A046363 -id:A046363 - cp's OEIS frontend

A134331 Sum of prime factors (counted with multiplicity) of the period numbers defined by A133900.

0, 4, 6, 8, 10, 12, 14, 12, 12, 18, 22, 19, 26, 22, 19, 16, 34, 22, 38, 22, 23, 32, 46, 23, 20, 36, 18, 26, 58, 37, 62, 20, 34, 46, 29, 29, 74, 50, 38, 31, 82, 38, 86, 36, 30, 58, 94, 30, 28, 32, 46, 40, 106, 30, 37, 37, 50, 70, 118, 41, 122, 74, 36, 24, 41, 48, 134, 50, 58, 50

Offset: 1

Hieronymus Fischer, Oct 20 2007

nonn

			a(6)=12, since A133900(6)=72=2*2*2*3*3 and 2+2+2+3+3=12.
a(12)=19, since A133900(12)=864=2*2*2*2*2*3*3*3 and 2+2+2+2+2+3+3+3=19.

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

Cf. A133872, A133880, A133890, A133900, A133910, A133911.

A134332 Integer part of the arithmetic mean of the prime factors (counted with multiplicity) of the period numbers defined by A133900.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 2, 3, 3, 11, 2, 13, 3, 3, 2, 17, 2, 19, 2, 4, 4, 23, 2, 5, 5, 3, 3, 29, 3, 31, 2, 5, 5, 5, 2, 37, 6, 6, 2, 41, 3, 43, 4, 3, 7, 47, 2, 7, 3, 7, 4, 53, 2, 7, 3, 8, 8, 59, 2, 61, 9, 4, 2, 8, 3, 67, 5, 9, 3, 71, 2, 73, 9, 4, 5, 8, 4, 79, 2, 3, 9, 83, 3, 9, 11, 10, 3, 89, 2, 9, 6, 11, 12

Offset: 1

Hieronymus Fischer, Oct 23 2007

nonn

			a(6)=2, since floor(A134331(6)/A133911(6))=floor(12/5)=2.
a(7)=7, since floor(A134331(7)/A133911(7))=floor(14/2)=7.

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

Cf. A133872, A133880, A133890, A133900, A133910, A133911, A134331.

a(n)=floor(A134331(n)/A133911(n)) for n>1, defining a(1):=1.

a(n)=n, if n is a prime or 1.

A078177 Composite numbers with an integer arithmetic mean of all prime factors.

Original entry on oeis.org

4, 8, 9, 15, 16, 20, 21, 25, 27, 32, 33, 35, 39, 42, 44, 49, 50, 51, 55, 57, 60, 64, 65, 68, 69, 77, 78, 81, 85, 87, 91, 92, 93, 95, 105, 110, 111, 112, 114, 115, 116, 119, 121, 123, 125, 128, 129, 133, 140, 141, 143, 145, 155, 156, 159, 161, 164, 169, 170, 177, 180

Offset: 1

Reinhard Zumkeller, Nov 20 2002

nonn

That is, composite numbers such that the arithmetic mean of their prime factors (counted with multiplicity) is an integer.

			60 = 2*2*3*5: (2+2+3+5)/4 = 3, therefore 60 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A078175, A001222, A100118, A046363, A133620, A133621.

Cf. A133880, A133890, A133900, A133910, A133911, A046346, A134331, A134332, A134333, A134334, A134344, A134376.

Select[Range[200], CompositeQ[#] && IntegerQ[Mean[Flatten[Table[#[[1]], #[[2]]]& /@ FactorInteger[#]]]]&] (* Jean-François Alcover, Aug 03 2018 *)

lista(nn) = {forcomposite(n=1, nn, my(f = factor(n)); if (! (sum(k=1, #f~, f[k,1]*f[k,2]) % vecsum(f[,2])), print1(n, ", ")););} \\ Michel Marcus, Feb 22 2016

A001414(a(n)) == 0 modulo A001222(a(n)).

Edited by N. J. A. Sloane, May 30 2008 at the suggestion of R. J. Mathar

A046365 Composite palindromes whose sum of prime factors is prime (counted with multiplicity).

Original entry on oeis.org

6, 22, 88, 99, 202, 252, 333, 414, 424, 454, 464, 595, 686, 747, 777, 808, 838, 848, 858, 909, 1001, 1551, 1771, 2442, 3553, 4114, 5335, 5775, 6336, 6996, 8008, 8228, 9009, 9559, 9669, 9889, 12121, 14241, 16261, 16761, 17171, 18081, 18381, 20102, 20602, 21012

Offset: 1

Patrick De Geest, Jun 15 1998

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A046363, A046364.

Select[Range[20125], !PrimeQ[#] && Reverse[x=IntegerDigits[#]] == x && PrimeQ[Total[Times@@@FactorInteger[#]]]&] (* Jayanta Basu, May 29 2013 *)

from itertools import product
from sympy import factorint, isprime
def pals(d, base=10): # all d-digit palindromes
    digits = "".join(str(i) for i in range(base))
    for p in product(digits, repeat=d//2):
        if d > 1 and p[0] == "0": continue
        left = "".join(p); right = left[::-1]
        for mid in [[""], digits][d%2]: yield int(left + mid + right)
def ok(pal):
    f = factorint(pal); return len(f)>1 and isprime(sum(p*f[p] for p in f))
print(list(filter(ok, (p for d in range(1, 6) for p in pals(d) if ok(p))))) # Michael S. Branicky, Jun 22 2021

A046363 INTERSECT A002113. - R. J. Mathar, Sep 09 2015

a(45) and beyond from Michael S. Branicky, Jun 22 2021

A066038 Numbers with at least two prime factors such that the sum of the prime factors is prime.

Original entry on oeis.org

6, 10, 12, 18, 20, 22, 24, 34, 36, 40, 44, 48, 50, 54, 58, 68, 72, 80, 82, 88, 96, 100, 108, 116, 118, 136, 142, 144, 160, 162, 164, 165, 176, 192, 200, 202, 210, 214, 216, 232, 236, 242, 250, 272, 273, 274, 284, 288, 298, 320, 324, 328, 345, 352, 358, 382, 384

Offset: 1

Joseph L. Pe, Dec 12 2001

nonn

Numbers with just one prime factor (prime powers) trivially satisfy the defining condition and are not included.

			The prime factors of 12 are 2 and 3, which add up to 5, a prime.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A046363, A000961, A008472.

Reap[For[n = 6, n <= 1000, n++, pp = FactorInteger[n][[All, 1]]; If[Length[pp] >= 2 && PrimeQ[Total[pp]], Sow[n]]]][[2, 1]] (* Jean-François Alcover, Oct 16 2016 *)

sopf(n)= { local(f,s=0); f=factor(n); for(i=1, matsize(f)[1], s+=f[i, 1]); return(s) }
{ n=0; for (m=1, 10^9, if (omega(m) > 1 && isprime(sopf(m)), write("b066038.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Nov 07 2009

isok(n) = (omega(n) > 1) && isprime(vecsum(factor(n)[,1])); \\ Michel Marcus, Dec 15 2018

More terms from Vladeta Jovovic, Dec 13 2001

A036350 Composite numbers such that the sum of the prime factors is odd (counted with multiplicity).

Original entry on oeis.org

6, 10, 12, 14, 20, 22, 24, 26, 27, 28, 34, 38, 40, 44, 45, 46, 48, 52, 54, 56, 58, 62, 63, 68, 74, 75, 76, 80, 82, 86, 88, 90, 92, 94, 96, 99, 104, 105, 106, 108, 112, 116, 117, 118, 122, 124, 125, 126, 134, 136, 142, 146, 147, 148, 150, 152, 153, 158, 160, 164

Offset: 1

Patrick De Geest, Dec 15 1998

			44 = 2 * 2 * 11 -> sum = 15 -> 15 is odd.

Cf. A001414, A046363, A036349.

Subsequence of A335657.

okQ[n_]:=!PrimeQ[n]&&OddQ[Total[Flatten[Table[First[#],{Last[#]}]&/@ FactorInteger[n]]]]; Select[Range[2,200],okQ] (* Harvey P. Dale, Dec 18 2011 *)

A046364 Odd composite numbers whose sum of prime factors is prime (counted with multiplicity).

Original entry on oeis.org

45, 63, 75, 99, 117, 147, 153, 165, 175, 207, 245, 273, 279, 325, 333, 345, 369, 385, 399, 405, 423, 435, 475, 477, 507, 549, 561, 567, 595, 603, 651, 657, 665, 675, 715, 747, 759, 775, 777, 795, 833, 845, 847, 867, 873, 885, 891, 903, 909, 925, 927, 957

Offset: 0

Patrick De Geest, Jun 15 1998

nonn

Cf. A046363, A046365.

isA046364 := proc(n)
    if isprime(n) then
        false;
    else
        isprime(A001414(n))
    end if;
end proc:
for n from 3 to 1001 by 2 do
    if isA046364(n) then
        printf("%d,",n) ;
     end if;
end do: # R. J. Mathar, Aug 14 2024

Select[Range[1,957,2], !PrimeQ[#] && PrimeQ[Total[Times@@@FactorInteger[#]]]&] (* Jayanta Basu, May 29 2013 *)

A134335 Numbers such that the arithmetic mean of their prime factors (counted with multiplicity) is an integer, but not a prime.

Original entry on oeis.org

15, 35, 39, 42, 50, 51, 55, 65, 77, 78, 87, 91, 92, 95, 110, 111, 114, 115, 119, 123, 140, 141, 143, 155, 159, 161, 164, 170, 183, 185, 186, 187, 189, 201, 203, 204, 209, 215, 219, 221, 222, 225, 230, 235, 236, 242, 247, 258, 259, 264, 267, 284, 285, 287, 290

Offset: 1

Hieronymus Fischer, Oct 23 2007

nonn

			a(1) = 15, since 15 = 3*5 and (3+5)/2 = 4 is not prime.
a(5) = 50, since 50 = 2*5*5 and (2+5+5)/3 = 4 is not prime.

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

Cf. A000040, A001222, A100118, A046363, A133620, A133621.

Cf. A133880, A133890, A133900, A133910, A133911, A046346, A134331, A134332, A134333, A134334.

fp[{a_,b_}]:=a*b;s={};Do[If[q=Total[fp/@FactorInteger[n]]/Total[Last/@FactorInteger[n]];IntegerQ[q]&&!PrimeQ[q],AppendTo[s,n]],{n,2,290}];s (* James C. McMahon, Apr 05 2025 *)

Definition clarified by the author, May 06 2013

A340060 Averages k of twin primes such that the sum (with multiplicity) of prime factors of k-1, k and k+1 is prime.

Original entry on oeis.org

6, 12, 108, 420, 432, 462, 1020, 1290, 1302, 1428, 1482, 2088, 3120, 3468, 3540, 4092, 4242, 5280, 5652, 5850, 5868, 6690, 7332, 8820, 9420, 9462, 9930, 10038, 10092, 11118, 11832, 11940, 14628, 16140, 16362, 17208, 17388, 17682, 18042, 18312, 20640, 20772, 22278, 22962, 23028, 23202, 25848

Offset: 1

J. M. Bergot and Robert Israel, Dec 27 2020

nonn

			a(3)=108 is a term because 107 and 109 are primes and the sum of prime factors of 107, 108=2^2*3^3 and 109 is 107+2+2+3+3+3+109 = 229, which is prime.

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

Subset of A014574. Cf. A001414, A046363.

a(n) = A330488(n)+1.

P:= select(isprime, {seq(i,i=3..10^6,2)}):
T:= map(`+`,P,1) intersect map(`-`,P,1):
filter:= proc(t) local s; isprime(2*t+add(s[1]*s[2],s=ifactors(t)[2])) end proc:
sort(convert(select(filter,T),list));

A353125 a(1)=2. If a(n) is a novel term, a(n+1) = sopfr(a(n)), else if there are k occurrences of a(j)=a(n), (1<=j<=n), a(n+1)=k*a(n).

Original entry on oeis.org

2, 2, 4, 4, 8, 6, 5, 5, 10, 7, 7, 14, 9, 6, 12, 7, 21, 10, 20, 9, 18, 8, 16, 8, 24, 9, 27, 9, 36, 10, 30, 10, 40, 11, 11, 22, 13, 13, 26, 15, 8, 32, 10, 50, 12, 24, 48, 11, 33, 14, 28, 11, 44, 15, 30, 60, 12, 36, 72, 12, 48, 96, 13, 39, 16, 32, 64, 12, 60, 120

Offset: 1

David James Sycamore and Michael De Vlieger, Apr 24 2022

nonn

1 and 3 cannot be terms. Let conditions 1, 2 refer to the effects of a novel and repeat term respectively, and let M(m) be the multiplicity of m in the sequence. M(m) >= 2 for all m in N \ {1,3}. All first occurrences of a prime p > 2 follow a novel term in A046363 (by condition 1).
If the first occurrence of composite m arises from condition 1, then so does the second. Proof: Suppose not, then the second m must be a multiple w*t of a prior term t (condition 2); w >= 2. The term following the second m must be (w+1)*t, and this must equal 2*m (condition 2). Thus 2*w*t = (w+1)*t, so then w=1. But w >= 2; contradiction. Corollary: Once composite m has occurred by condition 1, then all subsequent occurrences of m occur the same way.
Most composite terms appear first by condition 2 (e.g., 12 as 2 copies of 6), and then subsequently by condition 1. Thus for all k in the sequence M(k) >= A000607(k), with equality when k is prime > 2, or certain composite numbers.
Conjecture: The 10 composite numbers 6,9,15,25,35,49,77,121,143,169 behave as primes in this sequence, namely for any such m, M(m) = A000607(m). For all other composite m, M(m) > A000607(m), i.e., at least one (up front) copy by condition 2.
The first occurrences of primes appear in natural order initially, but this is not sustained (e.g., 61 appears before 59).

			a(1)=2, a novel term, so a(2)=sopfr(2)=2. 2 occurs twice and is the only prime p whose multiplicity is not A000607(p), simply because it is the seed term.
Since 2 has now appeared twice, a(3)=2*2=4, a novel term, so a(4)=sopfr(4)=4.
a(25)=24 (3 occurrences of 8), a(46)=24 (2 occurrences of 12). Subsequently all occurrences of 24 are from condition 1. Therefore M(24) = 2 + A000607(24) = 48.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated log-log scatterplot of a(n), n = 1..2^12, records in red and likely local minima in blue.
Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^16.

Cf. A001414 (sopfr), A046363, A000607, A000040.

Block[{a, c, j, k, nn}, nn = 120; c[] = 0; j = a[1] = 2; c[2]++; Do[If[c[j] == 1, Set[k, Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[j]]], Set[k, c[j] j]]; j = a[i] = k; c[k]++, {i, 2, nn}]; Array[a, nn] ] (* _Michael De Vlieger, Apr 24 2022 *)

sopfr(n) = (n=factor(n))[, 1]~*n[, 2]; \\ A001414
lista(nn) = {my(v=vector(nn), k); v[1] = 2; for (n=2, nn, if ((k=#select(x->(x==v[n-1]), Vec(v, n-1))) == 1, v[n] = sopfr(v[n-1]), v[n] = k*v[n-1]);); v;} \\ Michel Marcus, May 16 2022

More terms from Michael De Vlieger, Apr 24 2022

cp's OEIS Frontend

A134331 Sum of prime factors (counted with multiplicity) of the period numbers defined by A133900.

Views

Author

Keywords

Examples

Crossrefs

A134332 Integer part of the arithmetic mean of the prime factors (counted with multiplicity) of the period numbers defined by A133900.

Views

Author

Keywords

Examples

Crossrefs

Formula

A078177 Composite numbers with an integer arithmetic mean of all prime factors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A046365 Composite palindromes whose sum of prime factors is prime (counted with multiplicity).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A066038 Numbers with at least two prime factors such that the sum of the prime factors is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Extensions

A036350 Composite numbers such that the sum of the prime factors is odd (counted with multiplicity).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A046364 Odd composite numbers whose sum of prime factors is prime (counted with multiplicity).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

A134335 Numbers such that the arithmetic mean of their prime factors (counted with multiplicity) is an integer, but not a prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A340060 Averages k of twin primes such that the sum (with multiplicity) of prime factors of k-1, k and k+1 is prime.

Views

Author