cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A380409 Product of prime indices plus sum of prime factors.

Original entry on oeis.org

1, 3, 5, 5, 8, 7, 11, 7, 10, 10, 16, 9, 19, 13, 14, 9, 24, 12, 27, 12, 18, 18, 32, 11, 19, 21, 17, 15, 39, 16, 42, 11, 24, 26, 24, 14, 49, 29, 28, 14, 54, 20, 57, 20, 23, 34, 62, 13, 30, 21, 34, 23, 69, 19, 31, 17, 38, 41, 76, 18, 79, 44, 29, 13, 36, 26, 86
Offset: 1

Views

Author

Gus Wiseman, Jan 25 2025

Keywords

Comments

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, with product A003963.

Examples

			72 has prime factors {2,2,2,3,3} and prime indices {1,1,1,2,2}, so a(72) = 12 + 4 = 16.

Crossrefs

For factors instead of indices we have A075254.
For indices instead of factors we have A379681.
For minus instead of plus we have A380344, zeros A331384.
Triangles:
- A027746 = prime factors
- A112798 = prime indices
Statistics:
- A000027 = product of prime factors = row products of A027746
- A001414 = sum of prime factors = row sums of A027746
- A003963 = product of prime indices = row products of A112798
- A056239 = sum of prime indices = row sums of A112798
Combinations:
- A075254 = product of factors + sum of factors = A000027 + A001414
- A075255 = product of factors - sum of factors = A000027 - A001414
- A178503 = product of factors - sum of indices = A000027 - A056239
- A325036 = product of indices - sum of indices = A003963 - A056239
- A379681 = product of indices + sum of indices = A003963 + A056239
- A380344 = product of indices - sum of factors = A003963 - A001414
- A380345 = product of factors + sum of indices = A000027 + A056239
- A380409 = product of indices + sum of factors = A003963 + A001414
A000040 lists the primes, differences A001223.
A001222 counts prime factors with multiplicity.
A055396 gives least prime index, greatest A061395.
Cf. A000720, A175508, A319000, A325032, A325033, A325034, A325035, A325040, A379682, A380220, A380410.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[Prime/@prix[n]]+Times@@prix[n],{n,100}]

Formula

a(n) = A003963(n) + A001414(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

Views

Author

Gus Wiseman, Mar 29 2019

Keywords

Comments

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.

Crossrefs

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.

Programs

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

Formula

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

Views

Author

Gus Wiseman, Mar 29 2019

Keywords

Comments

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.

Crossrefs

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.

Programs

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

Formula

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

Views

Author

Gus Wiseman, Mar 29 2019

Keywords

Comments

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.

Crossrefs

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.

Programs

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

Formula

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

A325043 Heinz numbers of integer partitions, with at least three parts, whose product of parts is one fewer than their sum.

Original entry on oeis.org

18, 60, 168, 216, 400, 528, 1248, 2240, 2880, 3264, 7296, 14080, 17664, 25088, 32256, 41472, 44544, 66560, 95232, 153600, 227328, 315392, 348160, 405504, 503808, 1056768, 1556480, 2310144, 2981888, 3833856, 5210112, 6881280, 7536640, 7929856, 8847360, 11599872
Offset: 1

Views

Author

Gus Wiseman, Mar 25 2019

Keywords

Comments

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

Examples

			The sequence of terms together with their prime indices begins:
     18: {1,2,2}
     60: {1,1,2,3}
    168: {1,1,1,2,4}
    216: {1,1,1,2,2,2}
    400: {1,1,1,1,3,3}
    528: {1,1,1,1,2,5}
   1248: {1,1,1,1,1,2,6}
   2240: {1,1,1,1,1,1,3,4}
   2880: {1,1,1,1,1,1,2,2,3}
   3264: {1,1,1,1,1,1,2,7}
   7296: {1,1,1,1,1,1,1,2,8}
  14080: {1,1,1,1,1,1,1,1,3,5}
  17664: {1,1,1,1,1,1,1,1,2,9}
  25088: {1,1,1,1,1,1,1,1,1,4,4}
  32256: {1,1,1,1,1,1,1,1,1,2,2,4}
  41472: {1,1,1,1,1,1,1,1,1,2,2,2,2}
  44544: {1,1,1,1,1,1,1,1,1,2,10}
  66560: {1,1,1,1,1,1,1,1,1,1,3,6}
  95232: {1,1,1,1,1,1,1,1,1,1,2,11}

Crossrefs

Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.
Cf. A325032, A325033, A325036, A325037, A325038, A325041, A325042, A325044.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[10000],And[PrimeOmega[#]>2,Times@@primeMS[#]==Total[primeMS[#]]-1]&]

Formula

a(n) = 2 * A301988(n).

Extensions

More terms from Jinyuan Wang, Jun 27 2020
Previous Showing 21-25 of 25 results.

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