Judson Neer - cp's OEIS frontend

A349932 Product of n and its binary ones' complement.

0, 2, 0, 12, 10, 6, 0, 56, 54, 50, 44, 36, 26, 14, 0, 240, 238, 234, 228, 220, 210, 198, 184, 168, 150, 130, 108, 84, 58, 30, 0, 992, 990, 986, 980, 972, 962, 950, 936, 920, 902, 882, 860, 836, 810, 782, 752, 720, 686, 650, 612, 572, 530, 486, 440, 392, 342, 290

Offset: 1

Judson Neer, Dec 05 2021

			For n = 9, a(9) = 0b1001 * 0b0110 = 9 * 6 = 54.

Cf. A035327.

a(n) = n*bitneg(n, exponent(n)) \\ Andrew Howroyd, Dec 05 2021

def a(n): return 0 if n == 0 else n * (n^((1 << n.bit_length()) - 1))
print([a(n) for n in range(1, 58)]) # Michael S. Branicky, Dec 05 2021

a(n) = n*A035327(n).

A276115 Numbers whose digits have a permutation that is a palindrome.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 131, 133, 141, 144, 151, 155, 161, 166, 171, 177, 181, 188, 191, 199, 202, 211, 212, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 232, 233, 242, 244, 252, 255, 262, 266

Offset: 1

Judson Neer, Aug 19 2016

Permutations with leading zeros are not considered as palindromic, thus (for example) 10 is not included in the sequence.

Also numbers in which at most 1 digit occurs an odd number of times and (if there is more than one digit) at least 2 digits are nonzero. - David A. Corneth, Aug 21 2016, corrected by Robert Israel, Aug 31 2016

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

Cf. A084050 (for a sequence where leading zero numbers are included).

filter:= proc(n) local L,M;
  if n < 10 then return true fi;
  L:= convert(n,base,10);
  M:= [seq(numboccur(j,L),j=0..9)];
  convert(M mod 2, `+`) <= 1 and convert(M[2..-1],`+`)>=2
end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 31 2016

is(n) = {my(v = concat(vecsort(digits(n)), ["a"]), prev=1, odd=0); if(#v>2&&v[#v-2]==0,return(0)); for(i=1,#v-1, if(v[i]!=v[i+1], odd+=(i-prev+1)%2; if(odd==2,return(0)); prev = i + 1)); 1} \\ David A. Corneth, Aug 21 2016

101 inserted by Robert Israel, Aug 31 2016

A059160 Number of ordered ways of writing n as a sum of 5 generalized pentagonal numbers (A001318).

Original entry on oeis.org

1, 5, 15, 30, 45, 56, 65, 85, 115, 150, 171, 175, 185, 205, 260, 300, 325, 340, 350, 415, 440, 485, 500, 505, 580, 581, 650, 645, 675, 815, 815, 910, 845, 865, 985, 951, 1130, 1030, 1060, 1155, 1150, 1370, 1265, 1410, 1495, 1420, 1545, 1460, 1600, 1675, 1690

Offset: 0

Judson Neer, Feb 14 2001

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A001318, A008443, A014110, A080995.

a[n_] := SeriesCoefficient[(QPochhammer[-q, q^3]* QPochhammer[-q^2, q^3]*QPochhammer[q^3, q^3])^5, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jun 12 2017 *)

G.f.: f(x, x^2)^5 where f(,) is Ramanujan's two-variable theta function. - Michael Somos, Jun 08 2012

A051027 a(n) = sigma(sigma(n)) = sum of the divisors of the sum of the divisors of n.

Original entry on oeis.org

1, 4, 7, 8, 12, 28, 15, 24, 14, 39, 28, 56, 24, 60, 60, 32, 39, 56, 42, 96, 63, 91, 60, 168, 32, 96, 90, 120, 72, 195, 63, 104, 124, 120, 124, 112, 60, 168, 120, 234, 96, 252, 84, 224, 168, 195, 124, 224, 80, 128, 195, 171, 120, 360, 195, 360, 186, 234, 168, 480, 96

Offset: 1

Judson Neer

			a(2) = 4 because sigma(2)=1+2=3 and sigma(3)=1+3=4. - _Zak Seidov_, Aug 29 2012

József Sándor, On the composition of some arithmetic functions, Studia Univ. Babeș-Bolyai, Vol. 34, No. 1 (1989), pp. 7-14.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, p. 39.

T. D. Noe, Table of n, a(n) for n=1..5000

Cf. A000203.

with(numtheory): [seq(sigma(sigma(n)), n=1..100)];

DivisorSigma[1,DivisorSigma[1,Range[100]]] (* Zak Seidov, Aug 29 2012 *)

a(n)=sigma(sigma(n)); \\ Joerg Arndt, Feb 16 2014

from sympy import divisor_sigma as sigma
def a(n): return sigma(sigma(n))
print([a(n) for n in range(1, 62)]) # Michael S. Branicky, Dec 05 2021

a(n) = A000203(A000203(n)). - Zak Seidov, Aug 29 2012
a(p) = sigma(p+1) = A000203(p+1), for p prime. - Wesley Ivan Hurt, Feb 14 2014
a(n) = 2*n iff n = 2^q with M_(q+1) = 2^(q+1) - 1 is a Mersenne prime, hence iff n = 2^q with q in A090748. - Bernard Schott, Aug 08 2019
a(n) >= 2*n for even n, with equality only when n = 2^k and 2^(k+1) - 1 is prime (Sándor, 1989). - Amiram Eldar, Mar 09 2021

A049883 Primes in the Jacobsthal sequence (A001045).

Original entry on oeis.org

3, 5, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243

Offset: 1

Judson Neer

nonn

All terms, except a(2) = 5, are of the form (2^p + 1)/3 - the Wagstaff primes A000979 = {3, 11, 43, 683, 2731, 43691, 174763, ...}.
Indices of prime Jacobsthal numbers are listed in A107036 = {3, 4, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, ...}.
For n > 1, A107036(n) = A000978(n) (numbers m such that (2^m + 1)/3 is prime). - Alexander Adamchuk, Oct 10 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..23

Cf. A001045, A000978, A000979, A107036.

Select[Table[(2^n + (-1)^(n - 1))/3, {n, 200}], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Mar 29 2011 *)

A051357 Chernoff sequence A006939 divided by 2.

Original entry on oeis.org

1, 6, 180, 37800, 87318000, 2622159540000, 1338638666765400000, 12984380089637682726000000, 2896722619368127899492763620000000, 18740906719713843949122453226304292600000000

Offset: 1

Judson Neer

G. C. Greubel, Table of n, a(n) for n = 1..37

[(&*[NthPrime(k)^(n-k+1): k in [1..n]])/2: n in [1..15]]; // G. C. Greubel, Oct 14 2018

Table[Product[Prime[k]^(n - k + 1), {k, 1, n}]/2, {n, 1, 15}] (* G. C. Greubel, Oct 14 2018 *)

for(n=1, 15, print1(prod(k=1,n, prime(k)^(n-k+1))/2, ", ")) \\ G. C. Greubel, Oct 14 2018

a(n) = Product_{k=1..n} prime(k)^(n-k+1) / 2.

A051572 a(1) = 5, a(n) = sigma(a(n-1)).

Original entry on oeis.org

5, 6, 12, 28, 56, 120, 360, 1170, 3276, 10192, 24738, 61440, 196584, 491520, 1572840, 5433480, 20180160, 94859856, 355532800, 1040179456, 2143289344, 4966055344, 10092086208, 31800637440, 137371852800, 641012414823

Offset: 1

Judson Neer

N. J. A. Sloane, Table of n, a(n) for n = 1..523
G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 91-100.
Graeme L. Cohen and Herman J. J. te Riele, Iterating the Sum-of-Divisors Function, Experimental Mathematics, Vol. 5 (1996), No. 2, pp. 91-100.

Cf. A000203, A007497, A257348.

NestList[Plus@@Divisors[#] &, 5, 25] (* Alonso del Arte, Apr 28 2011 *)

a=[5];for(i=2,10,a=concat(a,sigma(a[#a]))); a \\ Charles R Greathouse IV, Oct 03 2011

from itertools import accumulate, repeat # requires Python 3.2 or higher
from sympy import divisor_sigma
A051572_list = list(accumulate(repeat(5,100), lambda x, _: divisor_sigma(x)))
# Chai Wah Wu, May 02 2015

cp's OEIS Frontend

User: Judson Neer

A349932 Product of n and its binary ones' complement.

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A276115 Numbers whose digits have a permutation that is a palindrome.

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A059160 Number of ordered ways of writing n as a sum of 5 generalized pentagonal numbers (A001318).

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A051027 a(n) = sigma(sigma(n)) = sum of the divisors of the sum of the divisors of n.

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A049883 Primes in the Jacobsthal sequence (A001045).

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A051357 Chernoff sequence A006939 divided by 2.

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A051572 a(1) = 5, a(n) = sigma(a(n-1)).

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