A000005 -id:A000005 - cp's OEIS frontend

A160812 a(n) = A161205(n)-A000005(n).

0, 0, 0, 0, 2, 0, 2, 0, 2, 2, 4, 0, 4, 2, 2, 2, 6, 2, 6, 2, 4, 4, 6, 0, 6, 6, 6, 4, 8, 2, 8, 4, 6, 6, 6, 2, 10, 8, 8, 4, 10, 4, 10, 6, 6, 8, 10, 2, 10, 8, 10, 8, 12, 6, 10, 6, 10, 10, 12, 2, 12, 10, 8, 8, 12, 8, 14, 10, 12, 8, 14, 4, 14, 12, 10, 10, 12, 8, 14, 6, 12, 14, 16, 6, 14, 14, 14, 10, 16, 6

Offset: 1

Omar E. Pol, Jun 19 2009

It appears that a(n)= 0 if and only if n divides 24 (See also the comments in A018253).

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos
Omar E. Pol, Illustration: Divisors and pi(x)

Cf. A000005, A000720, A018253, A160811, A161205.

a(n) = 2*(A000196(n) - A038548(n)) = 2*A236627(n). - Omar E. Pol, Feb 05 2014

Edited by Omar E. Pol, Aug 02 2009

A295664 Exponent of the highest power of 2 dividing number of divisors of n: a(n) = A007814(A000005(n)); 2-adic valuation of tau(n).

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, 2, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 3, 0, 2, 2, 1, 1, 3, 1, 1, 2, 2, 2, 0, 1, 2, 2, 3, 1, 3, 1, 1, 1, 2, 1, 1, 0, 1, 2, 1, 1, 3, 2, 3, 2, 2, 1, 2, 1, 2, 1, 0, 2, 3, 1, 1, 2, 3, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 0, 2, 1, 2, 2, 2, 2, 3, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 0, 1, 3, 1, 3, 3, 2, 1, 2, 1, 3, 2, 1, 1, 3, 2, 1, 1, 2, 2, 4, 0

Offset: 1

Antti Karttunen, Nov 28 2017

In the prime factorization of n = p1^e1 * ... pk^ek, add together the number of trailing 1-bits in each exponent e when they are written in binary.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A007814, A028234, A056169, A067029, A077761, A162642, A295663.

Cf. A000290 (positions of zeros).

Table[IntegerExponent[DivisorSigma[0, n], 2], {n, 120}] (* Michael De Vlieger, Nov 28 2017 *)

a(n) = valuation(numdiv(n), 2); \\ Michel Marcus, Nov 30 2017

from sympy import divisor_count
def A295664(n): return (~(m:=int(divisor_count(n))) & m-1).bit_length() # Chai Wah Wu, Jul 05 2022

Additive with a(p^e) = A007814(1+e).
a(1) = 0; for n > 1, a(n) = A007814(1+A067029(n)) + a(A028234(n)).
a(n) = A007814(A000005(n)).
a(n) >= A162642(n) >= A056169(n).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) =-0.223720656976344505701..., where f(x) = -x + (1-x) * Sum_{k>=1} x^(2^k-1)/(1-x^(2^k)). - Amiram Eldar, Sep 28 2023

A339258 Triangle read by rows T(n,k), (n >= 1, k > = 1), in which row n has length A000070(n-1) and every column gives A000005, the number of divisors function.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 1, 3, 2, 2, 2, 1, 1, 1, 2, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 4, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 4, 2, 2, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 2, 4, 4, 2, 2, 2, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Omar E. Pol, Nov 29 2020

Conjecture: the sum of row n equals A006128(n), the total number of parts in all partitions of n.

			Triangle begins:
  1;
  2, 1;
  2, 2, 1, 1;
  3, 2, 2, 2, 1, 1, 1;
  2, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1;
  4, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1;
  2, 4, 2, 2, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, ...
  ...

Paolo Xausa, Table of n, a(n) for n = 1..10980 (rows 1..21 of the triangle, flattened)

Row sums give A006128 (conjectured).

Cf. A000005, A000041, A176206, A221530, A221531, A337209.

A339258row[n_]:=Flatten[Table[ConstantArray[DivisorSigma[0,n-m],PartitionsP[m]],{m,0,n-1}]];Array[A339258row,10] (* Paolo Xausa, Sep 02 2023 *)

f(n) = sum(k=0, n-1, numbpart(k));
T(n, k) = {if (k > f(n), error("invalid k")); if (k==1, return (numdiv(n))); my(s=0); while (k <= f(n-1), s++; n--;); numdiv(1+s);}
tabf(nn) = {for (n=1, nn, for (k=1, f(n), print1(T(n,k), ", ");); print;);} \\ Michel Marcus, Jan 13 2021

T(n,k) = A000005(A176206(n,k)).

A360179 a(1) = 1. Thereafter if a(n-1) is a novel term, a(n) = d(a(n-1)); otherwise a(n) = a(n-1) + d(u), where d is the divisor function A000005 and u is the smallest unstarred prior term (each time we use a prior term we star it, and starred terms cannot be reused).

Original entry on oeis.org

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

Offset: 1

David James Sycamore, Jan 29 2023

Whilst the definition is subtly different from that of A345147, d(u) being used in place of u, the scatterplots are remarkably different, the one for this sequence displaying numerous precipitous "gorges" which are open to explanation. 1 is the only number which occurs precisely twice, all other numbers are repeated infinitely many times.
From Michael De Vlieger, Apr 04 2023: (Start)
The sequence is a series of nondecreasing cycles that reach a maximum M and then reset to start a new cycle. (See scatterplot B.)
The sequence is dynamic and responds to a bank of copies of the same number called a "prevailing low" L. When M < L, the sequence experiences a run of short or "crashed" cycles that make no headway at eliminating the copies of L, resulting in a "gorge" in the scatterplot.
Referring to scatterplot A:
The green line represents the smallest missing number u and is not actually a feature of the sequence. The red line represents the "prevailing low" L(n), also is not a feature of the sequence.
Dark blue terms a(n) = tau(a(n-1))..421 populate a "semi-coherent" phase (1A) of cycle c(i), where tau(n) = A000005(n).
Light blue terms a(n) = 422..L populate the "coherent" phase (1B) of cycle c(i). Black terms m > L populate phase (2) of c(i).
Magenta terms constitute a crashed cycle that has M < L; multiple consecutive crashed cycles constitute a gorge. In crashed cycles, we have only phase (1).
The "triple point" of the graph, where we first have phase (1B), appears to be a(14478) = 414, but is in actuality (given 2^20 terms) a(14786) = 422. (End)

			a(2) = 1 since a(1) = 1 is a novel term and d(1) = 1. Thus the sequence starts 1,1 and since a(2) is a repeated term, a(3) = a(2) + d(1) (1 = least unstarred prior term). Therefore a(3) = 1 + 1 = 2.

Michael De Vlieger, Table of n, a(n) for n = 1..40000 [The exceptionally large b-file was added at my request. - _N. J. A. Sloane_, Apr 07 2023]
Michael De Vlieger, Scatterplot A of a(n), n = 1..40000.
Michael De Vlieger, Scatterplot B of a(n) for n = 1..128. Records appear in red, local minima in blue, terms instigated by a(n) = m new to the sequence appear in gold, otherwise in green. The magenta line indicates the smallest missing number u not in a(1..n-1).
Michael De Vlieger, Scatterplot of a(n), n = 1500000.

Cf. A000005, A343887, A345147.
Cf. A362127 records, A362128 indices of records.
Cf. A362129 a(n) mod 2, A362130 d(a(n)) mod 2.
Cf. A362131 smallest missing number in a(1..n).
Cf. A362134 novel terms, A362135 indices of novel terms.
Cf. A362136 row lengths, if this sequence seen as rows of strictly increasing terms.
See A361511 for another version.

nn = 120; c[] := False; h[] := 0; f[n_] := DivisorSigma[0, n]; a[1] = j = u = 1; Do[If[c[j], k = j + f[u]; h[j]++; h[u]--, k = f[j]; c[j] = True; h[j]++]; u = Min[u, j]; Set[{a[n], j}, {k, k}]; While[h[u] == 0, u++], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Feb 02 2023 *)

A371165 Positive integers with as many divisors (A000005) as distinct divisors of prime indices (A370820).

Original entry on oeis.org

3, 5, 11, 17, 26, 31, 35, 38, 39, 41, 49, 57, 58, 59, 65, 67, 69, 77, 83, 86, 87, 94, 109, 119, 127, 129, 133, 146, 148, 157, 158, 179, 191, 202, 206, 211, 217, 235, 237, 241, 244, 253, 274, 277, 278, 283, 284, 287, 291, 298, 303, 319, 326, 331, 333, 334, 353

Offset: 1

Gus Wiseman, Mar 14 2024

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 terms together with their prime indices begin:
     3: {2}        67: {19}        158: {1,22}
     5: {3}        69: {2,9}       179: {41}
    11: {5}        77: {4,5}       191: {43}
    17: {7}        83: {23}        202: {1,26}
    26: {1,6}      86: {1,14}      206: {1,27}
    31: {11}       87: {2,10}      211: {47}
    35: {3,4}      94: {1,15}      217: {4,11}
    38: {1,8}     109: {29}        235: {3,15}
    39: {2,6}     119: {4,7}       237: {2,22}
    41: {13}      127: {31}        241: {53}
    49: {4,4}     129: {2,14}      244: {1,1,18}
    57: {2,8}     133: {4,8}       253: {5,9}
    58: {1,10}    146: {1,21}      274: {1,33}
    59: {17}      148: {1,1,12}    277: {59}
    65: {3,6}     157: {37}        278: {1,34}

For prime factors instead of divisors on both sides we get A319899.
For prime factors on LHS we get A370802, for distinct prime factors A371177.
The RHS is A370820, for prime factors instead of divisors A303975.
For (greater than) instead of (equal) we get A371166.
For (less than) instead of (equal) we get A371167.
Partitions of this type are counted by A371172.
Other inequalities: A370348 (A371171), A371168 (A371173), A371169, A371170.
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, A112798 indices, length A001222.
A239312 counts divisor-choosable partitions, ranks A368110.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A370320 counts non-divisor-choosable partitions, ranks A355740.
A370814 counts divisor-choosable factorizations, complement A370813.
Cf. A000792, A003963, A355529, A355739, A355741, A368100, A370808, A371127.

Select[Range[100],Length[Divisors[#]] == Length[Union@@Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]

A000005(a(n)) = A370820(a(n)).

A051281 Sum of divisors of n, sigma(n) (A000203), is a power of number of divisors of n, d(n) (A000005).

Original entry on oeis.org

1, 3, 7, 31, 127, 217, 889, 2667, 3937, 8191, 27559, 57337, 131071, 172011, 253921, 524287, 917497, 1040257, 1777447, 3670009, 4063201, 11010027, 12189603, 16252897, 16646017, 66584449, 113770279, 116522119, 225735769, 677207307, 1073602561, 2147483647, 3612185689, 4294434817, 7515217927

Offset: 1

Michel ten Voorde

All Mersenne primes (A000668) are terms.

Subsequence of A046528 (product of distinct Mersenne primes). - Michel Marcus, Feb 15 2020

			d(217) = 4; sigma(217) = 256 = 4^4.

David A. Corneth, Table of n, a(n) for n = 1..2000

Cf. A000005, A000203, A046528, A286628, A289276.

spdQ[n_]:=Module[{sd=DivisorSigma[1,n],nd=DivisorSigma[0,n]},sd == nd^IntegerExponent[sd,nd]]; Join[{1},Select[Range[2,226000000],spdQ]] (* Harvey P. Dale, May 02 2012 *)

is(n)=my(t,e=ispower(sigma(n),,&t)); if(!e,return(n==1),nd); nd=numdiv(n); fordiv(e,d,if(t^d==nd,return(1)));0 \\ Charles R Greathouse IV, Feb 19 2013

isA051281(n) = { if(n==1, return(1)); my(sig = sigma(n), ndiv = numdiv(n), v = valuation(sig, ndiv)); (ndiv^v == sig); } \\ Antti Karttunen, Jun 30 2017

More terms from Jud McCranie
a(30)-a(32) from Donovan Johnson, Oct 03 2012
a(33)-a(35) from Michel Marcus, Feb 14 2020

A061017 List in which n appears d(n) times, where d(n) [A000005] is the number of divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 20, 20, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 24

Offset: 1

Jont Allen (jba(AT)research.att.com), May 25 2001

The union of N, 2N, 3N, ..., where N = {1, 2, 3, 4, 5, 6, ...}. In other words, the numbers {m*n, m >= 1, n >= 1} sorted into nondecreasing order.
Considering the maximal rectangle in each of the Ferrers graphs of partitions of n, a(n) is the smallest such maximal rectangle; a(n) is also an inverse of A006218. - Henry Bottomley, Mar 11 2002
The numbers in A003991 arranged in numerical order. - Matthew Vandermast, Feb 28 2003
Least k such that tau(1) + tau(2) + tau(3) + ... + tau(k) >= n. - Michel Lagneau, Jan 04 2012
The number 1 appears only once, primes appear twice, squares of primes appear thrice. All other positive integers appear at least four times. - Alonso del Arte, Nov 24 2013

			Array begins:
   1
   2  2
   3  3
   4  4  4
   5  5
   6  6  6  6
   7  7
   8  8  8  8
   9  9  9
  10 10 10 10
  11 11
  12 12 12 12 12 12
  13 13
  14 14 14 14
  15 15 15 15
  16 16 16 16 16
  17 17
  18 18 18 18 18 18
  19 19
  20 20 20 20 20 20
  21 21 21 21
  22 22 22 22
  23 23
  24 24 24 24 24 24 24 24

N. J. A. Sloane, Table of n, a(n) for n = 1..7069
Hayato Kobayashi, Perplexity on Reduced Corpora, in: Proceedings of the 52nd Annual Meeting of the Association for Computational Linguistics, Baltimore, Maryland, USA, June 23-25 2014, Association for Computational Linguistics, 2014, pp. 797-806.

Cf. A000005. An inverse to A006218.

Cf. A027750, A056538, A072691.

with(numtheory); t1:=[]; for i from 1 to 1000 do for j from 1 to tau(i) do t1:=[op(t1),i]; od: od: t1:=sort(t1);

Flatten[Table[Table[n, {Length[Divisors[n]]}], {n, 30}]]

a(n)=if(n<0,0,t=1;while(sum(k=1,t,floor(t/k))Benoit Cloitre, Nov 08 2009

a(n) >= pi(n+1) for all n; a(n) >= pi(n) + 1 for all n >= 24 (cf. A098357, A088526, A006218, A052511). - N. J. A. Sloane, Oct 22 2008
a(n) = A027750(n) * A056538(n). - Charles Kusniec, Jan 21 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 (A072691). - Amiram Eldar, Jan 14 2024

More terms from Erich Friedman, Jun 01 2001

A112954 Number of numbers m such that phi(m) = n*tau(m), with phi=A000010 and tau=A000005.

Original entry on oeis.org

7, 9, 10, 9, 7, 17, 4, 17, 14, 15, 7, 19, 2, 16, 20, 21, 0, 29, 0, 29, 9, 13, 7, 32, 7, 11, 23, 21, 7, 39, 0, 19, 17, 4, 11, 44, 2, 0, 11, 41, 7, 24, 2, 19, 30, 11, 0, 55, 4, 23, 7, 21, 7, 46, 9, 27, 4, 11, 0, 61, 0, 0, 27, 29, 9, 30, 2, 10, 19, 31, 0, 57, 2, 9, 27, 4, 4, 30, 2, 50, 29, 9

Offset: 1

Reinhard Zumkeller, Oct 07 2005

nonn

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000

Cf. A020491, A020488, A062516, A063469, A063470, A112955, A175667.

More terms from Max Alekseyev, Mar 01 2010

A248397 Noncongruent squarefree numbers n with A248394(n)/d(n) = 1, where d(n) = A000005(n).

Original entry on oeis.org

1, 3, 33, 51, 57, 59, 83, 139, 177, 187, 209, 211, 267, 321, 339, 345, 379, 385, 411, 451, 489, 499, 515, 555, 587, 595, 649, 659, 665, 681, 707, 803, 811, 827, 835, 899, 921, 969, 1001, 1059, 1099, 1137, 1171, 1211, 1219, 1235, 1259, 1267, 1281, 1315, 1329, 1363

Offset: 1

N. J. A. Sloane, Oct 20 2014

nonn

J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

Cf. A000005, A003273, A165564, A248394, A248395, A248397-A248407.

More terms from Amiram Eldar, Oct 13 2019

A302051 An analog of A000005 for nonstandard factorization based on the sieve of Eratosthenes (A083221).

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

Offset: 1

Antti Karttunen, Apr 01 2018

nonn

See A302042, A302044 and A302045 for a description of the factorization process.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences generated by sieves

Cf. A000005, A083221, A302042, A302044, A302045, A302052 (reduced modulo 2), A302053 (gives the positions of odd numbers).

Cf. also A253557, A302041, A302050, A302052, A302039, A302055 for other similar analogs.

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A000265(n) = (n/2^valuation(n, 2));
A001511(n) = 1+valuation(n,2);
A302045(n) = A001511(A078898(n));
A302044(n) = { my(c = A000265(A078898(n))); if(1==c,1,my(p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); (k*p)); };
A302051(n) = if(1==n,n,(A302045(n)+1)*A302051(A302044(n)));

\\ Or, using also some of the code from above:
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A055396(n) = if(1==n,0,primepi(A020639(n)));
A250246(n) = if(1==n,n,my(k = 2*A250246(A078898(n)), r = A055396(n)); if(1==r, k, while(r>1, k = A003961(k); r--); (k)));
A302051(n) = numdiv(A250246(n));

a(1) = 1, for n > 1, a(n) = (A302045(n)+1) * a(A302044(n)).
a(n) = A000005(A250246(n)).
a(n) = A106737(A252754(n)).

cp's OEIS Frontend

A160812 a(n) = A161205(n)-A000005(n).

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A295664 Exponent of the highest power of 2 dividing number of divisors of n: a(n) = A007814(A000005(n)); 2-adic valuation of tau(n).

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A339258 Triangle read by rows T(n,k), (n >= 1, k > = 1), in which row n has length A000070(n-1) and every column gives A000005, the number of divisors function.

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A360179 a(1) = 1. Thereafter if a(n-1) is a novel term, a(n) = d(a(n-1)); otherwise a(n) = a(n-1) + d(u), where d is the divisor function A000005 and u is the smallest unstarred prior term (each time we use a prior term we star it, and starred terms cannot be reused).

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A371165 Positive integers with as many divisors (A000005) as distinct divisors of prime indices (A370820).

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A051281 Sum of divisors of n, sigma(n) (A000203), is a power of number of divisors of n, d(n) (A000005).

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A061017 List in which n appears d(n) times, where d(n) [A000005] is the number of divisors of n.

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