A300820 -id:A300820 - cp's OEIS frontend

A087207 A binary representation of the primes that divide a number, shown in decimal.

0, 1, 2, 1, 4, 3, 8, 1, 2, 5, 16, 3, 32, 9, 6, 1, 64, 3, 128, 5, 10, 17, 256, 3, 4, 33, 2, 9, 512, 7, 1024, 1, 18, 65, 12, 3, 2048, 129, 34, 5, 4096, 11, 8192, 17, 6, 257, 16384, 3, 8, 5, 66, 33, 32768, 3, 20, 9, 130, 513, 65536, 7, 131072, 1025, 10, 1, 36, 19, 262144, 65, 258

Offset: 1

Mitch Cervinka (puritan(AT)planetkc.com), Oct 26 2003

The binary representation of a(n) shows which prime numbers divide n, but not the multiplicities. a(2)=1, a(3)=10, a(4)=1, a(5)=100, a(6)=11, a(10)=101, a(30)=111, etc.
For n > 1, a(n) gives the (one-based) index of the column where n is located in array A285321. A008479 gives the other index. - Antti Karttunen, Apr 17 2017
From Antti Karttunen, Jun 18 & 20 2017: (Start)
A268335 gives all n such that a(n) = A248663(n); the squarefree numbers (A005117) are all the n such that a(n) = A285330(n) = A048675(n).
For all n > 1 for which the value of A285331(n) is well-defined, we have A285331(a(n)) <= floor(A285331(n)/2), because then n is included in the binary tree A285332 and a(n) is one of its ancestors (in that tree), and thus must be at least one step nearer to its root than n itself.
Conjecture: Starting at any n and iterating the map n -> a(n), we will always reach 0 (see A288569). This conjecture is equivalent to the conjecture that at any n that is neither a prime nor a power of two, we will eventually hit a prime number (which then becomes a power of two in the next iteration). If this conjecture is false then sequence A285332 cannot be a permutation of natural numbers. On the other hand, if the conjecture is true, then A285332 must be a permutation of natural numbers, because all primes and powers of 2 occur in definite positions in that tree. This conjecture also implies the conjectures made in A019565 and A285320 that essentially claim that there are neither finite nor infinite cycles in A019565.
If there are any 2-cycles in this sequence, then both terms of the cycle should be present in A286611 and the larger one should be present in A286612.
(End)
Binary rank of the distinct prime indices of n, where the binary rank of an integer partition y is given by Sum_i 2^(y_i-1). For all prime indices (with multiplicity) we have A048675. - Gus Wiseman, May 25 2024

			a(38) = 129 because 38 = 2*19 = prime(1)*prime(8) and 129 = 2^0 + 2^7 (in binary 10000001).
a(140) = 13, binary 1101 because 140 is divisible by the first, third and fourth primes and 2^(1-1) + 2^(3-1) + 2^(4-1) = 13.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000 [First 1000 terms from _T. D. Noe_]
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization

For partial sums see A288566.
Cf. A000040, A000120, A001221, A005117, A008479, A019565, A055396, A285320, A285321, A285329, A285330, A285332.
Sequences with related definitions: A007947, A008472, A027748, A048675, A248663, A276379 (same sequence shown in base 2), A288569, A289271, A297404.
Cf. A286608 (numbers n for which a(n) < n), A286609 (n for which a(n) > n), and also A286611, A286612.
A003986, A003961, A059896 are used to express relationship between terms of this sequence.
Related to A267116 via A225546.
Positions of particular values are: A000079\{1} (1), A000244\{1} (2), A033845 (3), A000351\{1} (4), A033846 (5), A033849 (6), A143207 (7), A000420\{1} (8), A033847 (9), A033850 (10), A033851 (12), A147576 (14), A147571 (15), A001020\{1} (16), A033848 (17).
A048675 gives binary rank of prime indices.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Binary indices (listed A048793):
- length A000120, complement A023416
- min A001511, opposite A000012
- sum A029931, product A096111
- max A029837 or A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
- opposite A371572, sum A230877
Cf. A000720, A005940, A018819, A023506, A071814, A225620, A277319, A277905, A304818, A372689, A372890.

a087207 = sum . map ((2 ^) . (subtract 1) . a049084) . a027748_row
-- Reinhard Zumkeller, Jul 16 2013

a[n_] := Total[ 2^(PrimePi /@ FactorInteger[n][[All, 1]] - 1)]; a[1] = 0; Table[a[n], {n, 1, 69}] (* Jean-François Alcover, Dec 12 2011 *)

a(n) = {if (n==1, 0, my(f=factor(n), v = []); forprime(p=2, vecmax(f[,1]), v = concat(v, vecsearch(f[,1], p)!=0);); fromdigits(Vecrev(v), 2));} \\ Michel Marcus, Jun 05 2017

A087207(n)=vecsum(apply(p->1<M. F. Hasler, Jun 23 2017

from sympy import factorint, primepi
def a(n):
    return sum(2**primepi(i - 1) for i in factorint(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 06 2017

(definec (A087207 n) (if (= 1 n) 0 (+ (A000079 (+ -1 (A055396 n))) (A087207 (A028234 n))))) ;; This uses memoization-macro definec
(define (A087207 n) (A048675 (A007947 n))) ;; Needs code from A007947 and A048675. - Antti Karttunen, Jun 19 2017

Additive with a(p^e) = 2^(i-1) where p is the i-th prime. - Vladeta Jovovic, Oct 29 2003
a(n) gives the m such that A019565(m) = A007947(n). - Naohiro Nomoto, Oct 30 2003
A000120(a(n)) = A001221(n); a(n) = Sum(2^(A049084(p)-1): p prime-factor of n). - Reinhard Zumkeller, Nov 30 2003
G.f.: Sum_{k>=1} 2^(k-1)*x^prime(k)/(1-x^prime(k)). - Franklin T. Adams-Watters, Sep 01 2009
From Antti Karttunen, Apr 17 2017, Jun 19 2017 & Dec 06 2018: (Start)
a(n) = A048675(A007947(n)).
a(1) = 0; for n > 1, a(n) = 2^(A055396(n)-1) + a(A028234(n)).
A000035(a(n)) = 1 - A000035(n). [a(n) and n are of opposite parity.]
A248663(n) <= a(n) <= A048675(n). [XOR-, OR- and +-variants.]
a(A293214(n)) = A218403(n).
a(A293442(n)) = A267116(n).
A069010(a(n)) = A287170(n).
A007088(a(n)) = A276379(n).
A038374(a(n)) = A300820(n) for n >= 1.
(End)
From Peter Munn, Jan 08 2020: (Start)
a(A059896(n,k)) = a(n) OR a(k) = A003986(a(n), a(k)).
a(A003961(n)) = 2*a(n).
a(n^2) = a(n).
a(n) = A267116(A225546(n)).
a(A225546(n)) = A267116(n).
(End)

More terms from Don Reble, Ray Chandler and Naohiro Nomoto, Oct 28 2003

Name clarified by Antti Karttunen, Jun 18 2017

A319630 Positive numbers that are not divisible by two consecutive prime numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 37, 38, 39, 40, 41, 43, 44, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 76, 79, 80, 81, 82, 83, 85, 86, 87

Offset: 1

Rémy Sigrist, Sep 25 2018

nonn

This sequence is the complement of A104210.
Equivalently, this sequence corresponds to the positive numbers k such that:
- A300820(k) <= 1,
- A087207(k) is a Fibbinary number (A003714).
For any n > 0 and k >= 0, a(n)^k belongs to the sequence.
The numbers of terms not exceeding 10^k, for k=1,2,..., are 9, 78, 758, 7544, 75368, 753586, 7535728, 75356719, 753566574, ... Apparently, the asymptotic density of this sequence is 0.75356... - Amiram Eldar, Apr 10 2021
Numbers not divisible by any term of A006094. - Antti Karttunen, Jul 29 2022

			The number 10 is only divisible by 2 and 5, hence 10 appears in the sequence.
The number 42 is divisible by 2 and 3, hence 42 does not appear in the sequence.

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

Cf. A003714, A006094, A087207, A104210, A300820, A356171 (odd terms only).

Positions of 1's in A322361 and in A356173 (characteristic function).

N:= 1000: # for terms <= N
R:= {}:
p:= 2:
do
  q:= p; p:= nextprime(p);
  if p*q > N then break fi;
  R:= R union {seq(i,i=p*q..N,p*q)}
od:
sort(convert({$1..N} minus R,list)); # Robert Israel, Apr 13 2020

q[n_] := SequenceCount[FactorInteger[n][[;; , 1]], {p1_, p2_} /; p2 == NextPrime[p1]] ==  0; Select[Range[100], q] (* Amiram Eldar, Apr 10 2021 *)

is(n) = my (f=factor(n)); for (i=1, #f~-1, if (nextprime(f[i,1]+1)==f[i+1,1], return (0))); return (1)

A300820(a(n)) <= 1.

A384890 Number of maximal anti-runs (increasing by more than 1) in the binary indices of n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jun 17 2025

nonn

First differs from A272604 at a(51) = 3, A272604(51) = 2.
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
Do all constant runs in this sequence have lengths 1, 2, or 3?

			The binary indices of 51 are {1,2,5,6}, with maximal anti-runs ((1),(2,5),(6)), so a(51) = 3.

For runs instead of anti-runs we have A069010 = run-lengths of A245563 (reverse A245562).
Row-lengths of A384877, firsts A384878.
For prime indices instead of binary indices we have A384906.
A000120 counts binary indices.
A356606 counts strict partitions without a neighborless part, complement A356607.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
Cf. A044813, A048793, A052499, A164707, A243815, A300820, A328592, A384177, A384879, A384893.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Split[bpe[n],#2!=#1+1&]],{n,0,100}]

A104210 Positive integers divisible by at least 2 consecutive primes.

Original entry on oeis.org

6, 12, 15, 18, 24, 30, 35, 36, 42, 45, 48, 54, 60, 66, 70, 72, 75, 77, 78, 84, 90, 96, 102, 105, 108, 114, 120, 126, 132, 135, 138, 140, 143, 144, 150, 154, 156, 162, 165, 168, 174, 175, 180, 186, 192, 195, 198, 204, 210, 216, 221, 222, 225, 228, 231, 234, 240

Offset: 1

Leroy Quet, Mar 13 2005

nonn

If a perfect square is in this sequence, then so is its square root (e.g., 144 and 12). - Alonso del Arte, May 07 2012

The numbers of terms not exceeding 10^k, for k=1,2,..., are 1, 22, 242, 2456, 24632, 246414, 2464272, 24643281, 246433426, ... Apparently, the asymptotic density of this sequence is 0.24643... - Amiram Eldar, Apr 10 2021

			35 is divisible by both 5 and 7, and 5 and 7 are consecutive primes.
77 is divisible by both 7 and 11, and 7 and 11 are consecutive primes.
110 is not in the sequence because, although it is divisible by 2, 5 and 11, it is not divisible by 3 or 7.

Cf. A003961, A296210 (characteristic function), A319630 (complement), A379230 [= A252748(a(n))].
Positions of terms larger than 1 in A300820 and in A322361.
Subsequences: A006094, A349169 (conjectured, after its initial 1), A349176, A355527 (squarefree terms), A372566, A378884, A379232.

N:= 1000: # for terms <= N
R:= {}:
p:= 2:
do
  q:= p; p:= nextprime(p);
  if p*q > N then break fi;
  R:= R union {seq(i,i=p*q..N,p*q)}
od:
sort(convert(R,list)); # Robert Israel, Apr 13 2020

fQ[n_] := Block[{lst = PrimePi /@ Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[n]]}, Count[ Drop[lst, 1] - Drop[lst, -1], 1] > 0]; Select[ Range[244], fQ[ # ] &] (* Robert G. Wilson v, Mar 16 2005 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
is_A104210(n) = (gcd(n,A003961(n))>1); \\ Antti Karttunen, Dec 24 2024

{k such that gcd(k, A003961(k)) > 1}. - Antti Karttunen, Dec 24 2024

More terms from Robert G. Wilson v, Mar 16 2005

A384906 Number of maximal anti-runs of consecutive parts not increasing by 1 in the prime indices of n (with multiplicity).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 22 2025

nonn

First differs from A300820 at a(462) = 3, A300820(462) = 2.

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 prime indices of 462 are {1,2,4,5}, with maximal anti-runs ((1),(2,4),(5)), so a(462) = 3.

For the strict case we have A356228.
For binary instead of prime indices we have A384890 (for runs A069010).
For runs instead of anti-runs we have A385213.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
Cf. A130091, A245562, A268193, A300820, A351202, A356607, A382525, A384177, A384321, A384893.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Split[prix[n],#2!=#1+1&]],{n,100}]

A385213 Number of maximal runs of consecutive parts increasing by 1 in the prime indices of n (with multiplicity).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 22 2025

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 prime indices of 24 are {1,1,1,2}, with maximal runs ((1),(1),(1,2)), so a(24) = 3.

Positions of first appearances are A000079.
For binary instead of prime indices we have A069010 (for anti-runs A384890).
For anti-runs instead of runs we have A384906.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
Cf. A002110, A048767, A130091, A300820, A356606, A351202, A382525, A384893.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Split[prix[n],#2==#1+1&]],{n,100}]

A167447 Number of divisors of n which are not multiples of 3 consecutive primes.

Original entry on oeis.org

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

Offset: 1

Matthew Vandermast, Nov 05 2009

nonn

If a number is a product of any number of consecutive primes, the number of its divisors which are not multiples of n consecutive primes is always a Fibonacci n-step number. See also A073485, A166469.

			Since 2 of 60's 12 divisors (30 and 60) are multiples of at least 3 consecutive primes, a(60) = 12 - 2 = 10.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Cf. A000040, A000073, A002110, A046301, A073485, A166469, A300820.

A300820(n) = if(omega(n)<=1, omega(n), my(pis=apply(p->primepi(p),factor(n)[,1]),el=1,m=1); for(i=2,#pis,if(pis[i] == (1+pis[i-1]),el++; m = max(m,el), el=1)); (m));
A167447(n) = sumdiv(n,d,(A300820(d)<3)); \\ Antti Karttunen, Mar 21 2018

a) If n has no prime gaps in its factorization (cf. A073491), then, if the canonical factorization of n into prime powers is the product of p_i^(e_i), a(n) is the sum of all products of exponents which do not include 3 consecutive exponents, plus 1. For example, if A001221(n)=3, a(n)=e_1*e_2 +e_1*e_3 +e_2*e_3 +e_1 +e_2 +e_3 +1. If A001221(n)=k, the total number of terms always equals A000073(k+3).
The answer can also be computed in k steps, by finding the answers for the products of the first i powers for i=1 to i=k. Let the result of the i-th step be called r(i). r(1)=e_1+1; r(2)=e_1*e_2+e_1+e_2+1; r(3)=e_1*e_2+e_1*e_3+e_2*e_3+e_1+e_2+e_3+1; for i>3, r(i)=r(i-1)+e_i*r(i-2)+e_i*e-(i-1)*r(i-3).
b) If n has prime gaps in its factorization, express it as a product of the minimum number of A073491's members possible. Then apply either of the above methods to each of those members, and multiply the results to get a(n). a(n)=A000005(n) iff n has no triple of consecutive primes as divisors.
a(A002110(n)) = A000073(n+2).
a(n) = Sum_{d|n} [A300820(d) < 3]. - Antti Karttunen, Mar 21 2018

A378886 The number of consecutive primes in the prime factorization of n starting from the smallest prime dividing n; a(1) = 0.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Dec 10 2024

First differs from A300820 at n = 70 = 2 * 5 * 7: A300820(70) = 2 while a(70) = 1.

			a(42) = 2 since 42 = 2 * 3 * 7 and 2 and 3 are 2 consecutive primes.
a(28) = 1 since 28 = 2^2 * 7 and 3 is not a divisor of 28.
a(4095) = 3 since 4095 = 3^2 * 5 * 7 * 13 and 3, 5 and 7 are 3 consecutive primes.

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

Cf. A001221, A020639, A073491, A276084, A300820, A378884, A378885.

a[n_] := Module[{p = FactorInteger[n][[;; , 1]], c = 1, q}, q = p[[1]]; Do[q = NextPrime[q]; If[q == p[[i]], c++, Break[]], {i, 2, Length[p]}]; c]; a[1] = 0; Array[a, 100]

a(n) = if(n == 1, 0, my(p = factor(n)[,1], c = 1, q); q = p[1]; for(i = 2, #p, q = nextprime(q+1); if(q == p[i], c++, break)); c);

a(n) >= A276084(n).
a(n) <= A300820(n).
a(n) = A001221(n) if and only if n is in A073491.
a(n) >= 1 for n >= 2.
a(n) >= 2 if and only if n is in A378884.
a(n) >= 3 if and only if n is in A378885.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} k * (d(k) - d(k+1)) = 1.2630925015039..., where d(1) = 1 and d(k) = Sum_{i>=1} (Product_{j=1..i-1} (1-1/prime(j)))/(Product_{j=0..k-1} prime(i+j)), for k >= 2. d(k) is the asymptotic density of numbers m for which a(m) >= k.

cp's OEIS Frontend

A087207 A binary representation of the primes that divide a number, shown in decimal.

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A319630 Positive numbers that are not divisible by two consecutive prime numbers.

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A384890 Number of maximal anti-runs (increasing by more than 1) in the binary indices of n.

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A104210 Positive integers divisible by at least 2 consecutive primes.

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A384906 Number of maximal anti-runs of consecutive parts not increasing by 1 in the prime indices of n (with multiplicity).

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A385213 Number of maximal runs of consecutive parts increasing by 1 in the prime indices of n (with multiplicity).

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A167447 Number of divisors of n which are not multiples of 3 consecutive primes.

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