A325103 -id:A325103 - cp's OEIS frontend

A325123 Number of divisible pairs of positive integers up to n with no binary carries.

0, 0, 1, 1, 3, 3, 4, 4, 7, 7, 9, 9, 12, 12, 13, 13, 17, 17, 19, 19, 22, 22, 23, 23, 28, 28, 29, 29, 31, 31, 32, 32, 37, 37, 39, 39, 44, 44, 45, 45, 50, 50, 52, 52, 54, 54, 55, 55, 62, 62, 64, 64, 66, 66, 68, 68, 72, 72, 73, 73, 76, 76, 77, 77, 83, 83, 85, 85

Offset: 0

Gus Wiseman, Mar 29 2019

nonn

Two positive integers are divisible if the first divides the second, and they have a binary carry if the positions of 1's in their reversed binary expansion overlap.

a(2k+1) = a(2k), since an odd number and any divisor will overlap in the last digit. Additionally, a(2k+2) > a(2k+1) because the pair {1,2k+2} is always valid. Therefore, every term appears exactly twice. - Charlie Neder, Apr 02 2019

			The a(2) = 1 through a(11) = 9 pairs:
  {1,2}  {1,2}  {1,2}  {1,2}  {1,2}  {1,2}  {1,2}  {1,2}  {1,2}   {1,2}
                {1,4}  {1,4}  {1,4}  {1,4}  {1,4}  {1,4}  {1,4}   {1,4}
                {2,4}  {2,4}  {1,6}  {1,6}  {1,6}  {1,6}  {1,6}   {1,6}
                              {2,4}  {2,4}  {1,8}  {1,8}  {1,8}   {1,8}
                                            {2,4}  {2,4}  {2,4}   {2,4}
                                            {2,8}  {2,8}  {2,8}   {2,8}
                                            {4,8}  {4,8}  {4,8}   {4,8}
                                                          {1,10}  {1,10}
                                                          {5,10}  {5,10}

Cf. A006218, A019565, A050315, A070939, A080572, A247935, A267610.

Cf. A325095, A325096, A325101, A325103, A325104, A325105, A325106, A325124.

Table[Length[Select[Tuples[Range[n],2],Divisible@@Reverse[#]&&Intersection[Position[Reverse[IntegerDigits[#[[1]],2]],1],Position[Reverse[IntegerDigits[#[[2]],2]],1]]=={}&]],{n,0,20}]

A325100 Heinz numbers of strict integer partitions with no binary carries.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 11, 13, 14, 17, 19, 21, 23, 26, 29, 31, 33, 35, 37, 38, 41, 42, 43, 47, 53, 57, 58, 59, 61, 67, 69, 71, 73, 74, 79, 83, 86, 89, 95, 97, 101, 103, 106, 107, 109, 111, 113, 114, 122, 123, 127, 131, 133, 137, 139, 142, 149, 151, 157, 158, 159

Offset: 1

Gus Wiseman, Mar 28 2019

nonn

A binary carry of two positive integers is an overlap of the positions of 1's in their reversed binary expansion.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are squarefree numbers whose prime indices have no carries. 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.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   6: {1,2}
   7: {4}
  11: {5}
  13: {6}
  14: {1,4}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  26: {1,6}
  29: {10}
  31: {11}
  33: {2,5}
  35: {3,4}
  37: {12}
  38: {1,8}
  41: {13}
  42: {1,2,4}

Cf. A050315, A056239, A080572, A112798, A247935, A267610.

Cf. A325095, A325096, A325097, A325100, A325101, A325103, A325110, A325119.

binpos[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[100],SquareFreeQ[#]&&stableQ[PrimePi/@First/@FactorInteger[#],Intersection[binpos[#1],binpos[#2]]!={}&]&]

A325124 Number of divisible pairs of positive integers up to n with at least one binary carry.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 10, 12, 13, 16, 18, 20, 23, 25, 28, 32, 33, 35, 39, 41, 44, 48, 51, 53, 56, 59, 62, 66, 70, 72, 79, 81, 82, 86, 88, 92, 96, 98, 101, 105, 108, 110, 116, 118, 122, 128, 131, 133, 136, 139, 143, 147, 151, 153, 159, 163, 167, 171, 174, 176, 185

Offset: 0

Gus Wiseman, Mar 29 2019

nonn

Two positive integers are divisible if the first divides the second, and they have a binary carry if the positions of 1's in their reversed binary expansion overlap.

			The a(1) = 1 through a(8) = 13 pairs:
  (1,1)  (1,1)  (1,1)  (1,1)  (1,1)  (1,1)  (1,1)  (1,1)
         (2,2)  (1,3)  (1,3)  (1,3)  (1,3)  (1,3)  (1,3)
                (2,2)  (2,2)  (1,5)  (1,5)  (1,5)  (1,5)
                (3,3)  (3,3)  (2,2)  (2,2)  (1,7)  (1,7)
                       (4,4)  (3,3)  (2,6)  (2,2)  (2,2)
                              (4,4)  (3,3)  (2,6)  (2,6)
                              (5,5)  (3,6)  (3,3)  (3,3)
                                     (4,4)  (3,6)  (3,6)
                                     (5,5)  (4,4)  (4,4)
                                     (6,6)  (5,5)  (5,5)
                                            (6,6)  (6,6)
                                            (7,7)  (7,7)
                                                   (8,8)

Cf. A006218, A019565, A050315, A070939, A080572, A247935, A267610, A307230.

Cf. A325095, A325096, A325101, A325103, A325104, A325105, A325106, A325123.

Table[Length[Select[Tuples[Range[n],2],Divisible@@Reverse[#]&&Intersection[Position[Reverse[IntegerDigits[#[[1]],2]],1],Position[Reverse[IntegerDigits[#[[2]],2]],1]]!={}&]],{n,0,20}]

a(n) = A307230(n) + n.

A325120 Sum of binary lengths of the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 29 2019

nonn

The binary length of n is the number of digits in its binary representation. 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.

Cf. A000120, A000720, A001222, A019565, A048881, A070939, A112798.
Cf. A325103, A325104, A325106, A325118, A325119.
Other totally additive sequences: A056239, A302242, A318994, A318995, A325033, A325034, A325121, A325122.

Table[Sum[pr[[2]]*IntegerLength[PrimePi[pr[[1]]],2],{pr,FactorInteger[n]}],{n,100}]

Totally additive with a(prime(n)) = A070939(n).

A325121 Sum of binary digits of the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 29 2019

nonn

The sum of binary digits of an integer is the number of 1's in its binary representation. 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.

Cf. A000120, A000720, A001222, A019565, A048881, A070939, A112798.
Cf. A325103, A325104, A325106, A325118, A325119.
Other totally additive sequences: A056239, A302242, A318994, A318995, A325033, A325034, A325120, A325122.

Table[Sum[pr[[2]]*DigitCount[PrimePi[pr[[1]]],2,1],{pr,FactorInteger[n]}],{n,100}]

Totally additive with a(prime(n)) = A000120(n).

A325122 Sum of binary digits of the prime indices of n, minus Omega(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0, 1, 1, 2, 0, 1, 2, 1, 0, 1, 0, 1, 1, 2, 0, 2, 1, 1, 1, 3, 0, 0, 2, 2, 1, 0, 0, 2, 0, 0, 1, 1, 1, 1, 2, 0, 0, 2, 1, 2, 2, 1, 1, 1, 0, 2, 1, 2, 0, 1, 1, 2, 1, 0, 2, 3, 0, 3, 2, 1

Offset: 1

Gus Wiseman, Mar 29 2019

nonn

The sum of binary digits of an integer is the number of 1's in its binary representation. 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.

Positions of zeros are A318400.
Cf. A000120, A018900, A019565, A048881, A051438, A070939, A112798.
Cf. A325103, A325104, A325106, A325118.
Other totally additive sequences: A056239, A302242, A318994, A318995, A325033, A325034, A325120, A325121.

Table[Sum[pr[[2]]*(DigitCount[PrimePi[pr[[1]]],2,1]-1),{pr,If[n==1,{},FactorInteger[n]]}],{n,100}]

Totally additive with a(prime(n)) = A048881(n).

A307230 Number of divisible pairs of distinct positive integers up to n with at least one binary carry.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 5, 5, 7, 8, 9, 11, 12, 14, 17, 17, 18, 21, 22, 24, 27, 29, 30, 32, 34, 36, 39, 42, 43, 49, 50, 50, 53, 54, 57, 60, 61, 63, 66, 68, 69, 74, 75, 78, 83, 85, 86, 88, 90, 93, 96, 99, 100, 105, 108, 111, 114, 116, 117, 125, 126, 128, 133, 133

Offset: 0

Gus Wiseman, Mar 29 2019

nonn

Two positive integers are divisible if the first divides the second, and have a binary carry if the positions of 1's in their reversed binary expansion overlap.

			The a(3) = 1 through a(12) = 11 pairs:
  {1,3}  {1,3}  {1,3}  {1,3}  {1,3}  {1,3}  {1,3}  {1,3}   {1,3}   {1,3}
                {1,5}  {1,5}  {1,5}  {1,5}  {1,5}  {1,5}   {1,5}   {1,5}
                       {2,6}  {1,7}  {1,7}  {1,7}  {1,7}   {1,7}   {1,7}
                       {3,6}  {2,6}  {2,6}  {1,9}  {1,9}   {1,9}   {1,9}
                              {3,6}  {3,6}  {2,6}  {2,6}   {2,6}   {2,6}
                                            {3,6}  {3,6}   {3,6}   {3,6}
                                            {3,9}  {3,9}   {3,9}   {3,9}
                                                   {2,10}  {1,11}  {1,11}
                                                           {2,10}  {2,10}
                                                                   {4,12}
                                                                   {6,12}

Cf. A006218, A019565, A050315, A070939, A080572, A247935, A267610.

Cf. A325095, A325096, A325101, A325103, A325104, A325105, A325106, A325124.

Table[Length[Select[Subsets[Range[n],{2}],Divisible@@Reverse[#]&&Intersection[Position[Reverse[IntegerDigits[#[[1]],2]],1],Position[Reverse[IntegerDigits[#[[2]],2]],1]]!={}&]],{n,0,20}]

a(n) = A325124(n) - n.

cp's OEIS Frontend

A325123 Number of divisible pairs of positive integers up to n with no binary carries.

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A325100 Heinz numbers of strict integer partitions with no binary carries.

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A325124 Number of divisible pairs of positive integers up to n with at least one binary carry.

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A325120 Sum of binary lengths of the prime indices of n.

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A325121 Sum of binary digits of the prime indices of n.

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A325122 Sum of binary digits of the prime indices of n, minus Omega(n).

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A307230 Number of divisible pairs of distinct positive integers up to n with at least one binary carry.

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