A066312 -id:A066312 - cp's OEIS frontend

A355527 Squarefree numbers having at least one pair of consecutive prime factors. Numbers n such that the minimal difference between adjacent prime indices of n is 1.

6, 15, 30, 35, 42, 66, 70, 77, 78, 102, 105, 114, 138, 143, 154, 165, 174, 186, 195, 210, 221, 222, 231, 246, 255, 258, 282, 285, 286, 318, 323, 330, 345, 354, 366, 385, 390, 402, 426, 429, 435, 437, 438, 442, 455, 462, 465, 474, 498, 510, 534, 546, 555, 570

Offset: 1

Gus Wiseman, Jul 10 2022

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.
A number is squarefree if it is not divisible by any perfect square > 1.
A number has consecutive prime factors if it is divisible by both prime(k) and prime(k+1) for some k.

			The terms together with their prime indices begin:
    6: {1,2}
   15: {2,3}
   30: {1,2,3}
   35: {3,4}
   42: {1,2,4}
   66: {1,2,5}
   70: {1,3,4}
   77: {4,5}
   78: {1,2,6}
  102: {1,2,7}
  105: {2,3,4}
  114: {1,2,8}
  138: {1,2,9}
  143: {5,6}
  154: {1,4,5}
  165: {2,3,5}
  174: {1,2,10}
  186: {1,2,11}
  195: {2,3,6}
  210: {1,2,3,4}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
All terms are in A005117, complement A013929.
For minimal difference <= 1 we have A055932.
For maximal instead of minimal difference = 1 we have A066312.
For minimal difference > 1 we have A325160.
If zero is considered a prime index we get A355530.
A001522 counts partitions with a fixed point (unproved), ranked by A352827.
A287352, A355533, A355534, A355536 list the differences of prime indices.
A355524 or A355525 give minimal difference between prime indices.
Cf. A000005, A000040, A056239, A120944, A130091, A238353, A238354, A286470, A325161, A352822, A355526, A355531.

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

Intersection of A005117 (squarefree) and A104210 (has consecutive primes).

A355532 Maximal augmented difference between adjacent reversed prime indices of n; a(1) = 0.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 1, 2, 3, 5, 2, 6, 4, 2, 1, 7, 2, 8, 3, 3, 5, 9, 2, 3, 6, 2, 4, 10, 2, 11, 1, 4, 7, 3, 2, 12, 8, 5, 3, 13, 3, 14, 5, 2, 9, 15, 2, 4, 3, 6, 6, 16, 2, 3, 4, 7, 10, 17, 2, 18, 11, 3, 1, 4, 4, 19, 7, 8, 3, 20, 2, 21, 12, 2, 8, 4, 5, 22, 3, 2

Offset: 1

Gus Wiseman, Jul 14 2022

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 augmented differences aug(q) of a (usually weakly decreasing) sequence q of length k are given by aug(q)i = q_i - q{i+1} + 1 if i < k and aug(q)_k = q_k. For example, we have aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

			The reversed prime indices of 825 are (5,3,3,2), with augmented differences (3,1,2,2), so a(825) = 3.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
Prepending 1 to the positions of 1's gives A000079.
Positions of first appearances are A008578.
Positions of 2's are A065119.
The non-augmented version is A286470, also A355526.
The non-augmented minimal version is A355524, also A355525.
The minimal version is A355531.
Row maxima of A355534, which has Heinz number A325351.
A001222 counts prime indices, distinct A001221.
A112798 lists prime indices, sum A056239.
Cf. A066312, A124010, A129654, A243055, A243056, A307824, A325366, A325394, A355533, A355536.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aug[y_]:=Table[If[i

A355523 Number of distinct differences between adjacent prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 10 2022

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.

			For example, the prime indices of 22770 are {1,2,2,3,5,9}, with differences (1,0,1,2,4), so a(22770) = 4.

Crossrefs found in the link are not repeated here.
Counting m such that A056239(m) = n and a(m) = k gives A279945.
With multiplicity we have A252736(n) = A001222(n) - 1.
The maximal difference is A286470, minimal A355524.
A008578 gives the positions of 0's.
A287352 lists differences between 0-prepended prime indices.
A355534 lists augmented differences between prime indices.
A355536 lists differences between prime indices.
Cf. A066312, A238353, A238710, A286469, A320348, A325388, A325406, A351294, A352827, A355525-A355533.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Differences[primeMS[n]]]],{n,1000}]

A355523(n) = if(1==n, 0, my(pis = apply(primepi,factor(n)[,1]), difs = vector(#pis-1, i, pis[i+1]-pis[i])); (#Set(difs)+!issquarefree(n))); \\ Antti Karttunen, Jan 20 2025

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

A355530 Squarefree numbers that are either even or have at least one pair of consecutive prime factors. Numbers n such that the minimal difference between adjacent 0-prepended prime indices of n is 1.

Original entry on oeis.org

2, 6, 10, 14, 15, 22, 26, 30, 34, 35, 38, 42, 46, 58, 62, 66, 70, 74, 77, 78, 82, 86, 94, 102, 105, 106, 110, 114, 118, 122, 130, 134, 138, 142, 143, 146, 154, 158, 165, 166, 170, 174, 178, 182, 186, 190, 194, 195, 202, 206, 210, 214, 218, 221, 222, 226, 230

Offset: 1

Gus Wiseman, Jul 10 2022

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.
A number is squarefree if it is not divisible by any perfect square > 1.
A number has consecutive prime factors if it is divisible by both prime(k) and prime(k+1) for some k.

			The terms together with their prime indices begin:
   2: {1}
   6: {1,2}
  10: {1,3}
  14: {1,4}
  15: {2,3}
  22: {1,5}
  26: {1,6}
  30: {1,2,3}
  34: {1,7}
  35: {3,4}
  38: {1,8}
  42: {1,2,4}
  46: {1,9}
  58: {1,10}
  62: {1,11}
  66: {1,2,5}
  70: {1,3,4}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
All terms are in A005117, complement A013929.
For maximal instead of minimal difference we have A055932 or A066312.
Not prepending zero gives A355527.
A001522 counts partitions with a fixed point (unproved), ranked by A352827.
A056239 adds up prime indices.
A238352 counts partitions by fixed points, rank statistic A352822.
A279945 counts partitions by number of distinct differences.
A287352, A355533, A355534, A355536 list the differences of prime indices.
A355524 gives minimal difference if singletons go to 0, to index A355525.
Cf. A000005, A000040, A120944, A238354, A286469, A286470, A325160, A325161, A355526, A355531.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Min@@Differences[Prepend[primeMS[#],0]]==1&]

Equals A005117 /\ (A005843 \/ A104210).

A356736 Heinz numbers of integer partitions with no neighborless parts.

Original entry on oeis.org

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]

A066309 Numbers k such that k > (product of digits of k) * (sum of digits of k).

Original entry on oeis.org

10, 11, 12, 13, 20, 21, 22, 30, 31, 32, 40, 41, 50, 51, 60, 61, 70, 71, 80, 81, 90, 91, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 130, 131, 132, 133, 134, 140, 141, 142

Offset: 1

Labos Elemer and Klaus Brockhaus, Dec 13 2001

			13 is in the sequence because (1*3)*(1+3) = 3*4 = 12 < 13.
125 is a term because (1*2*5)*(1+2+5) = 10*8 = 80 < 125.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A038369, A049101-A049106, A034710, A061672, A066306, A066307.

function a066312(a,b: integer); var n,k,j,p,d: integer; s: string; begin for n := a to b do s := itoa(n); k := 0; p := 1; for j := 0 to length(s) - 1 do d := atoi(s[j..j]); k := k + d; p := p*d; end; if n > p*k then write(n,","); end; end; end; a066312(0,150);

asum[x_] := Apply[Plus, IntegerDigits[x]] apro[x_] := Apply[Times, IntegerDigits[x]] sz[x_] := asu[x]*apro[x] Do[s=sz[n]; If[Greater[n, s], Print[n]], {n, 1, 1000}]
okQ[n_]:=Module[{idn=IntegerDigits[n]},n> Total[idn]Times@@idn];Select[Range[150],okQ]  (* Harvey P. Dale, Mar 12 2011 *)

isok(k) = {my(d=digits(k)); k > vecprod(d) * vecsum(d)} \\ Harry J. Smith, Feb 10 2010

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

A358171 The a(n)-th composition in standard order (A066099) is the first differences plus one of the prime indices of n (A112798).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 3, 1, 4, 0, 6, 0, 8, 2, 7, 0, 5, 0, 12, 4, 16, 0, 14, 1, 32, 3, 24, 0, 10, 0, 15, 8, 64, 2, 13, 0, 128, 16, 28, 0, 20, 0, 48, 6, 256, 0, 30, 1, 9, 32, 96, 0, 11, 4, 56, 64, 512, 0, 26, 0, 1024, 12, 31, 8, 40, 0, 192, 128, 18, 0, 29, 0

Offset: 1

Gus Wiseman, Dec 21 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

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 36 are {1,1,2,2}, with first differences plus one (1,2,1), which is the 13th composition in standard order, so a(36) = 13.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
Prepend 1 to indices: A253566 (cf. A358169), inverse A253565 (cf. A242628).
Taking Heinz number instead of standard composition number gives A325352.
These compositions minus one are listed by A355536, sums A243055.
A001222 counts prime indices, distinct A001221.
A066099 lists standard compositions, lengths A000120, sums A070939.
A112798 lists prime indices, sum A056239.
A355534 = augmented diffs. of rev. prime indices, Heinz numbers A325351.
Cf. A000720, A005940, A055932, A057335, A066312, A286470, A287352, A325390.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[Differences[primeMS[n]]+1],{n,100}]

cp's OEIS Frontend

A355527 Squarefree numbers having at least one pair of consecutive prime factors. Numbers n such that the minimal difference between adjacent prime indices of n is 1.

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A355532 Maximal augmented difference between adjacent reversed prime indices of n; a(1) = 0.

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A355523 Number of distinct differences between adjacent prime indices of n.

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A355530 Squarefree numbers that are either even or have at least one pair of consecutive prime factors. Numbers n such that the minimal difference between adjacent 0-prepended prime indices of n is 1.

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A356736 Heinz numbers of integer partitions with no neighborless parts.

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A066309 Numbers k such that k > (product of digits of k) * (sum of digits of k).

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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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