A067824 -id:A067824 - cp's OEIS frontend

A337074 Number of strict chains of divisors in A130091 (numbers with distinct prime multiplicities), starting with n!.

1, 1, 2, 0, 28, 0, 768, 0, 0, 0, 42155360, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Aug 16 2020

nonn

Support appears to be {0, 1, 2, 4, 6, 10}.

			The a(4) = 28 chains:
  24  24/1   24/2/1   24/4/2/1   24/8/4/2/1
      24/2   24/3/1   24/8/2/1   24/12/4/2/1
      24/3   24/4/1   24/8/4/1
      24/4   24/4/2   24/8/4/2
      24/8   24/8/1   24/12/2/1
      24/12  24/8/2   24/12/3/1
             24/8/4   24/12/4/1
             24/12/1  24/12/4/2
             24/12/2
             24/12/3
             24/12/4

A336867 is the complement of the support.
A336868 is the characteristic function (image under A057427).
A336942 is half the version for superprimorials (n > 1).
A337071 does not require distinct prime multiplicities.
A337104 is the case of chains ending with 1.
A000005 counts divisors.
A000142 lists factorial numbers.
A027423 counts divisors of factorial numbers.
A067824 counts chains of divisors starting with n.
A074206 counts chains of divisors from n to 1.
A076716 counts factorizations of factorial numbers.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts chains of divisors.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336414 counts divisors of n! with distinct prime multiplicities.
A336415 counts divisors of n! with equal prime multiplicities.
A336423 counts chains using A130091, with maximal case A336569.
A336571 counts chains of divisors 1 < d < n using A130091.
Cf. A000110, A001055, A002033, A007489, A022559, A048656, A071626, A098859, A124010, A167865, A325617, A336416, A336424.

chnsc[n_]:=If[!UnsameQ@@Last/@FactorInteger[n],{},If[n==1,{{1}},Prepend[Join@@Table[Prepend[#,n]&/@chnsc[d],{d,Most[Divisors[n]]}],{n}]]];
Table[Length[chnsc[n!]],{n,0,6}]

a(n) = 2*A337104(n) = 2*A336423(n!) for n > 1.

A337107 Irregular triangle read by rows where T(n,k) is the number of strict length-k chains of divisors from n! to 1.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 6, 9, 4, 0, 1, 14, 45, 52, 20, 0, 1, 28, 183, 496, 655, 420, 105, 0, 1, 58, 633, 2716, 5755, 6450, 3675, 840, 0, 1, 94, 1659, 11996, 46235, 106806, 155869, 145384, 84276, 27720, 3960

Offset: 1

Gus Wiseman, Aug 23 2020

Row n > 1 appears to be row n! of A334996.

			Triangle begins:
    1
    0    1
    0    1    2
    0    1    6    9    4
    0    1   14   45   52   20
    0    1   28  183  496  655  420  105
    0    1   58  633 2716 5755 6450 3675  840
Row n = 4 counts the following chains:
  24/1  24/2/1   24/4/2/1   24/8/4/2/1
        24/3/1   24/6/2/1   24/12/4/2/1
        24/4/1   24/6/3/1   24/12/6/2/1
        24/6/1   24/8/2/1   24/12/6/3/1
        24/8/1   24/8/4/1
        24/12/1  24/12/2/1
                 24/12/3/1
                 24/12/4/1
                 24/12/6/1

A097805 is the restriction to powers of 2.
A325617 is the maximal case.
A337105 gives row sums.
A337106 is column k = 3.
A000005 counts divisors.
A000142 lists factorial numbers.
A001055 counts factorizations.
A074206 counts chains of divisors from n to 1.
A027423 counts divisors of factorial numbers.
A067824 counts chains of divisors starting with n.
A076716 counts factorizations of factorial numbers.
A253249 counts chains of divisors.
A337071 counts chains starting with n!.
Cf. A022559, A124010, A167865, A251683, A337070, A337074.

b:= proc(n) option remember; expand(x*(`if`(n=1, 1, 0) +
      add(b(d), d=numtheory[divisors](n) minus {n})))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n!)):
seq(T(n), n=1..10);  # Alois P. Heinz, Aug 23 2020

nv=5;
chnsc[n_]:=Select[Prepend[Join@@Table[Prepend[#,n]&/@chnsc[d],{d,DeleteCases[Divisors[n],n]}],{n}],MemberQ[#,1]&];
Table[Length[Select[chnsc[n!],Length[#]==k&]],{n,nv},{k,1+PrimeOmega[n!]}]

A342522 Heinz numbers of integer partitions with constant (equal) first quotients.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 71, 73, 74, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97

Offset: 1

Gus Wiseman, Mar 23 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

			The prime indices of 2093 are {4,6,9}, with first quotients (3/2,3/2), so 2093 is in the sequence.
Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
   12: {1,1,2}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   30: {1,2,3}
   36: {1,1,2,2}
   40: {1,1,1,3}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   50: {1,3,3}
   52: {1,1,6}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}

Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Wikipedia, Arithmetic progression
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

For multiplicities (prime signature) instead of quotients we have A072774.
The version counting strict divisor chains is A169594.
For differences instead of quotients we have A325328 (count: A049988).
These partitions are counted by A342496 (strict: A342515, ordered: A342495).
The distinct instead of equal version is A342521.
A000005 count constant partitions.
A000041 counts partitions (strict: A000009).
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
A318991/A318992 rank reversed partitions with/without integer quotients.
A342086 counts strict chains of divisors with strictly increasing quotients.
Cf. A003242, A047966, A049980, A056239, A067824, A112798, A118914, A124010, A130091, A181819, A325351, A325352.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],SameQ@@Divide@@@Reverse/@Partition[primeptn[#],2,1]&]

A343343 Numbers with either no prime index dividing, or no prime index divisible by all the other prime indices.

Original entry on oeis.org

1, 15, 30, 33, 35, 45, 51, 55, 60, 66, 69, 70, 75, 77, 85, 90, 91, 93, 95, 99, 102, 105, 110, 119, 120, 123, 132, 135, 138, 140, 141, 143, 145, 150, 153, 154, 155, 161, 165, 170, 175, 177, 180, 182, 186, 187, 190, 195, 198, 201, 203, 204, 205, 207, 209, 210

Offset: 1

Gus Wiseman, Apr 15 2021

nonn

After 1, first differs from A318992 in lacking 390, with prime indices {1,2,3,6}.
First differs from A343337 in having 195, with prime indices {2,3,6}.
Alternative name: 1 and numbers where either the smallest prime index is not a divisor of all the other prime indices, or the greatest prime index is not divisible by all the other prime indices.
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.
Also Heinz numbers of partitions that either empty, have smallest part not dividing all the others, or have greatest part not divisible by all the others (counted by A343346). The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
      1: {}            90: {1,2,2,3}      141: {2,15}
     15: {2,3}         91: {4,6}          143: {5,6}
     30: {1,2,3}       93: {2,11}         145: {3,10}
     33: {2,5}         95: {3,8}          150: {1,2,3,3}
     35: {3,4}         99: {2,2,5}        153: {2,2,7}
     45: {2,2,3}      102: {1,2,7}        154: {1,4,5}
     51: {2,7}        105: {2,3,4}        155: {3,11}
     55: {3,5}        110: {1,3,5}        161: {4,9}
     60: {1,1,2,3}    119: {4,7}          165: {2,3,5}
     66: {1,2,5}      120: {1,1,1,2,3}    170: {1,3,7}
     69: {2,9}        123: {2,13}         175: {3,3,4}
     70: {1,3,4}      132: {1,1,2,5}      177: {2,17}
     75: {2,3,3}      135: {2,2,2,3}      180: {1,1,2,2,3}
     77: {4,5}        138: {1,2,9}        182: {1,4,6}
     85: {3,7}        140: {1,1,3,4}      186: {1,2,11}
For example, the prime indices of 90 are {1,2,2,3}, and, because 1 divides all the other parts, 90 is in the sequence, even though 3 is not divisible by all the other parts.

The partitions without these Heinz numbers are counted by A130714.
The first condition alone gives A342193.
The second condition alone gives A343337.
The "and" instead of "or" version is A343338.
The partitions with these Heinz numbers are counted by A343346.
A000005 counts divisors.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A056239 adds up prime indices, row sums of A112798.
A067824 counts strict chains of divisors starting with n.
A253249 counts strict chains of divisors.
A339564 counts factorizations with a selected factor.
Cf. A083710, A257993, A338470, A339562, A341450, A343339, A343340, A343341, A343342, A343378, A343379, A343382.

Select[Range[100],#==1||With[{p=PrimePi/@First/@FactorInteger[#]},!And@@IntegerQ/@(Max@@p/p)||!And@@IntegerQ/@(p/Min@@p)]&]

Equals the union of A342193 and A343337.

A343655 Number of pairwise coprime sets of divisors of n, where a singleton is not considered pairwise coprime unless it is {1}.

Original entry on oeis.org

1, 2, 2, 3, 2, 6, 2, 4, 3, 6, 2, 10, 2, 6, 6, 5, 2, 10, 2, 10, 6, 6, 2, 14, 3, 6, 4, 10, 2, 22, 2, 6, 6, 6, 6, 17, 2, 6, 6, 14, 2, 22, 2, 10, 10, 6, 2, 18, 3, 10, 6, 10, 2, 14, 6, 14, 6, 6, 2, 38, 2, 6, 10, 7, 6, 22, 2, 10, 6, 22, 2, 24, 2, 6, 10, 10, 6, 22, 2

Offset: 1

Gus Wiseman, Apr 26 2021

nonn

First differs from A015995 at a(210) = 88, A015995(210) = 86.

			For example, the a(n) subsets for n = 1, 2, 4, 6, 8, 12, 16, 24 are:
  {1}  {1}    {1}    {1}      {1}    {1}      {1}     {1}
       {1,2}  {1,2}  {1,2}    {1,2}  {1,2}    {1,2}   {1,2}
              {1,4}  {1,3}    {1,4}  {1,3}    {1,4}   {1,3}
                     {1,6}    {1,8}  {1,4}    {1,8}   {1,4}
                     {2,3}           {1,6}    {1,16}  {1,6}
                     {1,2,3}         {2,3}            {1,8}
                                     {3,4}            {2,3}
                                     {1,12}           {3,4}
                                     {1,2,3}          {3,8}
                                     {1,3,4}          {1,12}
                                                      {1,24}
                                                      {1,2,3}
                                                      {1,3,4}
                                                      {1,3,8}

The case of pairs is A063647.
The case of triples is A066620.
The version with empty sets and singletons is A225520.
A version for prime indices is A304711.
The version for strict integer partitions is A305713.
The version for subsets of {1..n} is A320426 = A276187 + 1.
The version for binary indices is A326675.
The version for integer partitions is A327516.
The version for standard compositions is A333227.
The maximal case is A343652.
The case without 1's is A343653.
The case without 1's with singletons is A343654.
The maximal case without 1's is A343660.
A018892 counts coprime unordered pairs of divisors.
A051026 counts pairwise indivisible subsets of {1..n}.
A100565 counts pairwise coprime unordered triples of divisors.
A325683 counts maximal Golomb rulers.
A326077 counts maximal pairwise indivisible sets.
Cf. A007360, A062319, A067824, A076078, A084422, A187106, A282935, A285572, A304709, A320423, A337485, A343659.

Table[Length[Select[Subsets[Divisors[n]],CoprimeQ@@#&]],{n,100}]

A336942 Number of strict chains of divisors in A130091 (numbers with distinct prime multiplicities) starting with the superprimorial A006939(n) and ending with 1.

Original entry on oeis.org

1, 1, 5, 95, 8823, 4952323, 20285515801, 714092378624317

Offset: 0

Gus Wiseman, Aug 14 2020

			The a(0) = 1 through a(2) = 5 chains:
  {1}  {2,1}  {12,1}
              {12,2,1}
              {12,3,1}
              {12,4,1}
              {12,4,2,1}

A076954 can be used instead of A006939 (cf. A307895, A325337).
A336423 and A336571 are not restricted to A006939.
A336941 is the version not restricted by A130091.
A337075 is the version for factorials.
A074206 counts chains of divisors from n to 1.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts chains of divisors.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336422 counts divisible pairs of divisors, both in A130091.
A336424 counts factorizations using A130091.
Cf. A000005, A001055, A002033, A032741, A067824, A124010, A167865, A336419, A336420, A336500, A336568.

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
chnstr[n_]:=If[n==1,1,Sum[chnstr[d],{d,Select[Most[Divisors[n]],UnsameQ@@Last/@FactorInteger[#]&]}]];
Table[chnstr[chern[n]],{n,0,3}]

a(n) = A336423(A006939(n)) = A336571(A006939(n)).

A343340 Numbers with a prime index dividing all the other prime indices, but with no prime index divisible by all the other prime indices.

Original entry on oeis.org

30, 60, 66, 70, 90, 102, 110, 120, 132, 138, 140, 150, 154, 170, 180, 182, 186, 190, 198, 204, 210, 220, 238, 240, 246, 264, 270, 273, 276, 280, 282, 286, 290, 300, 306, 308, 310, 322, 330, 340, 350, 354, 360, 364, 372, 374, 380, 396, 402, 406, 408, 410, 414

Offset: 1

Gus Wiseman, Apr 15 2021

nonn

Alternative name: Numbers > 1 whose smallest prime index divides all the other prime indices, but whose greatest prime index is not divisible by all the other prime indices.
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.
Also Heinz numbers of partitions with greatest part not divisible by all the others, but smallest part dividing all the others (counted by A343345). The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
     30: {1,2,3}        182: {1,4,6}          282: {1,2,15}
     60: {1,1,2,3}      186: {1,2,11}         286: {1,5,6}
     66: {1,2,5}        190: {1,3,8}          290: {1,3,10}
     70: {1,3,4}        198: {1,2,2,5}        300: {1,1,2,3,3}
     90: {1,2,2,3}      204: {1,1,2,7}        306: {1,2,2,7}
    102: {1,2,7}        210: {1,2,3,4}        308: {1,1,4,5}
    110: {1,3,5}        220: {1,1,3,5}        310: {1,3,11}
    120: {1,1,1,2,3}    238: {1,4,7}          322: {1,4,9}
    132: {1,1,2,5}      240: {1,1,1,1,2,3}    330: {1,2,3,5}
    138: {1,2,9}        246: {1,2,13}         340: {1,1,3,7}
    140: {1,1,3,4}      264: {1,1,1,2,5}      350: {1,3,3,4}
    150: {1,2,3,3}      270: {1,2,2,2,3}      354: {1,2,17}
    154: {1,4,5}        273: {2,4,6}          360: {1,1,1,2,2,3}
    170: {1,3,7}        276: {1,1,2,9}        364: {1,1,4,6}
    180: {1,1,2,2,3}    280: {1,1,1,3,4}      372: {1,1,2,11}

The first condition alone gives the complement of A342193.
The second condition alone gives A343337.
The partitions with these Heinz numbers are counted by A343345.
A000005 counts divisors.
A000070 counts partitions with a selected part.
A001055 counts factorizations.
A056239 adds up prime indices, row sums of A112798.
A067824 counts strict chains of divisors starting with n.
A253249 counts strict chains of divisors.
Cf. A083710, A083711, A097986, A098965, A130714, A339562, A339563, A343341, A343377, A343381.

Select[Range[2,100],With[{p=PrimePi/@First/@FactorInteger[#]},!And@@IntegerQ/@(Max@@p/p)&&And@@IntegerQ/@(p/Min@@p)]&]

Complement of A342193 in A343337.

A343659 Number of maximal pairwise coprime subsets of {1..n}.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 7, 9, 9, 10, 10, 12, 16, 19, 19, 20, 20, 22, 28, 32, 32, 33, 54, 61, 77, 84, 84, 85, 85, 94, 112, 123, 158, 161, 161, 176, 206, 212, 212, 214, 214, 229, 241, 260, 260, 263, 417, 428, 490, 521, 521, 526, 655, 674, 764, 818, 818, 820, 820, 874, 918, 975, 1182, 1189, 1189

Offset: 1

Gus Wiseman, Apr 26 2021

nonn

For this sequence, it does not matter whether singletons are considered pairwise coprime.
For n > 2, also the number of maximal pairwise coprime subsets of {2..n}.
For each prime p <= n, p divides exactly one element of each maximal subset. - Bert Dobbelaere, May 04 2021

			The a(1) = 1 through a(9) = 7 subsets:
  {1}  {12}  {123}  {123}  {1235}  {156}   {1567}   {1567}   {1567}
                    {134}  {1345}  {1235}  {12357}  {12357}  {12357}
                                   {1345}  {13457}  {13457}  {12579}
                                                    {13578}  {13457}
                                                             {13578}
                                                             {14579}
                                                             {15789}

Bert Dobbelaere, Table of n, a(n) for n = 1..500
Bert Dobbelaere, Python program

The case of pairs is A015614.
The case of triples is A015617.
The non-maximal version counting empty sets and singletons is A084422.
The non-maximal version counting singletons is A187106.
The non-maximal version is A320426(n) = A276187(n) + 1.
The version for indivisibility instead of coprimality is A326077.
The version for sets of divisors is A343652.
The version for sets of divisors > 1 is A343660.
A018892 counts coprime unordered pairs of divisors.
A051026 counts pairwise indivisible subsets of {1..n}.
A100565 counts pairwise coprime unordered triples of divisors.
Cf. A007360, A067824, A087087, A225520, A324837, A325683, A325859, A326358, A326496, A326675, A333227, A343653, A343655.

fasmax[y_]:=Complement[y,Union@@Most@*Subsets/@y];
Table[Length[fasmax[Select[Subsets[Range[n]],CoprimeQ@@#&]]],{n,15}]

More terms from Bert Dobbelaere, May 04 2021

A343662 Irregular triangle read by rows where T(n,k) is the number of strict length k chains of divisors of n, 0 <= k <= Omega(n) + 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 4, 5, 2, 1, 2, 1, 1, 4, 6, 4, 1, 1, 3, 3, 1, 1, 4, 5, 2, 1, 2, 1, 1, 6, 12, 10, 3, 1, 2, 1, 1, 4, 5, 2, 1, 4, 5, 2, 1, 5, 10, 10, 5, 1, 1, 2, 1, 1, 6, 12, 10, 3, 1, 2, 1, 1, 6, 12, 10, 3, 1, 4, 5, 2, 1, 4, 5, 2

Offset: 1

Gus Wiseman, May 01 2021

			Triangle begins:
   1:  1  1
   2:  1  2  1
   3:  1  2  1
   4:  1  3  3  1
   5:  1  2  1
   6:  1  4  5  2
   7:  1  2  1
   8:  1  4  6  4  1
   9:  1  3  3  1
  10:  1  4  5  2
  11:  1  2  1
  12:  1  6 12 10  3
  13:  1  2  1
  14:  1  4  5  2
  15:  1  4  5  2
  16:  1  5 10 10  5  1
For example, row n = 12 counts the following chains:
  ()  (1)   (2/1)   (4/2/1)   (12/4/2/1)
      (2)   (3/1)   (6/2/1)   (12/6/2/1)
      (3)   (4/1)   (6/3/1)   (12/6/3/1)
      (4)   (4/2)   (12/2/1)
      (6)   (6/1)   (12/3/1)
      (12)  (6/2)   (12/4/1)
            (6/3)   (12/4/2)
            (12/1)  (12/6/1)
            (12/2)  (12/6/2)
            (12/3)  (12/6/3)
            (12/4)
            (12/6)

Column k = 1 is A000005.
Row ends are A008480.
Row lengths are A073093.
Column k = 2 is A238952.
The case from n to 1 is A334996 or A251683 (row sums: A074206).
A non-strict version is A334997 (transpose: A077592).
The case starting with n is A337255 (row sums: A067824).
Row sums are A337256 (nonempty: A253249).
A001055 counts factorizations.
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A097805 counts compositions by sum and length.
A122651 counts strict chains of divisors summing to n.
A146291 counts divisors of n with k prime factors (with multiplicity).
A163767 counts length n - 1 chains of divisors of n.
A167865 counts strict chains of divisors > 1 summing to n.
A337070 counts strict chains of divisors starting with superprimorials.
Cf. A002033, A007425, A007426, A051026, A062319, A143773, A186972, A327527, A337074, A337105, A337107, A343658.

Table[Length[Select[Reverse/@Subsets[Divisors[n],{k}],And@@Divisible@@@Partition[#,2,1]&]],{n,15},{k,0,PrimeOmega[n]+1}]

A378647 Dirichlet convolution of A074206 and A103977, where A074206 is the number of ordered factorizations of n, and A103977 is the Zumkeller deficiency of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 13, 14, 15, 16, 17, 22, 19, 22, 21, 22, 23, 40, 25, 26, 27, 28, 29, 42, 31, 32, 33, 34, 35, 64, 37, 38, 39, 52, 41, 54, 43, 44, 45, 46, 47, 96, 49, 50, 51, 52, 53, 70, 55, 64, 57, 58, 59, 126, 61, 62, 63, 64, 65, 78, 67, 68, 69, 74, 71, 176, 73, 74, 75, 76, 77, 90, 79, 120, 81

Offset: 1

Antti Karttunen, Dec 03 2024

nonn

Möbius transform of A378648, which is the Dirichlet convolution of A067824 and A103977.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000027, A008683, A067824, A074206, A103977, A263837, A378648 (inverse Möbius transform), A378649 (Möbius transform), A378650 [= a(n)-n], A378655 (Dirichlet inverse).

A378647(n) = sumdiv(n,d,moebius(d)*A378648(n/d));

A378647(n) = sumdiv(n,d,A074206(d)*A103977(n/d));

a(n) = Sum_{d|n} A074206(d)*A103977(n/d).
a(n) = Sum_{d|n} A008683(d)*A378648(n/d).
a(n) = Sum_{d|n} A067824(d)*A378644(n/d).
a(n) = A378650(n)+n, with a(n) = n if and only if n is a non-abundant number (A263837).

cp's OEIS Frontend

A337074 Number of strict chains of divisors in A130091 (numbers with distinct prime multiplicities), starting with n!.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A337107 Irregular triangle read by rows where T(n,k) is the number of strict length-k chains of divisors from n! to 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

A342522 Heinz numbers of integer partitions with constant (equal) first quotients.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A343343 Numbers with either no prime index dividing, or no prime index divisible by all the other prime indices.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A343655 Number of pairwise coprime sets of divisors of n, where a singleton is not considered pairwise coprime unless it is {1}.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A336942 Number of strict chains of divisors in A130091 (numbers with distinct prime multiplicities) starting with the superprimorial A006939(n) and ending with 1.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A343340 Numbers with a prime index dividing all the other prime indices, but with no prime index divisible by all the other prime indices.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A343659 Number of maximal pairwise coprime subsets of {1..n}.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions