000027, in which the primes do not appear in order (prime order starts:2,5,7,3,13,11,19,17,31,23,43..).

			a(2)=2 is not a term in A046360, and has smallest prime factor 2, so a(3) = 4, the least novel multiple of 2. Likewise a(4)=6 since a(3)=4 is not in A046360 and the smallest prime factor of 4 is 2.
a(4)=6 is a term in A046360, so a(5)=A001414(6)=5.
a(6)=10 since 5 is the smallest prime factor of 5, and 10 is the smallest novel multiple of 5.
If a(n-1) = prime p, a(n) is the least novel multiple of p, for example a(12) = 3 and since a(4) = 6 it follows that a(13) = 9. Likewise a(19) = 13, and since no prior term is divisible by 13, a(20) = 36.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^12, showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, with purple additionally representing powerful numbers that are not squarefree.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20, with the same color function as immediately above. Note various trajectories of primes.

Cf. A000027, A001414, A300813.

nn = 120; c[] := False; m[] := 1;
a[1] = 1; j = a[2] = 2; c[1] = c[2] = True; m[1] = m[2] = 2;
f[x_] := f[x] = Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[x]];
Do[(If[PrimeQ[#2], k = #2, k = #1]; While[c[k*m[k]], m[k]++]; k *= m[k]) & @@
  {FactorInteger[j][[1, 1]], f[j]};
  Set[{a[i], c[k], j}, {k, True, k}], {i, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Sep 28 2024 *)

A133910 Period numbers of A133900 divided by n^2.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 6, 1, 4, 3, 1, 1, 8, 1, 4, 3, 8, 1, 6, 1, 8, 1, 4, 1, 360, 1, 1, 9, 16, 5, 24, 1, 16, 9, 20, 1, 144, 1, 8, 15, 16, 1, 18, 1, 16, 9, 8, 1, 16, 5, 28, 9, 16, 1, 360, 1, 16, 21, 1, 5, 288, 1, 16, 9, 1120, 1, 24, 1, 32, 9, 16, 7, 288, 1, 20, 1, 64, 1, 6048, 5, 32, 27, 8

Offset: 1

Hieronymus Fischer, Oct 20 2007

nonn

			a(6)=2, since A133900(6)/6^2=72/36=2.
a(18)=8, since A133900(18)/18^2=2592/324=8.

Cf. A000040, A100118, A046363, A133620, A133621, A133623, A133630, A133635.

Cf. A133872, A133880, A133890, A133900, A133911.

a(n)=A133900(n)/n^2.
a(n)=1, iff n is a prime or a power of a prime (including n=1).
If a prime p is a factor of a(n), then p is also a factor of n.

A100118 Numbers whose sum of prime factors is prime (counted with multiplicity).

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 12, 13, 17, 19, 22, 23, 28, 29, 31, 34, 37, 40, 41, 43, 45, 47, 48, 52, 53, 54, 56, 58, 59, 61, 63, 67, 71, 73, 75, 76, 79, 80, 82, 83, 88, 89, 90, 96, 97, 99, 101, 103, 104, 107, 108, 109, 113, 117, 118, 127, 131, 136, 137, 139, 142, 147, 148, 149

Offset: 1

Carlos Alves, Dec 26 2004

nonn

Numbers n such that integer log of n is a prime number.
As in A001414, denote sopfr(n) the integer log of n. Since sopfr(p)=p, the sequence includes all prime numbers.
See A046363 for the analog excluding prime numbers. - Hieronymus Fischer, Oct 20 2007
These numbers may be arranged in a family of posets of triangles of multiarrows (see link and example). - Gus Wiseman, Sep 14 2016

			40 = 2^3*5 and 2*3 + 5 = 11 is a prime number.
These numbers correspond to multiarrows in the multiorder of partitions of prime numbers into prime parts. For example: 2:2<=(2), 3:3<=(3), 6:5<=(2,3), 5:5<=(5), 12:7<=(2,2,3), 10:7<=(2,5), 7:7<=(7), 48:11<=(2,2,2,2,3), 52:11<=(2,3,3,3), 40:11<=(2,2,2,5), 45:11<=(3,3,5), 28:11<=(2,2,7), 11:11<=(11). - _Gus Wiseman_, Sep 14 2016

Jayanta Basu, Table of n, a(n) for n = 1..10000
Gus Wiseman, Lattice form posets indexed by A100118

Cf. A001414, A046363, A056768, A276687.

for n from 1 to 200 do
    if isprime(A001414(n)) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Sep 09 2015

L = {}; Do[ww = Transpose[FactorInteger[k]];w = ww[[1]].ww[[2]]; If[PrimeQ[w], AppendTo[L, k]], {k, 2, 500}];L
Select[Range[150], PrimeQ[Total[Times @@@ FactorInteger[#]]] &] (* Jayanta Basu, Aug 11 2013 *)

is(n)=my(f=factor(n)); isprime(sum(i=1,#f~,f[i,1]*f[i,2])) \\ Charles R Greathouse IV, Sep 21 2013

A134333 Numbers n whose number of prime factors (counted with multiplicity) is a prime factor of n.

Original entry on oeis.org

4, 6, 10, 12, 14, 18, 22, 26, 27, 30, 34, 38, 42, 45, 46, 58, 62, 63, 66, 74, 75, 78, 80, 82, 86, 94, 99, 102, 105, 106, 114, 117, 118, 120, 122, 134, 138, 142, 146, 147, 153, 158, 165, 166, 171, 174, 178, 180, 186, 194, 195, 200, 202, 206, 207, 214, 218, 222, 226

Offset: 1

Hieronymus Fischer, Oct 23 2007

nonn

			a(1) = 4, since 4 has 2 prime factors and 2 is a prime factor of 4.
a(4) = 12, since 12 = 2*2*3 has 3 prime factors, and 3 is a prime factor of 12.
a(21) = 75, since 75 = 3*3*5 has 3 prime factors. and 3 is a prime factor of 75.
9 = 3*3 is not a term, since the number of prime factors (=2) is not a divisor of 9.
28 = 2*2*7 is not a term, since the number of prime factors (=3) is not a divisor of 28.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A000040, A001222, A100118, A046363, A133620, A133621, A133880, A133890, A133900, A133910, A133911, A046346, A134331, A134332, A134334, A134344, A134376, A063989.

fQ[n_] := Module[{d = Total[Transpose[FactorInteger[n]][[2]]]}, PrimeQ[d] && Mod[n, d] == 0]; Select[Range[2, 226], fQ] (* T. D. Noe, Apr 05 2013 *)

a(n)=my(t=bigomega(n)); n%t==0 && isprime(t) \\ Charles R Greathouse IV, Sep 14 2015

a(n) << n log n/(log log n)^k for any fixed k. - Charles R Greathouse IV, Sep 14 2015

Sequence definition corrected and examples added by Hieronymus Fischer, Apr 05 2013

A134344 Composite numbers such that the arithmetic mean of their prime factors (counted with multiplicity) is prime.

Original entry on oeis.org

4, 8, 9, 16, 20, 21, 25, 27, 32, 33, 44, 49, 57, 60, 64, 68, 69, 81, 85, 93, 105, 112, 116, 121, 125, 128, 129, 133, 145, 156, 169, 177, 180, 188, 195, 205, 212, 213, 217, 220, 231, 237, 243, 249, 253, 256, 265, 272, 275, 289, 297, 309, 332, 336, 343, 356, 361

Offset: 1

Hieronymus Fischer, Oct 23 2007

nonn

Originally, the definition started with "Nonprime numbers ...". This may be misleading, since 1 is also nonprime, but has no prime factors. - Hieronymus Fischer, May 05 2013

			a(1) = 4, since 4 = 2*2 and the arithmetic mean (2+2)/2 = 2 is prime.
a(5) = 20, since 20 = 2*2*5 and the arithmetic mean (2+2+5)/3 = 3 is prime.

Harvey P. Dale and Hieronymus Fischer, Table of n, a(n) for n = 1..10000 (first 1000 terms from _Harvey P. Dale_)

Cf. A000040, A001222, A100118, A046363, A133620, A133621, A133880, A133890, A133900, A133910, A133911, A046346, A134331, A134332, A134333, A134334.

ampfQ[n_]:=PrimeQ[Mean[Flatten[Table[#[[1]],{#[[2]]}]&/@FactorInteger[ n]]]]; nn=400;Select[Complement[Range[nn],Prime[Range[ PrimePi[nn]]]], ampfQ] (* Harvey P. Dale, Nov 06 2012 *)

is(n)=if(n<4,return(0)); my(f=factor(n),s=sum(i=1,#f~,f[i,1]*f[i,2])/sum(i=1,#f~,f[i,2])); (#f~>1 || f[1,2]>1) && denominator(s)==1 && isprime(s) \\ Charles R Greathouse IV, Sep 14 2015

Definition clarified by Hieronymus Fischer, May 05 2013

A134376 Numbers whose sum of prime factors (counted with multiplicity) is not prime.

Original entry on oeis.org

1, 4, 8, 9, 14, 15, 16, 18, 20, 21, 24, 25, 26, 27, 30, 32, 33, 35, 36, 38, 39, 42, 44, 46, 49, 50, 51, 55, 57, 60, 62, 64, 65, 66, 68, 69, 70, 72, 74, 77, 78, 81, 84, 85, 86, 87, 91, 92, 93, 94, 95, 98, 100, 102, 105, 106, 110, 111, 112, 114, 115, 116, 119, 120, 121, 122

Offset: 1

Hieronymus Fischer, Oct 23 2007

nonn

The first term is 1, since 1 has no prime factors and so the sum of prime factors evaluates to zero.

Conjecture: a(n) ~ n. - Charles R Greathouse IV, Apr 28 2015

			a(2) = 4, since 4 = 2*2 and 2+2 = 4 is not prime.
a(5) = 14, since 14 = 2*7 and 2+7 = 9 is not prime.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A000040, A001222, A100118, A046363, A133620, A133621, A133880, A133890, A133900, A133910, A133911, A046346, A134331, A134332, A134333, A134334, A134344, A078177.

Select[Range[150],!PrimeQ[Total[Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[ #]]]]&] (* Harvey P. Dale, Jul 05 2021 *)

sopfr(n)=my(f=factor(n)); sum(i=1,#f~,f[i,1]*f[i,2])
is(n)=!isprime(sopfr(n)) \\ Charles R Greathouse IV, Apr 28 2015

Edited by the author at the suggestion of T. D. Noe, May 20 2013

A036349 Numbers whose sum of prime factors (taken with multiplicity) is even.

Original entry on oeis.org

1, 2, 4, 8, 9, 15, 16, 18, 21, 25, 30, 32, 33, 35, 36, 39, 42, 49, 50, 51, 55, 57, 60, 64, 65, 66, 69, 70, 72, 77, 78, 81, 84, 85, 87, 91, 93, 95, 98, 100, 102, 110, 111, 114, 115, 119, 120, 121, 123, 128, 129, 130, 132, 133, 135, 138, 140, 141, 143, 144, 145, 154, 155

Offset: 1

Patrick De Geest, Dec 15 1998

nonn

A multiplicative semigroup; if m and n are in the sequence then so is m*n. - David James Sycamore, Jul 17 2018
From Peter Munn, Jul 19 2020: (Start)
Also closed under the commutative binary operation A059897(.,.), forming a subgroup of the positive integers under A059897.
A number is listed if and only if it has an even number of odd prime factors, counting repetitions; equivalently, if and only if it is the product of a term of A046337 and a power of 2 (term of A000079).
(End)

			141 = 3 * 47 is a term since the sum 3 + 47 = 50 is even.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001414 (sopfr), A059897.
Complement of A335657.
Sequences with similar definitions: A036350, A046363, A289142.
Subsequences: A000079, A028982, A046337, A056913.

filter:= proc(n) local t; add(t[1]*t[2],t=ifactors(n)[2])::even end proc:
select(filter, [$1..200]); # Robert Israel, Jul 15 2020

Select[Range[160],EvenQ[Total[Times@@@FactorInteger[#]]]&] (* Harvey P. Dale, Sep 21 2011 *)

isok(n) = my(f=factor(n)); (sum(k=1, #f~, f[k,1]*f[k,2]) % 2) == 0; \\ Michel Marcus, Jul 19 2018

Sum_{n>=1} 1/a(n)^s = (zeta(s) + ((2^s + 1)/(2^s - 1))*zeta(2*s)/zeta(s))/2 for Re(s)>1. - Amiram Eldar, Nov 02 2020

First term (2) from Harvey P. Dale, Sep 21 2011

First term (1) from David James Sycamore, Jul 17 2018

A289142 Numbers whose sum of prime factors (taken with multiplicity) is divisible by 3.

Original entry on oeis.org

1, 3, 8, 9, 14, 20, 24, 26, 27, 35, 38, 42, 44, 50, 60, 62, 64, 65, 68, 72, 74, 77, 78, 81, 86, 92, 95, 105, 110, 112, 114, 116, 119, 122, 125, 126, 132, 134, 143, 146, 150, 155, 158, 160, 161, 164, 170, 180, 185, 186, 188, 192, 194, 195, 196, 203, 204

Offset: 1

David James Sycamore, Jun 26 2017

nonn

U{S(n); 3|n}, where S(n)= {x; sopfr(x)=n}; numbers placed in ascending order.
A multiplicative semigroup: if m and n are in the sequence, then so is m*n. - Robert Israel, Jul 03 2017
From Antti Karttunen, Jun 11 2024, with minor edits Jun 30 2024: (Start)
Numbers such that the multiplicities of prime factors of the forms 3m+1 (A002476) and 3m-1 (A003627) are equal modulo 3.
For n that is not a multiple of 3, sopfr(n) [= A001414(n)] is a multiple of 3 if and only if the arithmetic derivative of n [= A003415(n)] is a multiple of 3. See A373475 for a proof.
This sequence (as a multiplicative semigroup) is generated by the union of A369659 with {3}.
(End)

			sopfr(42) = 2 + 3 + 7 = 12 = 4*3, sopfr(95) = 5 + 19 = 24 = 8 * 3, sopfr(180) = 2 + 2 + 3 + 3 + 5 = 15 = 5 * 3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002476, A003627, A036349, A036350, A046363, A373371 (characteristic function).
Positions of multiples of 3 in A001414 (sopfr) and in A118503.
Subsequences that are formed by intersecting this sequence with other multiplicative semigroups: A102217, A369659, A373373, A373473, A373475, A373478, A373597.
Cf. also A373385, A373602, A374052.

select(n -> add(t[1]*t[2],t=ifactors(n)[2]) mod 3 = 0, [$1..1000]); # Robert Israel, Jul 03 2017

Join[{1},Select[Range[250],Mod[Total[Times@@@FactorInteger[#]],3]==0&]] (* Harvey P. Dale, Mar 16 2020 *)

s(n)=my(f=factor(n),p=f[,1],e=f[,2]);sum(k=1,#p,e[k]*p[k]);
for(n=1,200,if(s(n)%3==0,print1(n,","))); \\ Joerg Arndt, Jun 26 2017

isA289142 = A373371; \\ Antti Karttunen, Jun 08 2024

For n >= 2, a(n) = A102217(n-1)/3. - Antti Karttunen, Jun 08 2024

Corrected by Robert Israel, Jul 03 2017

A134334 Numbers which are not divisible by the number of their prime factors (counted with multiplicity).

Original entry on oeis.org

8, 9, 15, 20, 21, 25, 28, 32, 33, 35, 39, 44, 48, 49, 50, 51, 52, 54, 55, 57, 64, 65, 68, 69, 70, 72, 76, 77, 81, 85, 87, 90, 91, 92, 93, 95, 98, 108, 110, 111, 112, 115, 116, 119, 121, 123, 124, 125, 126, 128, 129, 130, 133, 135, 141, 143, 145, 148, 150, 154, 155, 159

Offset: 1

Hieronymus Fischer, Oct 23 2007

nonn

The asymptotic density of this sequence is 1 (Erdős and Pomerance, 1990). - Amiram Eldar, Jul 10 2020

			a(1) = 8, since 8 = 2*2*2 has 3 prime factors and 8 is not divisible by 3.
a(3) = 15, since 15 = 3*5 has 2 prime factors and 15 is not divisible by 2.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000
Paul Erdős and Carl Pomerance, On a theorem of Besicovitch: values of arithmetic functions that divide their arguments, Indian J. Math., Vol. 32 (1990), pp. 279-287.

Cf. A000040, A001222, A074946 (complement), A100118, A046363, A133620, A133621.

Cf. A133880, A133890, A133900, A133910, A133911, A046346, A134331, A134332, A134333, A134335, A134344, A134376.

Select[Range[2,200],Mod[#,PrimeOmega[#]]!=0&] (* Harvey P. Dale, May 13 2023 *)

isok(n) = (n % bigomega(n)) \\ Michel Marcus, Jul 15 2013

A133911 Number of prime factors (counted with multiplicity) of the period numbers defined by A133900.

Original entry on oeis.org

0, 2, 2, 4, 2, 5, 2, 6, 4, 6, 2, 8, 2, 6, 5, 8, 2, 9, 2, 8, 5, 7, 2, 10, 4, 7, 6, 8, 2, 12, 2, 10, 6, 8, 5, 12, 2, 8, 6, 11, 2, 12, 2, 9, 8, 8, 2, 13, 4, 10, 6, 9, 2, 12, 5, 11, 6, 8, 2, 14, 2, 8, 8, 12, 5, 13, 2, 10, 6, 13, 2, 14, 2, 9, 8, 10, 5, 13, 2, 13, 8, 10, 2, 17, 5, 9, 7, 11, 2, 16, 5, 10, 7, 9

Offset: 1

Hieronymus Fischer, Oct 20 2007

nonn

			a(6)=5, since A133900(6)=72=2*2*2*3*3.
a(12)=8, since A133900(12)=864=2*2*2*2*2*3*3*3.

Cf. A000040, A001222, A100118, A046363, A133620, A133621, A133623, A133630, A133635.

Cf. A133872, A133880, A133890, A133900, A133910, A134331.

a(n)=A001222(A133900(n)).

cp's OEIS Frontend

A376548 a(1)=1,a(2)=2. Let p be the smallest prime factor of j=a(n-1), then for n>2 a(n) is the smallest novel multiple of Sopfr(j) if j is in A046363, or of p if it is not.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A133910 Period numbers of A133900 divided by n^2.

Views

Author

Keywords

Examples

Crossrefs

Formula

A100118 Numbers whose sum of prime factors is prime (counted with multiplicity).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A134333 Numbers n whose number of prime factors (counted with multiplicity) is a prime factor of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A134344 Composite numbers such that the arithmetic mean of their prime factors (counted with multiplicity) is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A134376 Numbers whose sum of prime factors (counted with multiplicity) is not prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A036349 Numbers whose sum of prime factors (taken with multiplicity) is even.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions