A325033 -id:A325033 - cp's OEIS frontend

A357458 First differences of A325033 = "Sum of sums of the multiset of prime indices of each prime index of n.".

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

Offset: 1

Gus Wiseman, Sep 30 2022

sign

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 multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			We have A325033(5) - A325033(4) = 2 - 0, so a(4) = 2.

The partial sums are A325033, which has row-products A325032.
The version for standard compositions is A357187.
A000961 lists prime powers.
A003963 multiples prime indices.
A005117 lists squarefree numbers.
A056239 adds up prime indices.
Cf. A000720, A001221, A001222, A007716, A109082, A275024, A302242, A302243, A302505, A324926, A325034, A357139.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Differences[Table[Plus@@Join@@primeMS/@primeMS[n],{n,100}]]

a(n) = A325033(n + 1) - A325033(n).

A325037 Heinz numbers of integer partitions whose product of parts is greater than their sum.

Original entry on oeis.org

1, 15, 21, 25, 27, 33, 35, 39, 42, 45, 49, 50, 51, 54, 55, 57, 63, 65, 66, 69, 70, 75, 77, 78, 81, 85, 87, 90, 91, 93, 95, 98, 99, 100, 102, 105, 110, 111, 114, 115, 117, 119, 121, 123, 125, 126, 129, 130, 132, 133, 135, 138, 140, 141, 143, 145, 147, 150, 153

Offset: 1

Gus Wiseman, Mar 25 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose product of prime indices (A003963) is greater than their sum of prime indices (A056239).

The enumeration of these partitions by sum is given by A114324.

			The sequence of terms together with their prime indices begins:
   1: {}
  15: {2,3}
  21: {2,4}
  25: {3,3}
  27: {2,2,2}
  33: {2,5}
  35: {3,4}
  39: {2,6}
  42: {1,2,4}
  45: {2,2,3}
  49: {4,4}
  50: {1,3,3}
  51: {2,7}
  54: {1,2,2,2}
  55: {3,5}
  57: {2,8}
  63: {2,2,4}
  65: {3,6}
  66: {1,2,5}
  69: {2,9}
  70: {1,3,4}
  75: {2,3,3}
  77: {4,5}
  78: {1,2,6}
  81: {2,2,2,2}

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.

Cf. A325032, A325033, A325036, A325038, A325041, A325042, A325044.

q:= n-> (l-> mul(i, i=l)>add(i, i=l))(map(i->
    numtheory[pi](i[1])$i[2], ifactors(n)[2])):
select(q, [$1..200])[];  # Alois P. Heinz, Mar 27 2019

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Times@@primeMS[#]>Plus@@primeMS[#]&]

A003963(a(n)) > A056239(a(n)).

A325044 Heinz numbers of integer partitions whose sum of parts is greater than or equal to their product.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 26, 28, 29, 30, 31, 32, 34, 36, 37, 38, 40, 41, 43, 44, 46, 47, 48, 52, 53, 56, 58, 59, 60, 61, 62, 64, 67, 68, 71, 72, 73, 74, 76, 79, 80, 82, 83, 84, 86, 88, 89, 92, 94, 96, 97, 101

Offset: 1

Gus Wiseman, Mar 25 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose product of prime indices (A003963) is less than or equal to their sum of prime indices (A056239).

The enumeration of these partitions by sum is given by A096276.

			The sequence of terms together with their prime indices begins:
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   6: {1,2}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  10: {1,3}
  11: {5}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  16: {1,1,1,1}
  17: {7}
  18: {1,2,2}
  19: {8}
  20: {1,1,3}
  22: {1,5}
  23: {9}
  24: {1,1,1,2}

Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.

Cf. A325032, A325033, A325036, A325037, A325038, A325041, A325042.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Times@@primeMS[#]<=Plus@@primeMS[#]&]

A003963(a(n)) <= A056239(a(n)).
a(n) = A325038(n)/2.
Union of A301987 and A325038.

A325038 Heinz numbers of integer partitions whose sum of parts is greater than their product.

Original entry on oeis.org

4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 32, 34, 36, 38, 40, 44, 46, 48, 52, 56, 58, 60, 62, 64, 68, 72, 74, 76, 80, 82, 86, 88, 92, 94, 96, 104, 106, 112, 116, 118, 120, 122, 124, 128, 134, 136, 142, 144, 146, 148, 152, 158, 160, 164, 166, 168, 172

Offset: 1

Gus Wiseman, Mar 25 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose product of prime indices (A003963) is less than their sum of prime indices (A056239).

The enumeration of these partitions by sum is given by A096276 shifted once to the right.

			The sequence of terms together with their prime indices begins:
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  16: {1,1,1,1}
  18: {1,2,2}
  20: {1,1,3}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}
  32: {1,1,1,1,1}
  34: {1,7}
  36: {1,1,2,2}
  38: {1,8}
  40: {1,1,1,3}
  44: {1,1,5}
  46: {1,9}
  48: {1,1,1,1,2}

Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.

Cf. A325032, A325033, A325036, A325037, A325041, A325042, A325044.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Times@@primeMS[#]

A003963(a(n)) < A056239(a(n)).

a(n) = 2 * A325044(n).

A325036 Difference between product and sum of prime indices of n.

Original entry on oeis.org

1, 0, 0, -1, 0, -1, 0, -2, 0, -1, 0, -2, 0, -1, 1, -3, 0, -1, 0, -2, 2, -1, 0, -3, 3, -1, 2, -2, 0, 0, 0, -4, 3, -1, 5, -2, 0, -1, 4, -3, 0, 1, 0, -2, 5, -1, 0, -4, 8, 2, 5, -2, 0, 1, 7, -3, 6, -1, 0, -1, 0, -1, 8, -5, 9, 2, 0, -2, 7, 4, 0, -3, 0, -1, 10, -2, 11, 3, 0, -4, 8, -1, 0, 0, 11, -1, 8, -3, 0, 4, 14, -2, 9

Offset: 1

Gus Wiseman, Mar 25 2019

sign

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 prime indices of 45 are {2,2,3}, with product 12 and sum 7, so a(45) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Positions of zeros are A301987. Positions of ones are A325041. Positions of negative ones are A325042.
Cf. A000720, A001222, A003963, A056239, A112798, A122111, A178503, A175508, A319000.
Cf. A325032, A325033, A325034, A325035, A325037, A325038, A325040, A325044.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@primeMS[n]-Total[primeMS[n]],{n,100}]
dps[n_]:=Module[{pi=Flatten[Table[PrimePi[#[[1]]],#[[2]]]&/@FactorInteger[n]]},Times@@pi-Total[pi]]; Join[{1},Array[dps,100,2]] (* Harvey P. Dale, May 26 2023 *)

A003963(n) = { n=factor(n); n[, 1]=apply(primepi, n[, 1]); factorback(n) }; \\ From A003963
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); };
A325036(n) = (A003963(n) - A056239(n)); \\ Antti Karttunen, May 08 2022

a(n) = A003963(n) - A056239(n).

For all n >= 1, a(A325040(n)) = a(A122111(A325040(n))). - Antti Karttunen, May 08 2022

Data section extended up to a(93) by Antti Karttunen, May 08 2022

A325041 Heinz numbers of integer partitions whose product of parts is one greater than their sum.

Original entry on oeis.org

1, 15, 42, 54, 100, 132, 312, 560, 720, 816, 1824, 3520, 4416, 6272, 8064, 10368, 11136, 16640, 23808, 38400, 56832, 78848, 87040, 101376, 125952, 264192, 389120, 577536, 745472, 958464, 1302528, 1720320, 1884160, 1982464, 2211840, 2899968, 5996544

Offset: 1

Gus Wiseman, Mar 25 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose product of prime indices (A003963) is one more than their sum of prime indices (A056239).

			The sequence of terms together with their prime indices begins:
      1: {}
     15: {2,3}
     42: {1,2,4}
     54: {1,2,2,2}
    100: {1,1,3,3}
    132: {1,1,2,5}
    312: {1,1,1,2,6}
    560: {1,1,1,1,3,4}
    720: {1,1,1,1,2,2,3}
    816: {1,1,1,1,2,7}
   1824: {1,1,1,1,1,2,8}
   3520: {1,1,1,1,1,1,3,5}
   4416: {1,1,1,1,1,1,2,9}
   6272: {1,1,1,1,1,1,1,4,4}
   8064: {1,1,1,1,1,1,1,2,2,4}
  10368: {1,1,1,1,1,1,1,2,2,2,2}
  11136: {1,1,1,1,1,1,1,2,10}
  16640: {1,1,1,1,1,1,1,1,3,6}
  23808: {1,1,1,1,1,1,1,1,2,11}
  38400: {1,1,1,1,1,1,1,1,1,2,3,3}

Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.

Cf. A325032, A325033, A325036, A325037, A325038, A325042, A325044.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[10000],Times@@primeMS[#]==Total[primeMS[#]]+1&]

A003963(a(n)) = A056239(a(n)) + 1.

A325042 Heinz numbers of integer partitions whose product of parts is one fewer than their sum.

Original entry on oeis.org

4, 6, 10, 14, 18, 22, 26, 34, 38, 46, 58, 60, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 168, 178, 194, 202, 206, 214, 216, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 400, 422, 446, 454, 458, 466

Offset: 1

Gus Wiseman, Mar 25 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose product of prime indices (A003963) is one fewer than their sum of prime indices (A056239).

			The sequence of terms together with their prime indices begins:
    4: {1,1}
    6: {1,2}
   10: {1,3}
   14: {1,4}
   18: {1,2,2}
   22: {1,5}
   26: {1,6}
   34: {1,7}
   38: {1,8}
   46: {1,9}
   58: {1,10}
   60: {1,1,2,3}
   62: {1,11}
   74: {1,12}
   82: {1,13}
   86: {1,14}
   94: {1,15}
  106: {1,16}
  118: {1,17}
  122: {1,18}

Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.

Cf. A325032, A325033, A325036, A325037, A325038, A325041, A325044.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Times@@primeMS[#]==Total[primeMS[#]]-1&]

A003963(a(n)) = A056239(a(n)) - 1.

a(n) = 2 * A301987(n).

A325032 Product of products of the multisets of prime indices of each prime index of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 2, 1, 2, 1, 4, 1, 1, 2, 1, 3, 4, 1, 4, 2, 1, 1, 3, 2, 5, 1, 3, 4, 2, 1, 2, 1, 2, 2, 6, 1, 4, 3, 2, 4, 6, 1, 1, 4, 4, 2, 1, 1, 6, 1, 1, 3, 7, 2, 4, 5, 1, 1, 4, 3, 8, 4, 4, 2, 3, 1, 8, 2, 4, 1, 3, 2, 5, 2, 1, 6, 9, 1, 8, 4, 3

Offset: 1

Gus Wiseman, Mar 25 2019

nonn

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.

			94 has prime indices {1,15} with prime indices {{},{2,3}} with product a(94) = 6.

Cf. A000720, A001055, A001222, A003963, A056239, A112798, A302242, A320325.

Cf. A324850, A324926, A324930, A325031, A325033, A325034, A325035.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Join@@primeMS/@primeMS[n],{n,100}]

Fully multiplicative with a(prime(n)) = A003963(n).

A325034 Sum of products of the multisets of prime indices of each prime index of n.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 1, 3, 2, 3, 3, 3, 2, 2, 3, 4, 4, 3, 1, 4, 2, 4, 4, 4, 4, 3, 3, 3, 3, 4, 5, 5, 4, 5, 3, 4, 2, 2, 3, 5, 6, 3, 4, 5, 4, 5, 6, 5, 2, 5, 5, 4, 1, 4, 5, 4, 2, 4, 7, 5, 4, 6, 3, 6, 4, 5, 8, 6, 5, 4, 3, 5, 8, 3, 5, 3, 4, 4, 5, 6, 4, 7, 9, 4, 6, 5, 4

Offset: 1

Gus Wiseman, Mar 25 2019

nonn

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.

			94 has prime indices {1,15} with prime indices {{},{2,3}} with products {1,6} with sum a(94) = 7.

Cf. A000720, A001055, A001222, A003963, A056239, A066739, A112798, A302242.

Cf. A324926, A325031, A325032, A325033, A325035.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Plus@@Times@@@primeMS/@primeMS[n],{n,100}]

Totally additive with a(prime(n)) = A003963(n).

A380955 Sum of prime indices of n (with multiplicity) minus sum of distinct prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 11 2025

nonn

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 prime indices of 96 are {1,1,1,1,1,2}, with sum 7, and with distinct prime indices {1,2}, with sum 3, so a(96) = 7 - 3 = 4.

Positions of 0's are A005117, complement A013929.
For length instead of sum we have A046660.
Positions of 1's are A081770.
For factors instead of indices we have A280292, firsts A280286 (sorted A381075).
A multiplicative version is A290106.
Counting partitions by this statistic gives A364916.
Dominates A374248.
Positions of first appearances are A380956, sorted A380957.
For prime multiplicities instead of prime indices we have A380958.
For product instead of sum we have A380986.
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, length A001222.
A304038 lists distinct prime indices, sum A066328, length A001221.
Cf. A000720, A071625, A075254, A075255, A116861, A124010, A136565, A178503, A325033, A325034, A366528, A366749, A379681.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[prix[n]]-Total[Union[prix[n]]],{n,100}]

a(n) = A056239(n) - A066328(n).

Additive: a(m*n) = a(m) + a(n) if gcd(m,n) = 1.

cp's OEIS Frontend

A357458 First differences of A325033 = "Sum of sums of the multiset of prime indices of each prime index of n.".

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A325037 Heinz numbers of integer partitions whose product of parts is greater than their sum.

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A325044 Heinz numbers of integer partitions whose sum of parts is greater than or equal to their product.

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A325038 Heinz numbers of integer partitions whose sum of parts is greater than their product.

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A325036 Difference between product and sum of prime indices of n.

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A325041 Heinz numbers of integer partitions whose product of parts is one greater than their sum.

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A325042 Heinz numbers of integer partitions whose product of parts is one fewer than their sum.

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A325032 Product of products of the multisets of prime indices of each prime index of n.

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