1

Table of n, a(n) for n=1..121.

Index entries for characteristic functions

Index entries for sequences computed from exponents in factorization of n

a(n) = A080339(A000005(n)).

a[n_] := If[!CompositeQ[DivisorSigma[0, n]], 1, 0]; Array[a, 100] (* Amiram Eldar, Jul 21 2022 *)

(PARI) A355937(n) = ((1==n)||isprime(isprimepower(n)+1));

Characteristic function of {1} UNION A009087.

Cf. A000005, A080339.

Cf. also A355938.

nonn,new

Antti Karttunen, Jul 21 2022

approved

1

Table of n, a(n) for n=1..125.

Index entries for characteristic functions

a(n) = A080339(A000203(n^2)) = A080339(A065764(n)).

For all n >= 1, a(n) <= A010055(n).

a[n_] := If[!CompositeQ[DivisorSigma[1, n^2]], 1, 0]; Array[a, 100] (* Amiram Eldar, Jul 21 2022 *)

(PARI) A355938(n) = ((1==n)||isprime(sigma(n^2)));

Characteristic function of {1} UNION A055638.

Cf. A000203, A010055, A065764, A080339.

Cf. also A355937.

nonn,new

Antti Karttunen, Jul 21 2022

approved

1,8

Multiplicative because A295316 is.

Table of n, a(n) for n=1..105.

Index entries for sequences computed from exponents in factorization of n

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A295316(n/d) * a(d).

s[n_] := If[AllTrue[FactorInteger[n][[;; , -1]], OddQ], 1, 0]; a[1] = 1; a[n_] := -DivisorSum[n, a[#]*s[n/#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Jul 21 2022 *)

(PARI)

A295316(n) = factorback(apply(e -> (e%2), factorint(n)[, 2]));

memoA355936 = Map();

A355936(n) = if(1==n, 1, my(v); if(mapisdefined(memoA355936, n, &v), v, v = -sumdiv(n, d, if(d<n, A295316(n/d)*A355936(d), 0)); mapput(memoA355936, n, v); (v)));

Cf. A268335, A295316.

Cf. also A355826.

sign,mult,new

Antti Karttunen, Jul 21 2022

approved

0,3

Table of n, a(n) for n=0..14.

a(n) = A355733(A070826(n)).

The a(0) = 1 through a(4) = 10 multisets:

  {}  {1}  {1,1}  {1,1,1}  {1,1,1,1}

           {1,2}  {1,1,2}  {1,1,1,2}

                  {1,1,3}  {1,1,1,3}

                  {1,2,3}  {1,1,1,4}

                           {1,1,2,2}

                           {1,1,2,3}

                           {1,1,2,4}

                           {1,1,3,4}

                           {1,2,2,3}

                           {1,2,3,4}

Table[Length[Union[Sort/@Tuples[Divisors/@Range[n]]]], {n, 0, 10}]

The sum of the same integers is A000096.

The product of the same integers is A000142, Heinz number A070826.

Counting sequences instead of multisets gives A066843.

The integers themselves are the rows of A131818 (shifted).

For prime indices we have A355733, only prime factors A355744.

For prime factors instead of divisors we have A355746, factors A355537.

A000005 counts divisors.

A000040 lists the prime numbers.

A001221 counts distinct prime factors, with sum A001414.

A001222 counts prime factors with multiplicity.

Cf. A000720, A002110, A076610, A327486, A355538, A355731, A355737, A355741, A355745.

nonn,more,new

Gus Wiseman, Jul 20 2022

approved

1,6

Table of n, a(n) for n=1..27.

a(n) = A355744(A070826(n)).

The a(n) multisets for n = 2, 6, 10, 12:

  {1}  {1,1,1,2,3}  {1,1,1,1,1,2,2,3,4}  {1,1,1,1,1,1,2,2,3,4,5}

       {1,1,2,2,3}  {1,1,1,1,2,2,2,3,4}  {1,1,1,1,1,2,2,2,3,4,5}

                    {1,1,1,1,2,2,3,3,4}  {1,1,1,1,1,2,2,3,3,4,5}

                    {1,1,1,2,2,2,3,3,4}  {1,1,1,1,2,2,2,2,3,4,5}

                                         {1,1,1,1,2,2,2,3,3,4,5}

                                         {1,1,1,2,2,2,2,3,3,4,5}

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Table[Length[Union[Sort/@Tuples[primeMS/@Range[2, n]]]], {n, 15}]

The sum of the same integers is A000096.

The product of the same integers is A000142, Heinz number A070826.

The integers themselves are the rows of A131818 (shifted).

Counting sequences instead of multisets: A355537, with multiplicity A327486.

Using prime indices instead of 2..n gives A355744, for sequences A355741.

The version for divisors instead of prime factors is A355747.

A000040 lists the prime numbers.

A001221 counts distinct prime factors, with sum A001414.

A001222 counts prime factors with multiplicity.

A003963 multiplies together the prime indices of n.

A056239 adds up prime indices, row sums of A112798.

Cf. A000720, A002110, A076610, A355538, A355731, A355733, A355740, A355742, A355745.

nonn,more,new

Gus Wiseman, Jul 20 2022

approved

1,7

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.

Table of n, a(n) for n=1..87.

Wikipedia, Cartesian product.

Totally multiplicative with a(prime(k)) = A001222(k).

The prime indices of 49 are {4,4}, and the a(49) = 4 choices are: (2,2), (2,4), (4,2), (4,4).

The prime indices of 777 are {2,4,12}, and the a(777) = 6 choices are: (2,2,2), (2,2,3), (2,2,4), (2,4,2), (2,4,3), (2,4,4).

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Table[Times@@PrimeOmega/@primeMS[n], {n, 100}]

The unordered version is A001970, row-sums of A061260.

Positions of 1's are A076610, just primes A355743.

Positions of 0's are A299174.

Allowing all divisors (not just primes) gives A355731, firsts A355732.

Choosing only prime factors (not prime-powers) gives A355741.

Counting multisets of primes gives A355744.

The case of weakly increasing primes A355745, all divisors A355735.

A000688 counts factorizations into prime powers.

A001414 adds up distinct prime factors, counted by A001221.

A003963 multiplies together the prime indices of n.

A056239 adds up prime indices, row sums of A112798, counted by A001222.

Cf. A000720, A120383, A279784, A289509, A324850, A355733, A355739, A355746.

nonn,mult,new

Gus Wiseman, Jul 20 2022

approved

1,6

Also partial products of A001221 without the first term 0, sum A013939.

For initial terms up to n = 29 we have a(n) = 2^A355538(n). The first non-power of 2 is a(30) = 12288.

Table of n, a(n) for n=1..46.

The a(n) choices for n = 2, 6, 10, 12, with prime(k) replaced by k:

  (1)  (12131)  (121314121)  (12131412151)

       (12132)  (121314123)  (12131412152)

                (121324121)  (12131412351)

                (121324123)  (12131412352)

                             (12132412151)

                             (12132412152)

                             (12132412351)

                             (12132412352)

Table[Times@@PrimeNu/@Range[2, m], {m, 2, 30}]

The sum of the same integers is A000096.

The product of the same integers is A000142, Heinz number A070826.

The version for divisors instead of prime factors is A066843.

The integers themselves are the rows of A131818.

The version with multiplicity is A327486.

Using prime indices instead of 2..n gives A355741, for multisets A355744.

Counting sequences instead of multisets gives A355746.

A001221 counts distinct prime factors, with sum A001414.

A001222 counts prime factors with multiplicity.

A003963 multiplies together the prime indices of n.

A056239 adds up prime indices, row sums of A112798.

Cf. A000005, A000040, A000720, A002110, A013939, A076610, A355538, A355731, A355733, A355745, A355747.

nonn,new

Gus Wiseman, Jul 20 2022

approved

0,1

Table of n, a(n) for n=0..86.

Michael Ian Shamos, Shamos's Catalog of the Real Numbers, 2011, p. 564.

Equals Sum_{k>=2} arccoth(k)/k.

Equals Sum_{k>=1} (zeta(2*k)-1)/(2*k-1).

Equals log(Product_{k>=2} ((k+1)/(k-1))^(1/(2*k))).

0.67656513614796664082647201...

RealDigits[N[Sum[ArcTanh[1/k]/k, {k, 2, Infinity}], 30], 10, 26][[1]] (* Amiram Eldar, Jul 21 2022 *)

Cf. A016655, A355921, A355923.

nonn,cons,new

Amiram Eldar, Jul 21 2022

More terms from Jinyuan Wang, Jul 21 2022

approved

-1,1

Table of n, a(n) for n=-1..103.

Michael Ian Shamos, Shamos's Catalog of the Real Numbers, 2011, pp. 126-127.

Equals Sum_{k>=1} (zeta(2*k+1)-1)/(2*k+1).

Equals 1 - gamma - log(2)/2, where gamma is Euler's constant (A001620).

Equals Sum_{k>=2} ((1/2)*log((k+1)/(k-1)) - 1/k).

Equals 2 * Integral_{x>=0} x * exp(-x) * log(x) * sin(x) dx.

0.07621074481849448468487184918850928492009059687994877413...

RealDigits[1 - EulerGamma - Log[2]/2, 10, 100][[1]]

Cf. A001620, A002162, A016655, A239097, A352619, A355922.

nonn,cons,new

Amiram Eldar, Jul 21 2022

approved

1,3

Row sums see A001318.

Table of n, a(n) for n=1..92.

The triangle begins:

      k=1  2  3  4  5  6

  n=1:  1;

  n=2:  0, 2;

  n=3:  2, 0, 3;

  n=4:  0, 3, 0, 4;

  n=5:  3, 0, 4, 0, 5;

  n=6:  0, 4, 0, 5, 0, 6;

  and so on.

T[n_, k_] := If[EvenQ[n + k], (n + k)/2, 0]; Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Amiram Eldar, Jul 19 2022 *)

Cf. A001318, A138099 (without the zeros).

nonn,tabl,new

Ctibor O. Zizka, Jul 19 2022

approved