A030513 -id:A030513 - cp's OEIS frontend

A191369 Numbers n with k divisors such that n-1 and n+1 in binary representation have same number k of 0's as 1's.

11, 155, 203, 2164, 2228, 2252, 2276, 2348, 2404, 2452, 2468, 2588, 2612, 2636, 2644, 2675, 2708, 2763, 2836, 2891, 2979, 3148, 3179, 3236, 3275, 3283, 3411, 3475, 3716, 3971, 33723, 33755, 34235, 34523, 34539, 34715, 34771, 35315, 35563, 35571, 35787, 36155, 36411, 36507, 36555, 36579

Offset: 1

Juri-Stepan Gerasimov, Jun 04 2011

nonn

Cf. A030513, A030516, A030516, A191292.

isA191369(n)=my(b=vecsort(binary(n-1)),k=#b\2); #b==k+k & !b[k] & b[k+1] & b==vecsort(binary(n+1)) & numdiv(n)==k \\ Charles R Greathouse IV, Jun 05 2011

a(10) corrected, a(31)-a(46) added by Charles R Greathouse IV, Jun 05 2011

A338483 a(n) is the smallest number having n smaller numbers with the same number of divisors.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 201, 202, 203, 205, 206, 209, 213

Offset: 1

Ivan N. Ianakiev, Oct 30 2020

nonn

Inspired by A047983.

Are there prime terms greater than 31?

			The smallest number having two smaller numbers (2 and 3) with the same number of divisors is 5, so a(2) is 5.

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

Cf. A000005, A007422, A030513, A047983.

N:= 500: # for terms before the first term > N
T:= map(numtheory:-tau, [$1..N]):
M:= max(T):
V:= Vector(M):
for n from 1 to N do
  v:= T[n];
  V[v]:= V[v]+1;
  if not assigned(R[V[v]]) then R[V[v]]:= n fi
od:
for nn from 1 while assigned(R[nn]) do od:
seq(R[i],i=2..nn-1); # Robert Israel, Oct 30 2020

f[n_]:=With[{tau=DivisorSigma[0,n]},Length[Select[Range[n-1],DivisorSigma[0,#]==tau&]]];t=Table[f[n],{n,1,300}]; a[n_]:=FirstPosition[t,n]; Rest[a/@Range[0,65]]//Flatten (* f(n) by Jean-François Alcover at A047983 *)

f(n) = {my(d=numdiv(n)); sum(k=1, n-1, (numdiv(k)==d))} \\ A047983
a(n) = my(k=1); while (f(k)!= n, k++); k; \\ Michel Marcus, Oct 30 2020

A047983(a(n)) = n. - Rémy Sigrist, Dec 06 2020

A385350 Numbers j such that the product of odd proper divisors of j is j.

Original entry on oeis.org

1, 15, 21, 27, 33, 35, 39, 51, 55, 57, 65, 69, 77, 85, 87, 91, 93, 95, 111, 115, 119, 123, 125, 129, 133, 141, 143, 145, 155, 159, 161, 177, 183, 185, 187, 201, 203, 205, 209, 213, 215, 217, 219, 221, 235, 237, 247, 249, 253, 259, 265, 267, 287, 291, 295, 299

Offset: 1

Ilya Gutkovskiy, Jun 26 2025

Fixed points of A385349.
Odd terms in A007422.
Also 1 with odd numbers with exactly 4 divisors. - David A. Corneth, Jun 26 2025

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Multiplicative Perfect Number.

Cf. A007422, A030513, A073582, A385349.

q:= n-> n=1 or n::odd and numtheory[tau](n)=4:
select(q, [$1..500])[];  # Alois P. Heinz, Jun 26 2025

A385349[n_] := Times @@ Select[Divisors[n], # < n && OddQ[#] &]; Select[Range[300], A385349[#] == # &]

isok(k) = vecprod(select((x->((x%2)==1) && (xMichel Marcus, Jun 26 2025

is(n) = (n == 1) || (bitand(n, 1) && numdiv(n) == 4) \\ David A. Corneth, Jun 26 2025

from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A385350(n):
    def f(x): return int(n-1+x-primepi(integer_nthroot(x,3)[0])+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(3, s+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Jun 27 2025

A065985 Numbers k such that d(k) / 2 is prime, where d(k) = number of divisors of k.

Original entry on oeis.org

6, 8, 10, 12, 14, 15, 18, 20, 21, 22, 26, 27, 28, 32, 33, 34, 35, 38, 39, 44, 45, 46, 48, 50, 51, 52, 55, 57, 58, 62, 63, 65, 68, 69, 74, 75, 76, 77, 80, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 106, 111, 112, 115, 116, 117, 118, 119, 122, 123, 124, 125, 129, 133, 134

Offset: 1

Joseph L. Pe, Dec 10 2001

nonn

Numbers whose sorted prime signature (A118914) is either of the form {2*p-1} or {1, p-1}, where p is a prime. Equivalently, disjoint union of numbers of the form q^(2*p-1) where p and q are primes, and numbers of the form r * q^(p-1), where p, q and r are primes and r != q. - Amiram Eldar, Sep 09 2024

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

Cf. A000005, A118914.
Numbers with exactly 2*p divisors: A030513 (p=2), A030515 (p=3), A030628 \ {1} (p=5), A030632 (p=7), A137485 (p=11), A137489 (p=13), A175744 (p=17), A175747 (p=19).
Subsequences: A006881, A030078, A050997, A054753, A095990, A096156, A138031, A178739, A179665, A189987.

Select[Range[1, 1000], PrimeQ[DivisorSigma[0, # ] / 2] == True &]

n=0; for (m=1, 10^9, f=numdiv(m)/2; if (frac(f)==0 && isprime(f), write("b065985.txt", n++, " ", m); if (n==1000, return))) \\ Harry J. Smith, Nov 05 2009

is(n)=n=numdiv(n)/2; denominator(n)==1 && isprime(n) \\ Charles R Greathouse IV, Oct 15 2015

A079836 First column of the triangle in which the n-th row contains n numbers with n divisors that lie between A079835(n) and A079835(n+1).

Original entry on oeis.org

1, 3, 9, 51, 81, 28577, 117649, 594823330, 595067236, 596971504

Offset: 1

Amarnath Murthy, Feb 15 2003

1
3 5
9 25 49
51 55 57 58
81 625 2401 14641 28561
...
The 4th row consists of 4 consecutive elements of A030513, the 5th row 5 consecutive elements of A030514, the 6th and 7th rows consecutive elements of A030515 and A030516, the 8th of A030626, the 9th of A030627 etc. - R. J. Mathar, Mar 29 2007

Cf. A079835, A079837.

Cf. A030513, A030514, A030515, A030516, A030626, A030627.

a(6)-a(7) from R. J. Mathar, Mar 29 2007
a(8)-a(10) from Lambert Herrgesell (zero815(AT)googlemail.com), Feb 08 2008
a(2) and a(9) corrected by Pontus von Brömssen, Jan 14 2024

A137490 Numbers with 27 divisors.

Original entry on oeis.org

900, 1764, 2304, 4356, 4900, 6084, 6400, 10404, 11025, 12100, 12544, 12996, 16900, 19044, 23716, 26244, 27225, 28900, 30276, 30976, 33124, 34596, 36100, 38025, 43264, 49284, 52900, 53361, 56644, 60516, 65025, 66564, 70756, 73984, 74529

Offset: 1

R. J. Mathar, Apr 22 2008

nonn

Maple implementation: see A030513.

Numbers of the form p^26 (subset of A089081), p^2*q^2*r^2 (like 900, 1764, 4356, squares of A007304) or p^2*q^8 (like 2304, 6400, subset of the squares of A030628) where p, q and r are distinct primes. - R. J. Mathar, Mar 01 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000005, A030513-A030516, A030626, A030627, A030634-A030638, A005179, A003680, A096932, A061286, A061283.

Select[Range[150000],DivisorSigma[0,#]==27&] (* Vladimir Joseph Stephan Orlovsky, May 06 2011 *)

isA137490(n) = numdiv(n)==27 \\ Michael B. Porter, Apr 10 2010

A000005(a(n)) = 27.

Sum_{n>=1} 1/a(n) = (P(2)^3 + 2*P(6) - 3*P(2)*P(4))/6 + P(2)*P(8) - P(10) + P(26) = 0.00453941..., where P is the prime zeta function. - Amiram Eldar, Jul 03 2022

A189986 Numbers of the form 4k+1 having exactly 4 divisors.

Original entry on oeis.org

21, 33, 57, 65, 69, 77, 85, 93, 125, 129, 133, 141, 145, 161, 177, 185, 201, 205, 209, 213, 217, 221, 237, 249, 253, 265, 301, 305, 309, 321, 329, 341, 365, 377, 381, 393, 413, 417, 437, 445, 453, 469, 473, 481, 485, 489, 493, 497, 501, 505, 517, 533, 537, 545

Offset: 1

Juri-Stepan Gerasimov, May 03 2011

nonn

Intersection of A016813 and A030513; subsequence of A007422.

Numbers p*q with p == q (mod 4) together with p^3 with p == 1 (mod 4), p and q distinct primes. - Charles R Greathouse IV, May 03 2011

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A007422, A016813, A030513.

[ n: n in [1..600 by 4] | #Divisors(n) eq 4 ]; // Klaus Brockhaus, May 04 2011

Select[4Range[200]+1,DivisorSigma[0,#]==4&] (* Harvey P. Dale, May 11 2011 *)

Corrected (497 inserted) by Klaus Brockhaus, May 04 2011

A306510 Numbers k such that twice the number of divisors of k is equal to the number of divisors of the sum of digits of k.

Original entry on oeis.org

17, 19, 37, 53, 59, 71, 73, 107, 109, 127, 149, 163, 167, 181, 233, 239, 251, 257, 271, 293, 307, 347, 383, 419, 431, 433, 491, 499, 503, 509, 521, 523, 541, 563, 613, 617, 631, 653, 699, 701, 743, 761, 769, 787, 789, 811, 859, 877, 879, 941, 967

Offset: 1

Ctibor O. Zizka, Feb 20 2019

From Robert Israel, Jul 28 2020: (Start)
The first even term is a(2747)=68998.
Includes primes p such that A007953(p) is in A030513. (End)

			For k = 19, 2*A000005(19) = A000005(A007953(19)), 2*A000005(19) = A000005(10), thus k = 19 is a member of the sequence.

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

Cf. A000005, A007953, A030513, A306509.

filter:= proc(n) 2*numtheory:-tau(n) = numtheory:-tau(convert(convert(n,base,10),`+`)) end proc:
select(filter, [$1..1000]); # Robert Israel, Jul 28 2020

isok(k) = (k >= 1) && (2*numdiv(k) == numdiv(sumdigits(k, 10))); \\ Daniel Suteu, Feb 20 2019

2*A000005(k) = A000005(A007953(k)).

A307980 Numbers k whose number of divisors is the square of the number of decimal digits of k.

Original entry on oeis.org

1, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 196, 225, 256, 441, 484, 676, 1000, 1026, 1032, 1064, 1110, 1122, 1128, 1144, 1155, 1160, 1190, 1218, 1230, 1240, 1242, 1254, 1272, 1288, 1290, 1302, 1326, 1330, 1365, 1408

Offset: 1

Bernard Schott, May 08 2019

The terms with an odd number of digits are squares.
The terms with 2 digits are squarefree semiprimes (cf. A006881) Union {27}. The terms with 3 digits belong to A030627 (numbers with 9 divisors) and the ones with 4 digits belong to A030634 (numbers with 16 divisors).
The number of terms b(n) with n digits begins with: 1, 30, 7, 753, 3, 11409, 2, ... When there are an odd number of digits, the number of terms decreases from b(3) = 7, b(5) = 3, b(7) = 2. Is there a 2q+1 such that b(2q+1) = 0?
The sequence is infinite because 10^k is the term for each k. We have tau(10^k) = tau(2^k)*tau(5^k) = (k + 1)^2 and 10^k has k + 1 digits. - Marius A. Burtea, May 09 2019
a(n) >= 1, for any n, so b(2q+1)>= 1 for any q. - Marius A. Burtea, May 09 2019

			65 is a term with 2 digits and 4 divisors: {1, 5, 13, 65}.
484 is a term with 3 digits and 9 divisors: {1, 2, 4, 11, 22, 44, 121, 242, 484}.

Sean A. Irvine, Java program (github)

Cf. A095862 (number of decimal digits = number of divisors).
Cf. A006881 (squarefree semiprimes).
Cf. A030513 (numbers with 4 divisors), A030627 (numbers with 9 divisors), A030634 (numbers with 16 divisors).
Cf. A011557 (subsequence).

[n:n in [1..1500]|NumberOfDivisors(n) eq (#Intseq(n))^2]; // Marius A. Burtea, May 09 2019

is(n) = numdiv(n) == #digits(n)^2 \\ David A. Corneth, May 08 2019

A334802 Positive integers of the form x^4 - y^4 that have exactly 4 divisors.

Original entry on oeis.org

15, 65, 671, 3439, 12209, 102719, 113521, 178991, 246559, 515201, 1124111, 1342879, 2964961, 3940399, 9951391, 21254449, 27220159, 34209169, 45259649, 48986321, 70710641, 92110289, 93084991, 125620111, 131687681, 144402721, 201792079, 211782751, 276694241

Offset: 1

C. Kenneth Fan, May 12 2020

nonn

If a(n) = pq, where p > q are both prime, then p is the hypotenuse and q is a leg of a primitive Pythagorean triple. (x^4-y^4 = (x^2+y^2)(x+y)(x-y), hence x-y=1 and x^2+y^2 and x+y are both prime. Note that x^2+y^2 can never be (x+y)^2 so a(n) is never the cube of a prime.)

			2^4 - 1^4 = 15 = 3*5 and (3, 4, 5) is a Pythagorean triple, so 15 is a term.
6^4 - 5^4 = 671 = 11*61 and (11, 60, 61) is a Pythagorean triple, so 671 is a term.

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

Cf. A068501.

Intersection of A030513 and A147857.

f:= proc(y) if isprime(2*y+1) and isprime(2*y^2 + 2*y+1) then (2*y+1)*(2*y^2+2*y+1) fi end proc:
map(f, [$1..1000]); # Robert Israel, Jun 16 2020

Select[(#^4 - (#-1)^4) & /@ Range[420], DivisorSigma[0, #] == 4 &] (* Giovanni Resta, May 12 2020 *)

a(n) = (b(n)+1)^4 - b(n)^4 with b(n)=A068501(n).

a(n) = A048161(n)*A067756(n).

cp's OEIS Frontend

A191369 Numbers n with k divisors such that n-1 and n+1 in binary representation have same number k of 0's as 1's.

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A338483 a(n) is the smallest number having n smaller numbers with the same number of divisors.

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A385350 Numbers j such that the product of odd proper divisors of j is j.

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A065985 Numbers k such that d(k) / 2 is prime, where d(k) = number of divisors of k.

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A079836 First column of the triangle in which the n-th row contains n numbers with n divisors that lie between A079835(n) and A079835(n+1).

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A137490 Numbers with 27 divisors.

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A189986 Numbers of the form 4k+1 having exactly 4 divisors.

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Magma

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A306510 Numbers k such that twice the number of divisors of k is equal to the number of divisors of the sum of digits of k.

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