A328335 -id:A328335 - cp's OEIS frontend

A356736 Heinz numbers of integer partitions with no neighborless parts.

1, 6, 12, 15, 18, 24, 30, 35, 36, 45, 48, 54, 60, 72, 75, 77, 90, 96, 105, 108, 120, 135, 143, 144, 150, 162, 175, 180, 192, 210, 216, 221, 225, 240, 245, 270, 288, 300, 315, 323, 324, 360, 375, 384, 385, 405, 420, 432, 437, 450, 462, 480, 486, 525, 539, 540

Offset: 1

Gus Wiseman, Aug 31 2022

nonn

First differs from A066312 in having 1 and lacking 462.
First differs from A104210 in having 1 and lacking 42.
A part x is neighborless iff neither x - 1 nor x + 1 are parts.
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 terms together with their prime indices begin:
   1: {}
   6: {1,2}
  12: {1,1,2}
  15: {2,3}
  18: {1,2,2}
  24: {1,1,1,2}
  30: {1,2,3}
  35: {3,4}
  36: {1,1,2,2}
  45: {2,2,3}
  48: {1,1,1,1,2}
  54: {1,2,2,2}
  60: {1,1,2,3}
  72: {1,1,1,2,2}
  75: {2,3,3}
  77: {4,5}
  90: {1,2,2,3}
  96: {1,1,1,1,1,2}

These partitions are counted by A355394.
The singleton case is the complement of A356237.
The singleton case is counted by A355393, complement A356235.
The strict complement is A356606, counted by A356607.
The complement is A356734, counted by A356236.
A000041 counts integer partitions, strict A000009.
A001221 counts distinct prime factors, sum A001414.
A003963 multiplies together the prime indices of n.
A007690 counts partitions with no singletons, complement A183558.
A056239 adds up prime indices, row sums of A112798, lengths A001222.
A073491 lists numbers with gapless prime indices, complement A073492.
Cf. A066312, A286470, A287170 (firsts A066205), A328171, A328187, A328221, A328335, A356231, A356234.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Function[ptn,!Or@@Table[!MemberQ[ptn,x-1]&&!MemberQ[ptn,x+1],{x,Union[ptn]}]]@*primeMS]

A328674 Numbers whose distinct prime indices have no consecutive divisible parts.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 25, 27, 29, 31, 32, 33, 35, 37, 41, 43, 45, 47, 49, 51, 53, 55, 59, 61, 64, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 113, 119, 121, 123, 125, 127, 128, 131, 135

Offset: 1

Gus Wiseman, Oct 29 2019

nonn

First differs from A316476 in having 105, with prime indices {2, 3, 4}.

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}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   15: {2,3}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
For example, 45 is in the sequence because its distinct prime indices are {2,3} and 2 is not a divisor of 3.

These are the Heinz numbers of the partitions counted by A328675.
The strict version is A328603.
Partitions without consecutive divisibilities are A328171.
Compositions without consecutive divisibilities are A328460.
Cf. A056239, A112798, A316476, A318729, A328335, A328336, A328593, A328598.

Select[Range[100],!MatchQ[PrimePi/@First/@FactorInteger[#],{_,x_,y_,_}/;Divisible[y,x]]&]

A356221 Position of second appearance of 2n in the sequence of prime gaps A001223; if 2n does not appear at least twice, a(n) = -1.

Original entry on oeis.org

3, 6, 11, 72, 42, 47, 62, 295, 180, 259, 297, 327, 446, 462, 650, 1315, 1059, 1532, 4052, 2344, 3732, 3861, 8805, 7234, 4754, 2810, 4231, 14124, 5949, 9834, 17200, 10229, 19724, 25248, 15927, 30765, 42673, 28593, 24554, 50523, 44227, 44390, 29040, 89715, 47350

Offset: 1

Gus Wiseman, Aug 02 2022

nonn

Prime gaps (A001223) are the differences between consecutive prime numbers. They begin: 1, 2, 2, 4, 2, 4, 2, 4, 6, ...

The position of the first (instead of second) appearance of 2n is A038664.
Column k = 2 of A356222.
The position of the n-th appearance of 2n is A356223.
A001223 lists the prime gaps, reduced A028334.
A073491 lists numbers with gapless prime indices.
A274121 counts appearances of the n-th prime gap in those prior.
A356226 gives the lengths of maximal gapless intervals of prime indices.
Cf. A029709, A066205, A137921, A193829, A287170, A328335, A328457, A356224, A356225.

nn=1000;
gaps=Differences[Array[Prime,nn]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
Table[Position[gaps,2*n][[2,1]],{n,mnrm[Select[Range[nn],Length[Position[gaps,2*#]]>=2&]]}]

A356733 Number of neighborless parts in the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 26 2022

nonn

A part x is neighborless if neither x - 1 nor x + 1 are parts.

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 prime indices of 42 are {1,2,4}, of which only 4 is neighborless, so a(42) = 1.
The prime indices of 462 are {1,2,4,5}, all of which have neighbors, so a(462) = 0.
The prime indices of 1300 are {1,1,3,3,6}, with neighborless parts {1,3,6}, so a(1300) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to prime indices in the factorization of n.

Positions of first appearances are 1 followed by A066205.
Dominated by A287170 (firsts also A066205).
Positions of terms > 0 are A356734.
The complement is counted by A356735.
A001221 counts distinct prime factors, sum A001414.
A003963 multiplies together prime indices.
A007690 counts partitions with no singletons, complement A183558.
A056239 adds up prime indices, row sums of A112798, lengths A001222.
A073491 lists numbers with gapless prime indices, complement A073492.
A132747 counts non-isolated divisors, complement A132881.
A355393 counts partitions w/o a neighborless singleton, complement A356235.
A355394 counts partitions w/o a neighborless part, complement A356236.
A356069 counts gapless divisors, initial A356224 (complement A356225).
A356607 counts strict partitions w/ a neighborless part, complement A356606.
Cf. A000005, A066312, A286470, A289508, A325160, A328166, A328335, A356231, A356233, A356234, A356237.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Union[primeMS[n]],!MemberQ[primeMS[n],#-1]&&!MemberQ[primeMS[n],#+1]&]],{n,100}]

A356733(n) = if(1==n,0,my(pis=apply(primepi,factor(n)[,1])); sum(i=1, #pis, ((n%prime(pis[i]+1)) && (pis[i]==1 || (n%prime(pis[i]-1)))))); \\ Antti Karttunen, Jan 28 2025

a(n) = A001221(n) - A356735(n).

Data section extended to a(105) by Antti Karttunen, Jan 28 2025

A356735 Number of distinct parts that have neighbors in the integer partition with Heinz number n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 31 2022

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.

Also the number of distinct prime indices x of n such that either x - 1 or x + 1 is also a prime index of n, where a prime index of n is a number x such that prime(x) divides n.

			The prime indices of 42 are {1,2,4}, of which 1 and 2 have neighbors, so a(42) = 2.
The prime indices of 462 are {1,2,4,5}, all of which have neighbors, so a(462) = 4.
The prime indices of 990 are {1,2,2,3,5}, of which 1, 2, and 3 have neighbors, so a(990) = 3.
The prime indices of 1300 are {1,1,3,3,6}, none of which have neighbors, so a(1300) = 0.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to prime indices in the factorization of n.

Positions of first appearances are A002110 without 1 (or A231209).
The complement is counted by A356733.
Positions of zeros are A356734.
Positions of positive terms are A356736.
A001221 counts distinct prime factors, sum A001414.
A007690 counts partitions with no singletons, complement A183558.
A056239 adds up prime indices, row sums of A112798, lengths A001222.
A073491 lists numbers with gapless prime indices, complement A073492.
A355393 counts partitions w/o a neighborless singleton, complement A356235.
A355394 counts partitions w/o a neighborless part, complement A356236.
A356226 lists the lengths of maximal gapless submultisets of prime indices:
- length: A287170 (firsts A066205)
- minimum: A356227
- maximum: A356228
- bisected length: A356229
- standard composition: A356230
- Heinz number: A356231
- positions of first appearances: A356232
Cf. A000005, A066312, A286470, A289508, A325160, A328166, A328335, A356233, A356234, A356237.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Union[primeMS[n]], MemberQ[primeMS[n],#-1]|| MemberQ[primeMS[n],#+1]&]],{n,100}]

A356735(n) = if(1==n,0,my(pis=apply(primepi,factor(n)[,1])); omega(n)-sum(i=1, #pis, ((n%prime(pis[i]+1)) && (pis[i]==1 || (n%prime(pis[i]-1)))))); \\ Antti Karttunen, Jan 28 2025

a(n) + A356733(n) = A001221(n).

Data section extended to a(105) by Antti Karttunen, Jan 28 2025

cp's OEIS Frontend

A356736 Heinz numbers of integer partitions with no neighborless parts.

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A328674 Numbers whose distinct prime indices have no consecutive divisible parts.

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A356221 Position of second appearance of 2n in the sequence of prime gaps A001223; if 2n does not appear at least twice, a(n) = -1.

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A356733 Number of neighborless parts in the integer partition with Heinz number n.

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A356735 Number of distinct parts that have neighbors in the integer partition with Heinz number n.

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