A336422 -id:A336422 - cp's OEIS frontend

A336866 Number of integer partitions of n without all distinct multiplicities.

0, 0, 0, 1, 1, 2, 4, 5, 9, 15, 21, 28, 46, 56, 80, 114, 149, 192, 269, 337, 455, 584, 751, 943, 1234, 1527, 1944, 2422, 3042, 3739, 4699, 5722, 7100, 8668, 10634, 12880, 15790, 19012, 23093, 27776, 33528, 40102, 48264, 57469, 68793, 81727, 97372, 115227

Offset: 0

Gus Wiseman, Aug 09 2020

nonn

			The a(0) = 0 through a(9) = 15 partitions (empty columns shown as dots):
  .  .  .  (21)  (31)  (32)  (42)    (43)    (53)     (54)
                       (41)  (51)    (52)    (62)     (63)
                             (321)   (61)    (71)     (72)
                             (2211)  (421)   (431)    (81)
                                     (3211)  (521)    (432)
                                             (3221)   (531)
                                             (3311)   (621)
                                             (4211)   (3321)
                                             (32111)  (4221)
                                                      (4311)
                                                      (5211)
                                                      (32211)
                                                      (42111)
                                                      (222111)
                                                      (321111)

A098859 counts the complement.
A130092 gives the Heinz numbers of these partitions.
A001222 counts prime factors with multiplicity.
A013929 lists nonsquarefree numbers.
A047966 counts uniform partitions.
A047967 counts non-strict partitions.
A071625 counts distinct prime multiplicities.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A327498 gives the maximum divisor with distinct prime multiplicities.
Cf. A000009, A000041, A008284, A116608, A118914, A124010, A242882, A255231, A325280, A325242, A329739, A336422, A336571.

Table[Length[Select[IntegerPartitions[n],!UnsameQ@@Length/@Split[#]&]],{n,0,30}]

a(n) = A000041(n) - A098859(n).

A336500 Number of divisors d|n with distinct prime multiplicities such that the quotient n/d also has distinct prime multiplicities.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 2, 4, 3, 2, 2, 4, 2, 2, 2, 5, 2, 4, 2, 4, 2, 2, 2, 6, 3, 2, 4, 4, 2, 0, 2, 6, 2, 2, 2, 6, 2, 2, 2, 6, 2, 0, 2, 4, 4, 2, 2, 8, 3, 4, 2, 4, 2, 6, 2, 6, 2, 2, 2, 4, 2, 2, 4, 7, 2, 0, 2, 4, 2, 0, 2, 8, 2, 2, 4, 4, 2, 0, 2, 8, 5, 2, 2, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Aug 06 2020

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

			The a(1) = 1 through a(16) = 5 divisors:
  1  1  1  1  1  2  1  1  1  2  1  1  1  2  3  1
     2  3  2  5  3  7  2  3  5 11  3 13  7  5  2
           4           4  9        4           4
                       8          12           8
                                              16

A336419 is the version for superprimorials.
A336568 gives positions of zeros.
A336869 is the restriction to factorials.
A007425 counts divisors of divisors.
A056924 counts divisors greater than their quotient.
A074206 counts chains of divisors from n to 1.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime multiplicities.
A336424 counts factorizations using A130091.
A336422 counts divisible pairs of divisors, both in A130091.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336423 counts chains in A130091, with maximal version A336569.
A336568 gives numbers not a product of two elements of A130091.
A336571 counts divisor sets using A130091, with maximal version A336570.
Cf. A000005, A001055, A002033, A098859, A124010, A167865, A253249, A336870.

Table[Length[Select[Divisors[n],UnsameQ@@Last/@FactorInteger[#]&&UnsameQ@@Last/@FactorInteger[n/#]&]],{n,25}]

A336568 Numbers that are not a product of two numbers each having distinct prime multiplicities.

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 210, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 330, 345, 354, 357, 366, 370, 374, 385, 390, 399, 402, 406, 410, 418, 420, 426, 429

Offset: 1

Gus Wiseman, Aug 06 2020

nonn

First differs from A007304 and A093599 in having 210.
First differs from A287483 in having 222.
First differs from A350352 in having 420.
A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

			Selected terms together with their prime indices:
   660: {1,1,2,3,5}
   798: {1,2,4,8}
   840: {1,1,1,2,3,4}
  3120: {1,1,1,1,2,3,6}
  9900: {1,1,2,2,3,3,5}

A336500 has zeros at these positions.
A007425 counts divisors of divisors.
A056924 counts divisors greater than their quotient.
A074206 counts strict chains of divisors from n to 1.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A336424 counts factorizations using A130091.
A336422 counts divisible pairs of divisors, both in A130091.
A327498 is the maximum divisor with distinct prime multiplicities.
A336423 counts chains in A130091, with maximal version A336569.
A336571 counts divisor sets using A130091, with maximal version A336570.
Cf. A001055, A002033, A098859, A124010, A167865, A253249, A336420.

strsig[n_]:=UnsameQ@@Last/@FactorInteger[n]
Select[Range[100],Function[n,Select[Divisors[n],strsig[#]&&strsig[n/#]&]=={}]]

A336571 Number of sets of divisors d|n, 1 < d < n, all belonging to A130091 (numbers with distinct prime multiplicities) and forming a divisibility chain.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 5, 1, 3, 3, 8, 1, 5, 1, 5, 3, 3, 1, 14, 2, 3, 4, 5, 1, 4, 1, 16, 3, 3, 3, 17, 1, 3, 3, 14, 1, 4, 1, 5, 5, 3, 1, 36, 2, 5, 3, 5, 1, 14, 3, 14, 3, 3, 1, 16, 1, 3, 5, 32, 3, 4, 1, 5, 3, 4, 1, 35, 1, 3, 5, 5, 3, 4, 1, 36, 8, 3, 1

Offset: 1

Gus Wiseman, Jul 29 2020

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

			The a(n) sets for n = 4, 6, 12, 16, 24, 84, 36:
  {}   {}   {}     {}       {}        {}        {}
  {2}  {2}  {2}    {2}      {2}       {2}       {2}
       {3}  {3}    {4}      {3}       {3}       {3}
            {4}    {8}      {4}       {4}       {4}
            {2,4}  {2,4}    {8}       {7}       {9}
                   {2,8}    {12}      {12}      {12}
                   {4,8}    {2,4}     {28}      {18}
                   {2,4,8}  {2,8}     {2,4}     {2,4}
                            {4,8}     {2,12}    {3,9}
                            {2,12}    {2,28}    {2,12}
                            {3,12}    {3,12}    {2,18}
                            {4,12}    {4,12}    {3,12}
                            {2,4,8}   {4,28}    {3,18}
                            {2,4,12}  {7,28}    {4,12}
                                      {2,4,12}  {9,18}
                                      {2,4,28}  {2,4,12}
                                                {3,9,18}

A336423 is the version for chains containing n.
A336570 is the maximal version.
A000005 counts divisors.
A001055 counts factorizations.
A007425 counts divisors of divisors.
A032741 counts proper divisors.
A045778 counts strict factorizations.
A071625 counts distinct prime multiplicities.
A074206 counts strict 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.
A336422 counts divisible pairs of divisors, both in A130091.
A336424 counts factorizations using A130091.
A336500 counts divisors of n in A130091 with quotient also in A130091.
Cf. A001222, A002033, A005117, A098859, A118914, A124010, A294068, A305149, A327498, A327523, A336568, A336569.

strchns[n_]:=If[n==1,1,Sum[strchns[d],{d,Select[Most[Divisors[n]],UnsameQ@@Last/@FactorInteger[#]&]}]];
Table[strchns[n],{n,100}]

A336423 Number of strict chains of divisors from n to 1 using terms of A130091 (numbers with distinct prime multiplicities).

Original entry on oeis.org

1, 1, 1, 2, 1, 0, 1, 4, 2, 0, 1, 5, 1, 0, 0, 8, 1, 5, 1, 5, 0, 0, 1, 14, 2, 0, 4, 5, 1, 0, 1, 16, 0, 0, 0, 0, 1, 0, 0, 14, 1, 0, 1, 5, 5, 0, 1, 36, 2, 5, 0, 5, 1, 14, 0, 14, 0, 0, 1, 0, 1, 0, 5, 32, 0, 0, 1, 5, 0, 0, 1, 35, 1, 0, 5, 5, 0, 0, 1, 36, 8, 0, 1, 0

Offset: 1

Gus Wiseman, Jul 27 2020

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

			The a(n) chains for n = 4, 8, 12, 16, 24, 32:
  4/1    8/1      12/1      16/1        24/1         32/1
  4/2/1  8/2/1    12/2/1    16/2/1      24/2/1       32/2/1
         8/4/1    12/3/1    16/4/1      24/3/1       32/4/1
         8/4/2/1  12/4/1    16/8/1      24/4/1       32/8/1
                  12/4/2/1  16/4/2/1    24/8/1       32/16/1
                            16/8/2/1    24/12/1      32/4/2/1
                            16/8/4/1    24/4/2/1     32/8/2/1
                            16/8/4/2/1  24/8/2/1     32/8/4/1
                                        24/8/4/1     32/16/2/1
                                        24/12/2/1    32/16/4/1
                                        24/12/3/1    32/16/8/1
                                        24/12/4/1    32/8/4/2/1
                                        24/8/4/2/1   32/16/4/2/1
                                        24/12/4/2/1  32/16/8/2/1
                                                     32/16/8/4/1
                                                     32/16/8/4/2/1

A336569 is the maximal case.
A336571 does not require n itself to have distinct prime multiplicities.
A000005 counts divisors.
A007425 counts divisors of divisors.
A074206 counts strict chains of divisors from n to 1.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts nonempty strict 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.
A336500 counts divisors of n in A130091 with quotient also in A130091.
A337256 counts strict chains of divisors.
Cf. A001055, A002033, A005117, A032741, A067824, A071625, A118914, A124010, A167865, A336568, A336570, A336941.

strchns[n_]:=If[n==1,1,If[!UnsameQ@@Last/@FactorInteger[n],0,Sum[strchns[d],{d,Select[Most[Divisors[n]],UnsameQ@@Last/@FactorInteger[#]&]}]]];
Table[strchns[n],{n,100}]

A336569 Number of maximal strict chains of divisors from n to 1 using elements of A130091 (numbers with distinct prime multiplicities).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 2, 1, 2, 0, 0, 1, 3, 1, 0, 1, 2, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 3, 1, 0, 1, 2, 2, 0, 1, 4, 1, 2, 0, 2, 1, 3, 0, 3, 0, 0, 1, 0, 1, 0, 2, 1, 0, 0, 1, 2, 0, 0, 1, 5, 1, 0, 2, 2, 0, 0, 1, 4, 1, 0, 1, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jul 29 2020

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

			The a(n) chains for n = 12, 72, 144, 192 (ones not shown):
  12/3    72/18/2       144/72/18/2       192/96/48/24/12/3
  12/4/2  72/18/9/3     144/72/18/9/3     192/64/32/16/8/4/2
          72/24/12/3    144/48/24/12/3    192/96/32/16/8/4/2
          72/24/8/4/2   144/72/24/12/3    192/96/48/16/8/4/2
          72/24/12/4/2  144/48/16/8/4/2   192/96/48/24/8/4/2
                        144/48/24/8/4/2   192/96/48/24/12/4/2
                        144/72/24/8/4/2
                        144/48/24/12/4/2
                        144/72/24/12/4/2

A336423 is the non-maximal version.
A336570 is the version for chains not necessarily containing n.
A000005 counts divisors.
A001055 counts factorizations.
A001222 counts prime factors with multiplicity.
A007425 counts divisors of divisors.
A032741 counts proper divisors.
A045778 counts strict factorizations.
A071625 counts distinct prime multiplicities.
A074206 counts strict 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.
A336422 counts divisible pairs of divisors, both in A130091.
A336424 counts factorizations using A130091.
A336571 counts divisor sets of elements of A130091.
Cf. A002033, A005117, A098859, A118914, A124010, A305149, A327498, A327523, A336414, A336425, A336500, A336568.

strsigQ[n_]:=UnsameQ@@Last/@FactorInteger[n];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
strchs[n_]:=If[n==1,{{}},If[!strsigQ[n],{},Join@@Table[Prepend[#,d]&/@strchs[d],{d,Select[Most[Divisors[n]],strsigQ]}]]];
Table[Length[fasmax[strchs[n]]],{n,100}]

A336421 Number of ways to choose a divisor of a divisor, both having distinct prime exponents, of the n-th superprimorial number A006939(n).

Original entry on oeis.org

1, 3, 13, 76, 571, 5309, 59341, 780149

Offset: 0

Gus Wiseman, Jul 25 2020

A number has distinct prime exponents iff its prime signature is strict.

The n-th superprimorial or Chernoff number is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).

			The a(2) = 13 ways:
  12/1/1  12/2/1  12/3/1  12/4/1  12/12/1
          12/2/2  12/3/3  12/4/2  12/12/2
                          12/4/4  12/12/3
                                  12/12/4
                                  12/12/12

A000258 shifted once to the left is dominated by this sequence.
A336422 is the generalization to non-superprimorials.
A000110 counts divisors of superprimorials with distinct prime exponents.
A006939 lists superprimorials or Chernoff numbers.
A008302 counts divisors of superprimorials by bigomega.
A022915 counts permutations of prime indices of superprimorials.
A076954 can be used instead of A006939.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime exponents.
A181818 gives products of superprimorials.
A317829 counts factorizations of superprimorials.
Cf. A000005, A008278, A027423, A071625, A118914, A124010, A327498, A336417, A336419, A336420, A336426, A336500, A336568.

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
strsig[n_]:=UnsameQ@@Last/@FactorInteger[n];
Table[Total[Cases[Divisors[chern[n]],d_?strsig:>Count[Divisors[d],e_?strsig]]],{n,0,5}]

A336570 Number of maximal sets of proper divisors d|n, d < n, all belonging to A130091 (numbers with distinct prime multiplicities) and forming a divisibility chain.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 4, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 5, 1, 2, 2, 2, 2, 3, 1, 4, 1, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Jul 29 2020

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

			The a(n) sets for n = 36, 120, 144, 180 (ones not shown):
  {2,18}    {3,12,24}    {2,18,72}       {2,18}
  {3,12}    {5,20,40}    {3,9,18,72}     {3,12}
  {2,4,12}  {2,4,8,24}   {3,12,24,48}    {5,20}
  {3,9,18}  {2,4,8,40}   {3,12,24,72}    {5,45}
            {2,4,12,24}  {2,4,8,16,48}   {2,4,12}
            {2,4,20,40}  {2,4,8,24,48}   {2,4,20}
                         {2,4,8,24,72}   {3,9,18}
                         {2,4,12,24,48}  {3,9,45}
                         {2,4,12,24,72}

A336569 is the version for chains containing n.
A336571 is the non-maximal version.
A000005 counts divisors.
A001055 counts factorizations.
A007425 counts divisors of divisors.
A032741 counts proper divisors.
A045778 counts strict factorizations.
A071625 counts distinct prime multiplicities.
A074206 counts strict 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.
A336422 counts divisible pairs of divisors, both in A130091.
A336424 counts factorizations using A130091.
A336500 counts divisors of n in A130091 with quotient also in A130091.
Cf. A002033, A005117, A098859, A118914, A124010, A305149, A327498, A327523, A336414, A336423, A336425, A336568.

strsigQ[n_]:=UnsameQ@@Last/@FactorInteger[n];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
strses[n_]:=If[n==1,{{}},Join@@Table[Append[#,d]&/@strses[d],{d,Select[Most[Divisors[n]],strsigQ]}]];
Table[Length[fasmax[strses[n]]],{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)).

A336425 Number of ways to choose a divisor with distinct prime exponents of a divisor with distinct prime exponents of n!.

Original entry on oeis.org

1, 1, 3, 5, 24, 38, 132, 195, 570, 1588, 4193, 6086, 14561, 19232, 37142, 106479, 207291, 266871, 549726, 674330, 1465399, 3086598, 5939574, 7182133, 12324512, 28968994, 46819193, 82873443, 165205159, 196666406, 350397910, 406894074, 593725529, 1229814478, 1853300600, 4024414209, 6049714096, 6968090487, 9700557121, 16810076542, 26339337285

Offset: 0

Gus Wiseman, Aug 06 2020

nonn

			The a(4) = 24 divisors of divisors:
  1/1  2/1  3/1  4/1  8/1  12/1   24/1
       2/2  3/3  4/2  8/2  12/2   24/2
                 4/4  8/4  12/3   24/3
                      8/8  12/4   24/4
                           12/12  24/8
                                  24/12
                                  24/24

Max Alekseyev, Table of n, a(n) for n = 0..85

A336422 is the non-factorial generalization.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime exponents.
A327526 gives the maximum divisor of n with equal prime exponents.
A327498 gives the maximum divisor of n with distinct prime exponents.
A336414 counts divisors of n! with distinct prime exponents.
A336415 counts divisors of n! with equal prime exponents.
A336423 counts chains in A130091, with maximal version A336569.
Cf. A000005, A000110, A098859, A118914, A124010, A336424, A336500, A336568, A336570, A336571, A336865, A336866, A336869.
Factorial numbers: A000142, A022559, A027423, A048656, A048742, A071626, A325272, A325273, A325617, A327499, A336416, A336418, A336617.

strsigQ[n_]:=UnsameQ@@Last/@FactorInteger[n];
Table[Total[Cases[Divisors[n!],d_?strsigQ:>Count[Divisors[d],e_?strsigQ]]],{n,0,20}]

Terms a(21) onward from Max Alekseyev, Nov 07 2024

cp's OEIS Frontend

A336866 Number of integer partitions of n without all distinct multiplicities.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A336500 Number of divisors d|n with distinct prime multiplicities such that the quotient n/d also has distinct prime multiplicities.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A336568 Numbers that are not a product of two numbers each having distinct prime multiplicities.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A336571 Number of sets of divisors d|n, 1 < d < n, all belonging to A130091 (numbers with distinct prime multiplicities) and forming a divisibility chain.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A336423 Number of strict chains of divisors from n to 1 using terms of A130091 (numbers with distinct prime multiplicities).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A336569 Number of maximal strict chains of divisors from n to 1 using elements of A130091 (numbers with distinct prime multiplicities).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A336421 Number of ways to choose a divisor of a divisor, both having distinct prime exponents, of the n-th superprimorial number A006939(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A336570 Number of maximal sets of proper divisors d|n, d < n, all belonging to A130091 (numbers with distinct prime multiplicities) and forming a divisibility chain.

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