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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A366530 Heinz numbers of integer partitions of even numbers with at least one odd part.

Original entry on oeis.org

4, 10, 12, 16, 22, 25, 28, 30, 34, 36, 40, 46, 48, 52, 55, 62, 64, 66, 70, 75, 76, 82, 84, 85, 88, 90, 94, 100, 102, 108, 112, 115, 116, 118, 120, 121, 130, 134, 136, 138, 144, 146, 148, 154, 155, 156, 160, 165, 166, 172, 175, 184, 186, 187, 190, 192, 194, 196
Offset: 1

Views

Author

Gus Wiseman, Oct 16 2023

Keywords

Comments

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.

Examples

			The terms together with their prime indices are the following. Each multiset has even sum and at least one odd part.
    4: {1,1}
   10: {1,3}
   12: {1,1,2}
   16: {1,1,1,1}
   22: {1,5}
   25: {3,3}
   28: {1,1,4}
   30: {1,2,3}
   34: {1,7}
   36: {1,1,2,2}
   40: {1,1,1,3}
   46: {1,9}
   48: {1,1,1,1,2}
   52: {1,1,6}
   55: {3,5}
   62: {1,11}
   64: {1,1,1,1,1,1}

Crossrefs

These partitions are counted by A182616, even bisection of A086543.
Not requiring at least one odd part gives A300061.
Allowing partitions of odd numbers gives A366322.
A031368 lists primes of odd index.
A066207 ranks partitions with all even parts, counted by A035363.
A066208 ranks partitions with all odd parts, counted by A000009.
A112798 list prime indices, sum A056239.
A257991 counts odd prime indices, distinct A324966.
Cf. A000720, A001222, A003963, A047967, A257992, A324929, A358137.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100], EvenQ[Total[prix[#]]]&&Or@@OddQ/@prix[#]&]

A366748 Numbers k such that (sum of odd prime indices of k) = (sum of even prime indices of k).

Original entry on oeis.org

1, 12, 70, 90, 112, 144, 286, 325, 462, 520, 525, 594, 646, 675, 832, 840, 1045, 1080, 1326, 1334, 1344, 1666, 1672, 1728, 1900, 2142, 2145, 2294, 2465, 2622, 2695, 2754, 3040, 3432, 3465, 3509, 3526, 3900, 3944, 4186, 4255, 4312, 4455, 4845, 4864, 4900, 4982
Offset: 1

Views

Author

Gus Wiseman, Oct 23 2023

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.

Examples

			The terms together with their prime indices begin:
     1: {}
    12: {1,1,2}
    70: {1,3,4}
    90: {1,2,2,3}
   112: {1,1,1,1,4}
   144: {1,1,1,1,2,2}
   286: {1,5,6}
   325: {3,3,6}
   462: {1,2,4,5}
   520: {1,1,1,3,6}
   525: {2,3,3,4}
   594: {1,2,2,2,5}
   646: {1,7,8}
   675: {2,2,2,3,3}
   832: {1,1,1,1,1,1,6}
   840: {1,1,1,2,3,4}
For example, 525 has prime indices {2,3,3,4}, and 3+3 = 2+4, so 525 is in the sequence.

Crossrefs

For prime factors instead of indices we have A019507.
Partitions of this type are counted by A239261.
For count instead of sum we have A325698, distinct A325700.
The LHS (sum of odd prime indices) is A366528, triangle A113685.
The RHS (sum of even prime indices) is A366531, triangle A113686.
These are the positions of zeros in A366749.
A000009 counts partitions into odd parts, ranked by A066208.
A035363 counts partitions into even parts, ranked by A066207.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A257991 counts odd prime indices, even A257992.
A300061 lists numbers with even sum of prime indices, odd A300063.
Cf. A028260, A045931, A106529, A239241, A241638, A365067, A366533.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000], Total[Select[prix[#],OddQ]]==Total[Select[prix[#],EvenQ]]&]

Formula

These are numbers k such that A346697(k) = A346698(k).

A366847 Numbers whose halved even prime indices are nonempty and relatively prime.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 91, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168, 171, 174
Offset: 1

Views

Author

Gus Wiseman, Oct 31 2023

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.
Consists of powers of 2 times elements of the odd restriction A366849.

Examples

			The even prime indices of 91 are {4,6}, halved {2,3}, which are relatively prime, so 91 is in the sequence.
The prime indices of 665 are {3,4,8}, even {4,8}, halved {2,4}, which are not relatively prime, so 665 is not in the sequence.
The terms together with their prime indices begin:
    3: {2}
    6: {1,2}
    9: {2,2}
   12: {1,1,2}
   15: {2,3}
   18: {1,2,2}
   21: {2,4}
   24: {1,1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   33: {2,5}
   36: {1,1,2,2}
   39: {2,6}
   42: {1,2,4}
   45: {2,2,3}
   48: {1,1,1,1,2}

Crossrefs

Including odd indices gives A289509, ones of A289508, counted by A000837.
The complement including odd indices is A318978, counted by A018783.
The partitions with these ranks are counted by A366845.
A version for odd indices A366846, counted by A366850.
The odd restriction is A366849.
A000041 counts integer partitions, strict A000009 (also into odds).
A035363 counts partitions into all even parts, ranks A066207.
A112798 lists prime indices, length A001222, sum A056239.
A162641 counts even prime exponents, odd A162642.
A257992 counts even prime indices, odd A257991.
A366528 adds up odd prime indices, partition triangle A113685.
A366531 = 2*A366533 adds up even prime indices, triangle A113686/A174713.
Cf. A000720, A055396, A061395, A066208, A168532, A302696, A302697, A325698, A366842, A366843, A366844, A366848.

Programs

  • Mathematica
    Select[Range[100],GCD@@Select[PrimePi/@First/@FactorInteger[#],EvenQ]/2==1&]

A367107 Numbers m not divisible by prime(bigomega(m)). Heinz numbers of integer partitions whose length is not a part (counted by A229816).

Original entry on oeis.org

3, 4, 5, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 52, 53, 54, 55, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 85
Offset: 1

Views

Author

Gus Wiseman, Nov 21 2023

Keywords

Comments

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.

Crossrefs

Partitions of this type are counted by A229816.
The complement is A325761, counted by A002865.
If length is not a subset-sum: A367225, count A367213, complement A367224.
A005117 ranks strict integer partitions, counted by A000009.
A066208 ranks partitions into odd parts, also counted by A000009.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A237667 counts sum-free partitions, ranks A364531.
A237668 counts sum-full partitions, sum-free A364532.
Cf. A000720, A088902, A093641, A106529, A110295, A325762.

Programs

  • Mathematica
    Select[Range[2,100],!Divisible[#,Prime[PrimeOmega[#]]]&]

A319522 Completely multiplicative with a(prime(2*k)) = prime(k) and a(prime(2*k-1)) = 1 for any k > 0 (where prime(k) denotes the k-th prime number).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 3, 1, 4, 1, 1, 2, 5, 3, 2, 1, 1, 4, 7, 1, 6, 1, 1, 2, 1, 5, 8, 3, 11, 2, 1, 1, 2, 1, 3, 4, 13, 7, 10, 1, 1, 6, 17, 1, 4, 1, 1, 2, 9, 1, 2, 5, 19, 8, 1, 3, 14, 11, 1, 2, 23, 1, 12, 1, 5, 2, 1, 1, 2, 3, 29, 4, 1, 13, 2, 7, 3, 10, 31, 1, 16, 1
Offset: 1

Views

Author

Rémy Sigrist, Sep 22 2018

Keywords

Comments

See A319521 for a similar sequence and additional comments.

Examples

			a(42) = a(prime(1)) * a(prime(2)) * a(prime(4)) = 1 * prime(1) * prime(2) = 6.

Crossrefs

Cf. A007814, A007949, A064989, A066208, A319521.

Programs

  • PARI
    a(n) = my (f=factor(n)); prod(i=1, #f~, my (pi=primepi(f[i,1])); if (pi%2==0, prime(pi/2)^f[i,2], 1))

Formula

a(n) = 1 iff n belongs to A066208.
a(n) <= n with equality iff n = 1.
A007814(a(n)) = A007949(n).
a(n) = A319521(A064989(n)).

A338469 Products of three odd prime numbers of odd index.

Original entry on oeis.org

125, 275, 425, 575, 605, 775, 935, 1025, 1175, 1265, 1331, 1445, 1475, 1675, 1705, 1825, 1955, 2057, 2075, 2255, 2425, 2575, 2585, 2635, 2645, 2725, 2783, 3175, 3179, 3245, 3425, 3485, 3565, 3685, 3725, 3751, 3925, 3995, 4015, 4175, 4301, 4475, 4565, 4715
Offset: 1

Views

Author

Gus Wiseman, Nov 08 2020

Keywords

Comments

Also Heinz numbers of integer partitions with 3 parts, all of which are odd and > 1. These partitions are counted by A001399.

Examples

			The sequence of terms together with their prime indices begins:
     125: {3,3,3}     1825: {3,3,21}    3425: {3,3,33}
     275: {3,3,5}     1955: {3,7,9}     3485: {3,7,13}
     425: {3,3,7}     2057: {5,5,7}     3565: {3,9,11}
     575: {3,3,9}     2075: {3,3,23}    3685: {3,5,19}
     605: {3,5,5}     2255: {3,5,13}    3725: {3,3,35}
     775: {3,3,11}    2425: {3,3,25}    3751: {5,5,11}
     935: {3,5,7}     2575: {3,3,27}    3925: {3,3,37}
    1025: {3,3,13}    2585: {3,5,15}    3995: {3,7,15}
    1175: {3,3,15}    2635: {3,7,11}    4015: {3,5,21}
    1265: {3,5,9}     2645: {3,9,9}     4175: {3,3,39}
    1331: {5,5,5}     2725: {3,3,29}    4301: {5,7,9}
    1445: {3,7,7}     2783: {5,5,9}     4475: {3,3,41}
    1475: {3,3,17}    3175: {3,3,31}    4565: {3,5,23}
    1675: {3,3,19}    3179: {5,7,7}     4715: {3,9,13}
    1705: {3,5,11}    3245: {3,5,17}    4775: {3,3,43}

Links

Crossrefs

A046316 allows all primes (strict: A046389).
A338471 allows all odd primes (strict: A307534).
A338556 is the version for evens (strict: A338557).
A000009 counts partitions into odd parts (strict: A000700).
A001399(n-3) counts 3-part partitions (strict: A001399(n-6)).
A005408 lists odds (strict: A056911).
A008284 counts partitions by sum and length.
A014311 is a ranking of 3-part compositions (strict: A337453).
A014612 lists Heinz numbers of 3-part partitions (strict: A007304).
A023023 counts 3-part relatively prime partitions (strict: A101271).
A066207 lists numbers with all even prime indices (strict: A258117).
A066208 lists numbers with all odd prime indices (strict: A258116).
A075818 lists even Heinz numbers of 3-part partitions (strict: A075819).
A285508 lists Heinz numbers of non-strict 3-part partitions.
Cf. A001221, A001222, A002620, A005117, A037144, A056239, A112798.

Programs

  • Maple
    N:= 10000: # for terms <= N
P0:= [seq(ithprime(i),i=3..numtheory:-pi(floor(N/25)),2)]:
sort(select(`<=`,[seq(seq(seq(P0[i]*P0[j]*P0[k],k=1..j),j=1..i),i=1..nops(P0))], N)); # Robert Israel, Nov 12 2020
  • Mathematica
    Select[Range[1,1000,2],PrimeOmega[#]==3&&OddQ[Times@@PrimePi/@First/@FactorInteger[#]]&]
  • PARI
    isok(m) = my(f=factor(m)); (m%2) && (bigomega(f)==3) && (#select(x->!(x%2), apply(primepi, f[,1]~)) == 0); \\ Michel Marcus, Nov 10 2020
  • Python
    from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A338469(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum((primepi(x//(k*m))+1>>1)-(b+1>>1)+1 for a,k in filter(lambda x:x[0]&1,enumerate(primerange(5,integer_nthroot(x,3)[0]+1),3)) for b,m in filter(lambda x:x[0]&1,enumerate(primerange(k,isqrt(x//k)+1),a))))
    return bisection(f,n,n) # Chai Wah Wu, Oct 18 2024

A341447 Heinz numbers of integer partitions whose only even part is the smallest.

Original entry on oeis.org

3, 7, 13, 15, 19, 29, 33, 37, 43, 51, 53, 61, 69, 71, 75, 77, 79, 89, 93, 101, 107, 113, 119, 123, 131, 139, 141, 151, 161, 163, 165, 173, 177, 181, 193, 199, 201, 217, 219, 221, 223, 229, 239, 249, 251, 255, 263, 271, 281, 287, 291, 293, 299, 309, 311, 317
Offset: 1

Views

Author

Gus Wiseman, Feb 13 2021

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers whose only even prime index (counting multiplicity) is the smallest.

Examples

			The sequence of partitions together with their Heinz numbers begins:
      3: (2)         77: (5,4)     165: (5,3,2)
      7: (4)         79: (22)      173: (40)
     13: (6)         89: (24)      177: (17,2)
     15: (3,2)       93: (11,2)    181: (42)
     19: (8)        101: (26)      193: (44)
     29: (10)       107: (28)      199: (46)
     33: (5,2)      113: (30)      201: (19,2)
     37: (12)       119: (7,4)     217: (11,4)
     43: (14)       123: (13,2)    219: (21,2)
     51: (7,2)      131: (32)      221: (7,6)
     53: (16)       139: (34)      223: (48)
     61: (18)       141: (15,2)    229: (50)
     69: (9,2)      151: (36)      239: (52)
     71: (20)       161: (9,4)     249: (23,2)
     75: (3,3,2)    163: (38)      251: (54)

Crossrefs

These partitions are counted by A087897, shifted left once.
Terms of A340933 can be factored into elements of this sequence.
The odd version is A341446.
A000009 counts partitions into odd parts, ranked by A066208.
A001222 counts prime factors.
A005843 lists even numbers.
A026804 counts partitions whose least part is odd, ranked by A340932.
A026805 counts partitions whose least part is even, ranked by A340933.
A027187 counts partitions with even length/max, ranked by A028260/A244990.
A031215 lists even-indexed primes.
A055396 selects least prime index.
A056239 adds up prime indices.
A058696 counts partitions of even numbers, ranked by A300061.
A061395 selects greatest prime index.
A066207 lists numbers with all even prime indices.
A112798 lists the prime indices of each positive integer.
Cf. A026804, A035363, A036554, A160786, A244991, A257991, A257992, A300272, A300063, A340854/A340855.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],EvenQ[First[primeMS[#]]]&&And@@OddQ[Rest[primeMS[#]]]&]

A342082 Numbers with an inferior odd divisor > 1.

Original entry on oeis.org

9, 12, 15, 18, 21, 24, 25, 27, 30, 33, 35, 36, 39, 40, 42, 45, 48, 49, 50, 51, 54, 55, 56, 57, 60, 63, 65, 66, 69, 70, 72, 75, 77, 78, 80, 81, 84, 85, 87, 90, 91, 93, 95, 96, 98, 99, 100, 102, 105, 108, 110, 111, 112, 114, 115, 117, 119, 120, 121, 123, 125
Offset: 1

Views

Author

Gus Wiseman, Mar 06 2021

Keywords

Comments

We define a divisor d|n to be inferior if d <= n/d. Inferior divisors are counted by A038548 and listed by A161906.
Numbers n with an odd prime factor <= sqrt(n). - Chai Wah Wu, Mar 09 2021

Examples

			The divisors > 1 of 72 are {2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72}, of which {3, 9} are odd and {2, 3, 4, 6, 8} are inferior, with intersection {3}, so 72 is in the sequence.

Links

Crossrefs

The strictly inferior version is the same with A001248 removed.
Positions of terms > 1 in A069288.
The superior version is A116882, with complement A116883.
The complement is A342081.
A006530 selects the greatest prime factor.
A020639 selects the smallest prime factor.
A038548 counts superior (or inferior) divisors, with strict case A056924.
- Odd -
A000009 counts partitions into odd parts, ranked by A066208.
A001227 counts odd divisors.
A026424 lists numbers with odd Omega.
A027193 counts odd-length partitions.
A058695 counts partitions of odd numbers.
A067659 counts strict partitions of odd length, ranked by A030059.
A340101 counts factorizations into odd factors; A340102 also has odd length.
A340854/A340855 cannot/can be factored with odd minimum factor.
A341594 counts strictly superior odd divisors
A341675 counts superior odd divisors.
- Inferior: A033676, A066839, A161906.
- Superior: A033677, A051283, A059172, A063538/A063539.
- Strictly Inferior A333805, A341674.
- Strictly Superior: A064052/A048098, A341645/A341646.
Cf. A000005, A000203, A001055, A001221, A001222, A001414, A207375, A244991, A300272, A340832, A340931.

Programs

  • Mathematica
    Select[Range[100],Function[n,Select[Divisors[n]//Rest,OddQ[#]&&#<=n/#&]!={}]]
  • PARI
    is(n) = #select(x -> x > 2 && x^2 <= n, factor(n)[, 1]) > 0; \\ Amiram Eldar, Nov 01 2024
  • Python
    from sympy import primefactors
A342082_list = [n for n in range(1,10**3) if len([p for p in primefactors(n) if p > 2 and p*p <= n]) > 0] # Chai Wah Wu, Mar 09 2021

A352493 Number of non-constant integer partitions of n into prime parts with prime multiplicities.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 3, 0, 1, 4, 5, 3, 1, 3, 5, 7, 3, 5, 6, 8, 8, 11, 7, 6, 8, 15, 14, 14, 10, 15, 17, 21, 18, 23, 20, 28, 25, 31, 27, 35, 32, 33, 37, 46, 41, 50, 45, 58, 56, 63, 59, 78, 69, 76, 81, 85, 80, 103, 107, 111, 114, 127
Offset: 0

Views

Author

Gus Wiseman, Mar 24 2022

Keywords

Examples

			The a(n) partitions for selected n (B = 11):
n = 10    16       19        20         25          28
   ---------------------------------------------------------------
    3322  5533     55333     7733       77722       BB33
          55222    55522     77222      5533333     BB222
          3322222  3333322   553322     5553322     775522
                   33322222  5522222    55333222    55533322
                             332222222  55522222    772222222
                                        333333322   3322222222222
                                        3333322222

Crossrefs

Constant partitions are counted by A001221, ranked by A000961.
Non-constant partitions are counted by A144300, ranked A024619.
The constant version is A230595, ranked by A352519.
This is the non-constant case of A351982, ranked by A346068.
These partitions are ranked by A352518.
A000040 lists the primes.
A000607 counts partitions into primes, ranked by A076610.
A001597 lists perfect powers, complement A007916.
A038499 counts partitions of prime length.
A053810 lists primes to primes.
A055923 counts partitions with prime multiplicities, ranked by A056166.
A257994 counts prime indices that are themselves prime.
A339218 counts powerful partitions into prime parts, ranked by A352492.
Cf. A000005, A007690, A031368, A035444, A052485, A056239, A066208, A089723, A114639, A320628, A330945.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], !SameQ@@#&&And@@PrimeQ/@#&& And@@PrimeQ/@Length/@Split[#]&]],{n,0,30}]

A356605 Number of integer compositions of n into odd parts covering an interval of odd positive integers.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 6, 10, 15, 26, 41, 65, 104, 164, 262, 424, 687, 1112, 1792, 2898, 4677, 7556, 12197, 19699, 31836, 51466, 83234, 134593, 217674, 352057, 569452, 921165, 1490173, 2410784, 3900288, 6310436, 10210358, 16521108, 26733020, 43258086, 69999295
Offset: 0

Views

Author

Gus Wiseman, Aug 31 2022

Keywords

Examples

			The a(1) = 1 through a(8) = 15 compositions:
  (1)  (11)  (3)    (13)    (5)      (33)      (7)        (35)
             (111)  (31)    (113)    (1113)    (133)      (53)
                    (1111)  (131)    (1131)    (313)      (1133)
                            (311)    (1311)    (331)      (1313)
                            (11111)  (3111)    (11113)    (1331)
                                     (111111)  (11131)    (3113)
                                               (11311)    (3131)
                                               (13111)    (3311)
                                               (31111)    (111113)
                                               (1111111)  (111131)
                                                          (111311)
                                                          (113111)
                                                          (131111)
                                                          (311111)
                                                          (11111111)

Crossrefs

These compositions are ranked by the intersection of A060142 and A356841.
Before restricting to odds we have A107428, initial A107429.
The not necessarily gapless version is A324969 (essentially A000045).
The strict case is A332032.
The initial case is A356604.
The case of partitions is A356737, initial A053251 (ranked by A356232).
A000041 counts partitions, compositions A011782.
A066208 lists numbers with all odd prime indices, counted by A000009.
A073491 lists numbers with gapless prime indices, initial A055932.
Cf. A001227, A066205, A137921, A333217, A356224, A356846.

Programs

  • Mathematica
    nogapQ[m_]:=m=={}||Union[m]==Range[Min[m],Max[m]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], And@@OddQ/@#&&nogapQ[(#+1)/2]&]],{n,0,15}]

Extensions

More terms from Alois P. Heinz, Sep 01 2022
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