A355536 -id:A355536 - cp's OEIS frontend

A048675 If n = p_i^e_i * ... * p_k^e_k, p_i < ... < p_k primes (with p_i = prime(i)), then a(n) = (1/2) * (e_i * 2^i + ... + e_k * 2^k).

0, 1, 2, 2, 4, 3, 8, 3, 4, 5, 16, 4, 32, 9, 6, 4, 64, 5, 128, 6, 10, 17, 256, 5, 8, 33, 6, 10, 512, 7, 1024, 5, 18, 65, 12, 6, 2048, 129, 34, 7, 4096, 11, 8192, 18, 8, 257, 16384, 6, 16, 9, 66, 34, 32768, 7, 20, 11, 130, 513, 65536, 8, 131072, 1025, 12, 6, 36, 19

Offset: 1

Antti Karttunen, Jul 14 1999

nonn

The original motivation for this sequence was to encode the prime factorization of n in the binary representation of a(n), each such representation being unique as long as this map is restricted to A005117 (squarefree numbers, resulting a permutation of nonnegative integers A048672) or any of its subsequence, resulting an injective function like A048623 and A048639.
However, also the restriction to A260443 (not all terms of which are squarefree) results a permutation of nonnegative integers, namely A001477, the identity permutation.
When a polynomial with nonnegative integer coefficients is encoded with the prime factorization of n (e.g., as in A206296, A260443), then a(n) gives the evaluation of that polynomial at x=2.
The primitive completely additive integer sequence that satisfies a(n) = a(A225546(n)), n >= 1. By primitive, we mean that if b is another such sequence, then there is an integer k such that b(n) = k * a(n) for all n >= 1. - Peter Munn, Feb 03 2020
If the binary rank of an integer partition y is given by Sum_i 2^(y_i-1), and the Heinz number is Product_i prime(y_i), then a(n) is the binary rank of the integer partition with Heinz number n. Note the function taking a set s to Sum_i 2^(s_i-1) is the inverse of A048793 (binary indices), and the function taking a multiset m to Product_i prime(m_i) is the inverse of A112798 (prime indices). - Gus Wiseman, May 22 2024

			From _Gus Wiseman_, May 22 2024: (Start)
The A018819(7) = 6 cases of binary rank 7 are the following, together with their prime indices:
   30: {1,2,3}
   40: {1,1,1,3}
   54: {1,2,2,2}
   72: {1,1,1,2,2}
   96: {1,1,1,1,1,2}
  128: {1,1,1,1,1,1,1}
(End)

Antti Karttunen, Table of n, a(n) for n = 1..1024
Hans Havermann, Factorization of the first 10000 terms, in format [[primes], [exponents]]

Row 2 of A104244.
Similar logarithmic functions: A001414, A056239, A090880, A289506, A293447.
Left inverse of the following sequences: A000079, A019565, A038754, A068911, A134683, A260443, A332824.
A003961, A028234, A032742, A055396, A064989, A067029, A225546, A297845 are used to express relationship between terms of this sequence.
Cf. also A048623, A048676, A099884, A277896 and tables A277905, A285325.
Cf. A297108 (Möbius transform), A332813 and A332823 [= a(n) mod 3].
Pairs of sequences (f,g) that satisfy a(f(n)) = g(n), possibly with offset change: (A000203,A331750), (A005940,A087808), (A007913,A248663), (A007947,A087207), (A097248,A048675), (A206296,A000129), (A248692,A056239), (A283477,A005187), (A284003,A006068), (A285101,A028362), (A285102,A068052), (A293214,A001065), (A318834,A051953), (A319991,A293897), (A319992,A293898), (A320017,A318674), (A329352,A069359), (A332461,A156552), (A332462,A156552), (A332825,A000010) and apparently (A163511,A135529).
See comments/formulas in A277333, A331591, A331740 giving their relationship to this sequence.
The formula section details how the sequence maps the terms of A329050, A329332.
A277892, A322812, A322869, A324573, A324575 give properties of the n-th term of this sequence.
The term k appears A018819(k) times.
The inverse transformation is A019565 (Heinz number of binary indices).
The version for distinct prime indices is A087207.
Numbers k such that a(k) is prime are A277319, counts A372688.
Grouping by image gives A277905.
A014499 lists binary indices of prime numbers.
A061395 gives greatest prime index, least A055396.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Binary indices:
- listed A048793, sum A029931
- reversed A272020
- opposite A371572, sum A230877
- length A000120, complement A023416
- min A001511, opposite A000012
- max A070939, opposite A070940
- complement A368494, sum A359400
- opposite complement A371571, sum A359359
Cf. A000720, A035100, A071814, A096111, A225620, A243055, A304818, A355536, A358136, A372890.
Cf. A087207, A097248.

nthprime := proc(n) local i; if(isprime(n)) then for i from 1 to 1000000 do if(ithprime(i) = n) then RETURN(i); fi; od; else RETURN(0); fi; end; # nthprime(2) = 1, nthprime(3) = 2, nthprime(5) = 3, etc. - this is also A049084.
A048675 := proc(n) local s,d; s := 0; for d in ifactors(n)[ 2 ] do s := s + d[ 2 ]*(2^(nthprime(d[ 1 ])-1)); od; RETURN(s); end;
# simpler alternative
f:= n -> add(2^(numtheory:-pi(t[1])-1)*t[2], t=ifactors(n)[2]):
map(f, [$1..100]); # Robert Israel, Oct 10 2016

a[1] = 0; a[n_] := Total[ #[[2]]*2^(PrimePi[#[[1]]]-1)& /@ FactorInteger[n] ]; Array[a, 100] (* Jean-François Alcover, Mar 15 2016 *)

a(n) = my(f = factor(n)); sum(k=1, #f~, f[k,2]*2^primepi(f[k,1]))/2; \\ Michel Marcus, Oct 10 2016

\\ The following program reconstructs terms (e.g. for checking purposes) from the factorization file prepared by Hans Havermann:
v048675sigs = readvec("a048675.txt");
A048675(n) = if(n<=2,n-1,my(prsig=v048675sigs[n],ps=prsig[1],es=prsig[2]); prod(i=1,#ps,ps[i]^es[i])); \\ Antti Karttunen, Feb 02 2020

from sympy import factorint, primepi
def a(n):
    if n==1: return 0
    f=factorint(n)
    return sum([f[i]*2**(primepi(i) - 1) for i in f])
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jun 19 2017

a(1) = 0, a(n) = 1/2 * (e1*2^i1 + e2*2^i2 + ... + ez*2^iz) if n = p_{i1}^e1*p_{i2}^e2*...*p_{iz}^ez, where p_i is the i-th prime. (e.g. p_1 = 2, p_2 = 3).
Totally additive with a(p^e) = e * 2^(PrimePi(p)-1), where PrimePi(n) = A000720(n). [Missing factor e added to the comment by Antti Karttunen, Jul 29 2015]
From Antti Karttunen, Jul 29 2015: (Start)
a(1) = 0; for n > 1, a(n) = 2^(A055396(n)-1) + a(A032742(n)). [Where A055396(n) gives the index of the smallest prime dividing n and A032742(n) gives the largest proper divisor of n.]
a(1) = 0; for n > 1, a(n) = (A067029(n) * (2^(A055396(n)-1))) + a(A028234(n)).
Other identities. For all n >= 0:
a(A019565(n)) = n.
a(A260443(n)) = n.
a(A206296(n)) = A000129(n).
a(A005940(n+1)) = A087808(n).
a(A007913(n)) = A248663(n).
a(A007947(n)) = A087207(n).
a(A283477(n)) = A005187(n).
a(A284003(n)) = A006068(n).
a(A285101(n)) = A028362(1+n).
a(A285102(n)) = A068052(n).
Also, it seems that a(A163511(n)) = A135529(n) for n >= 1. (End)
a(1) = 0, a(2n) = 1+a(n), a(2n+1) = 2*a(A064989(2n+1)). - Antti Karttunen, Oct 11 2016
From Peter Munn, Jan 31 2020: (Start)
a(n^2) = a(A003961(n)) = 2 * a(n).
a(A297845(n,k)) = a(n) * a(k).
a(n) = a(A225546(n)).
a(A329332(n,k)) = n * k.
a(A329050(n,k)) = 2^(n+k).
(End)
From Antti Karttunen, Feb 02-25 2020, Feb 01 2021: (Start)
a(n) = Sum_{d|n} A297108(d) = Sum_{d|A225546(n)} A297108(d).
a(n) = a(A097248(n)).
For n >= 2:
A001221(a(n)) = A322812(n), A001222(a(n)) = A277892(n).
A000203(a(n)) = A324573(n), A033879(a(n)) = A324575(n).
For n >= 1, A331750(n) = a(A000203(n)).
For n >= 1, the following chains hold:
A293447(n) >= a(n) >= A331740(n) >= A331591(n).
a(n) >= A087207(n) >= A248663(n).
(End)
a(n) = A087207(A097248(n)). - Flávio V. Fernandes, Jul 16 2025

Entry revised by Antti Karttunen, Jul 29 2015

More linking formulas added by Antti Karttunen, Apr 18 2017

A358137 Heinz number of the partial sums of the prime indices of n.

Original entry on oeis.org

1, 2, 3, 6, 5, 10, 7, 30, 21, 14, 11, 42, 13, 22, 33, 210, 17, 110, 19, 66, 39, 26, 23, 330, 65, 34, 273, 78, 29, 130, 31, 2310, 51, 38, 85, 546, 37, 46, 57, 390, 41, 170, 43, 102, 357, 58, 47, 2730, 133, 238, 69, 114, 53, 1870, 95, 510, 87, 62, 59, 714, 61

Offset: 1

Gus Wiseman, Oct 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.

			The terms together with their prime indices begin:
      1: {}
      2: {1}
      3: {2}
      6: {1,2}
      5: {3}
     10: {1,3}
      7: {4}
     30: {1,2,3}
     21: {2,4}
     14: {1,4}
     11: {5}
     42: {1,2,4}
     13: {6}
     22: {1,5}
     33: {2,5}
    210: {1,2,3,4}
     17: {7}
    110: {1,3,5}

The sorted version is A325362.
The prime indices are rows of A358136, partial sums of rows of A112798.
A000040 lists the primes.
A000041 counts partitions, strict A000009.
A003963 multiplies prime indices.
A056239 adds up prime indices.
Cf. A000720, A001221, A001222, A215366, A296150, A318283, A355536, A358134.

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

A001222(a(n)) = A001222(n).

A358136 Irregular triangle read by rows whose n-th row lists the partial sums of the prime indices of n (row n of A112798).

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 3, 4, 1, 2, 3, 2, 4, 1, 4, 5, 1, 2, 4, 6, 1, 5, 2, 5, 1, 2, 3, 4, 7, 1, 3, 5, 8, 1, 2, 5, 2, 6, 1, 6, 9, 1, 2, 3, 5, 3, 6, 1, 7, 2, 4, 6, 1, 2, 6, 10, 1, 3, 6, 11, 1, 2, 3, 4, 5, 2, 7, 1, 8, 3, 7, 1, 2, 4, 6, 12, 1, 9, 2, 8, 1, 2, 3, 6, 13

Offset: 2

Gus Wiseman, Oct 31 2022

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.

			Triangle begins:
   2: 1
   3: 2
   4: 1 2
   5: 3
   6: 1 3
   7: 4
   8: 1 2 3
   9: 2 4
  10: 1 4
  11: 5
  12: 1 2 4
  13: 6
  14: 1 5
  15: 2 5
  16: 1 2 3 4
  17: 7
  18: 1 3 5

Row-lengths are A001222.
First element in each row is A055396.
Last element in each row is A056239.
Rows are the partial sums of rows of A112798.
Row-sums are A318283.
Sorted Heinz numbers of the rows are A325362.
The version for standard compositions is A358134.
Rows are ranked by A358137.
A000041 counts partitions, strict A000009.
A003963 multiplies prime indices.
A056239 adds up prime indices.
Cf. A000720, A001221, A355536, A358133.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Accumulate[primeMS[n]],{n,30}]

A359361 Irregular triangle read by rows whose n-th row lists the partial sums of the integer partition with Heinz number n.

Original entry on oeis.org

1, 2, 1, 2, 3, 2, 3, 4, 1, 2, 3, 2, 4, 3, 4, 5, 2, 3, 4, 6, 4, 5, 3, 5, 1, 2, 3, 4, 7, 2, 4, 5, 8, 3, 4, 5, 4, 6, 5, 6, 9, 2, 3, 4, 5, 3, 6, 6, 7, 2, 4, 6, 4, 5, 6, 10, 3, 5, 6, 11, 1, 2, 3, 4, 5, 5, 7, 7, 8, 4, 7, 2, 4, 5, 6, 12, 8, 9, 6, 8, 3, 4, 5, 6, 13

Offset: 2

Gus Wiseman, Dec 30 2022

The partial sums of a sequence (a, b, c, ...) are (a, a+b, a+b+c, ...).

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The partition with Heinz number n is the reversed n-th row of A112798.

			Triangle begins:
   2: 1
   3: 2
   4: 1 2
   5: 3
   6: 2 3
   7: 4
   8: 1 2 3
   9: 2 4
  10: 3 4
  11: 5
  12: 2 3 4
  13: 6
  14: 4 5
  15: 3 5
  16: 1 2 3 4
For example, the integer partition with Heinz number 90 is (3,2,2,1), so row n = 90 is (3,5,7,8).

Row-lengths are A001222.
The version for standard compositions is A048793, non-reversed A358134.
Last element in each row is A056239.
First element in each row is A061395
Rows are the partial sums of rows of A296150.
Row-sums are A304818.
A reverse version is A358136, row sums A318283, Heinz numbers A358137.
The sorted Heinz numbers of rows are A359397.
A000041 counts partitions, strict A000009.
A112798 lists prime indices, product A003963.
A355536 lists differences of prime indices.
Cf. A000720, A001221, A055396, A261079, A325362.

T:= n-> ListTools[PartialSums](sort([seq(numtheory
       [pi](i[1])$i[2], i=ifactors(n)[2])], `>`))[]:
seq(T(n), n=2..50);  # Alois P. Heinz, Jan 01 2023

Table[Accumulate[Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{n,2,30}]

A242628 Irregular table enumerating partitions; n-th row has partitions in previous row with each part incremented, followed by partitions in previous row with an additional part of size 1.

Original entry on oeis.org

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

Offset: 1

Franklin T. Adams-Watters, May 19 2014

This can be calculated using the binary expansion of n; see the PARI program.
The n-th row consists of all partitions with hook size (maximum + number of parts - 1) equal to n.
The partitions in row n of this sequence are the conjugates of the partitions in row n of A125106 taken in reverse order.
Row n is also the reversed partial sums plus one of the n-th composition in standard order (A066099) minus one. - Gus Wiseman, Nov 07 2022

			The table starts:
  1;
  2; 1,1;
  3; 2,2; 2,1; 1,1,1;
  4; 3,3; 3,2; 2,2,2; 3,1 2,2,1 2,1,1 1,1,1,1;
  ...

Alois P. Heinz, Rows n = 1..12, flattened
Gus Wiseman, Statistics, classes, and transformations of standard compositions

Cf. A241596 (another version of this list of partitions), A125106, A240837, A112531, A241597 (compositions).
For other schemes to list integer partitions, please see for example A227739, A112798, A241918, A114994.
First element in each row is A008687.
Last element in each row is A065120.
Heinz numbers of rows are A253565.
Another version is A358134.
Cf. A000120, A029837, A029931, A030303, A066099, A070939, A253566, A355536, A358135, A358136.

b:= proc(n) option remember; `if`(n=1, [[1]],
      [map(x-> map(y-> y+1, x), b(n-1))[],
       map(x-> [x[], 1], b(n-1))[]])
    end:
T:= n-> map(x-> x[], b(n))[]:
seq(T(n), n=1..7);  # Alois P. Heinz, Sep 25 2015

T[1] = {{1}};
T[n_] := T[n] = Join[T[n-1]+1, Append[#, 1]& /@ T[n-1]];
Array[T, 7] // Flatten (* Jean-François Alcover, Jan 25 2021 *)

apart(n) = local(r=[1]); while(n>1,if(n%2==0,for(k=1,#r,r[k]++),r=concat(r,[1]));n\=2);r \\ Generates n-th partition.

A372427 Numbers whose binary indices and prime indices have the same sum.

Original entry on oeis.org

19, 33, 34, 69, 74, 82, 130, 133, 305, 412, 428, 436, 533, 721, 755, 808, 917, 978, 1036, 1058, 1062, 1121, 1133, 1143, 1341, 1356, 1630, 1639, 1784, 1807, 1837, 1990, 2057, 2115, 2130, 2133, 2163, 2260, 2324, 2328, 2354, 2358, 2512, 2534, 2627, 2771, 2825

Offset: 1

Gus Wiseman, May 01 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

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 binary indices of 130 are {2,8}, and the prime indices are {1,3,6}. Both sum to 10, so 130 is in the sequence.
The terms together with their prime indices begin:
   19: {8}
   33: {2,5}
   34: {1,7}
   69: {2,9}
   74: {1,12}
   82: {1,13}
  130: {1,3,6}
  133: {4,8}
  305: {3,18}
  412: {1,1,27}
  428: {1,1,28}
The terms together with their binary expansions and binary indices begin:
   19:      10011 ~ {1,2,5}
   33:     100001 ~ {1,6}
   34:     100010 ~ {2,6}
   69:    1000101 ~ {1,3,7}
   74:    1001010 ~ {2,4,7}
   82:    1010010 ~ {2,5,7}
  130:   10000010 ~ {2,8}
  133:   10000101 ~ {1,3,8}
  305:  100110001 ~ {1,5,6,9}
  412:  110011100 ~ {3,4,5,8,9}
  428:  110101100 ~ {3,4,6,8,9}

John Tyler Rascoe, Table of n, a(n) for n = 1..10000

For length instead of sum we get A071814.
Positions of zeros in A372428.
For maximum instead of sum we have A372436.
A003963 gives product of prime indices.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
A326031 gives weight of the set-system with BII-number n.
Cf. A000720, A001221, A014499, A030101, A066099, A304818, A318283, A355536, A359401, A359402, A372429-A372433, A372441.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],Total[prix[#]]==Total[bix[#]]&]

A372436 Numbers whose binary indices and prime indices have the same maximum.

Original entry on oeis.org

3, 5, 14, 22, 39, 52, 68, 85, 102, 119, 133, 152, 171, 190, 209, 228, 247, 276, 299, 322, 345, 368, 391, 414, 437, 460, 483, 506, 522, 551, 580, 609, 638, 667, 696, 725, 754, 783, 812, 841, 870, 928, 957, 986, 1015, 1054, 1085, 1116, 1178, 1209, 1240, 1302

Offset: 1

Gus Wiseman, May 04 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
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.
Note that a number's binary and prime indices cannot have the same minimum; see A372437.

			The binary indices of 345 are {1,4,5,7,9}, and the prime indices are {2,3,9}. Both have maximum 9, so 345 is in the sequence.
The terms together with their prime indices begin:
     3: {2}
     5: {3}
    14: {1,4}
    22: {1,5}
    39: {2,6}
    52: {1,1,6}
    68: {1,1,7}
    85: {3,7}
   102: {1,2,7}
   119: {4,7}
   133: {4,8}
   152: {1,1,1,8}
   171: {2,2,8}
The terms together with their binary expansions and binary indices begin:
     3:           11 ~ {1,2}
     5:          101 ~ {1,3}
    14:         1110 ~ {2,3,4}
    22:        10110 ~ {2,3,5}
    39:       100111 ~ {1,2,3,6}
    52:       110100 ~ {3,5,6}
    68:      1000100 ~ {3,7}
    85:      1010101 ~ {1,3,5,7}
   102:      1100110 ~ {2,3,6,7}
   119:      1110111 ~ {1,2,3,5,6,7}
   133:     10000101 ~ {1,3,8}
   152:     10011000 ~ {4,5,8}
   171:     10101011 ~ {1,2,4,6,8}

For length instead of maximum we have A071814.
For sum instead of maximum we have A372427.
Positions of zeros in A372442, for minimum instead of maximum A372437.
A003963 gives product of prime indices.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A014499, A030101, A066099, A096111, A304818, A355536, A359401, A359402, A372428-A372433, A372441.

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max[prix[#]]==Max[bix[#]]&]

A070939(a(n)) = A061395(a(n)).

A355534 Irregular triangle read by rows where row n lists the augmented differences of the reversed prime indices of n.

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 4, 1, 1, 1, 1, 2, 3, 1, 5, 2, 1, 1, 6, 4, 1, 2, 2, 1, 1, 1, 1, 7, 1, 2, 1, 8, 3, 1, 1, 3, 2, 5, 1, 9, 2, 1, 1, 1, 1, 3, 6, 1, 1, 1, 2, 4, 1, 1, 10, 2, 2, 1, 11, 1, 1, 1, 1, 1, 4, 2, 7, 1, 2, 3, 1, 2, 1, 1, 12, 8, 1, 5, 2, 3, 1, 1, 1

Offset: 2

Gus Wiseman, Jul 12 2022

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).
One could argue that row n = 1 is empty, but adding it changes only the offset, not the data.

			Triangle begins:
   2: 1
   3: 2
   4: 1 1
   5: 3
   6: 2 1
   7: 4
   8: 1 1 1
   9: 1 2
  10: 3 1
  11: 5
  12: 2 1 1
  13: 6
  14: 4 1
  15: 2 2
  16: 1 1 1 1
For example, the reversed prime indices of 825 are (5,3,3,2), which have augmented differences (3,1,2,2).

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

Crossrefs found in the link are not repeated here.
Row-lengths are A001222.
Row-sums are A252464
Other similar triangles are A287352, A091602.
Constant rows have indices A307824.
The Heinz numbers of the rows are A325351.
Strict rows have indices A325366.
Row minima are A355531, non-augmented A355524, also A355525.
Row maxima are A355535, non-augmented A286470, also A355526.
The non-augmented version is A355536, also A355533.
A112798 lists prime indices, sum A056239.
Cf. A066312, A124010, A129654, A243055, A243056, A325352, A325394, A355523, A355528, A355531.

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

A358134 Triangle read by rows whose n-th row lists the partial sums of the n-th composition in standard order (row n of A066099).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 31 2022

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.

			Triangle begins:
  1
  2
  1 2
  3
  2 3
  1 3
  1 2 3
  4
  3 4
  2 4
  2 3 4
  1 4
  1 3 4
  1 2 4
  1 2 3 4

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

See link for sequences related to standard compositions.
First element in each row is A065120.
Rows are the partial sums of rows of A066099.
Last element in each row is A070939.
An adjusted version is A242628, ranked by A253565.
The first differences instead of partial sums are A358133.
The version for Heinz numbers of partitions is A358136, ranked by A358137.
Row sums are A359042.
A011782 counts compositions.
A351014 counts distinct runs in standard compositions.
A358135 gives last minus first of standard compositions.
Cf. A000120, A001511, A029837, A029931, A048896, A070939, A133494, A253566, A355536, A357135.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Join@@Table[Accumulate[stc[n]],{n,100}]

A372442 (Greatest binary index of n) minus (greatest prime index of n).

Original entry on oeis.org

1, 0, 2, 0, 1, -1, 3, 2, 1, -1, 2, -2, 0, 1, 4, -2, 3, -3, 2, 1, 0, -4, 3, 2, -1, 3, 1, -5, 2, -6, 5, 1, -1, 2, 4, -6, -2, 0, 3, -7, 2, -8, 1, 3, -3, -9, 4, 2, 3, -1, 0, -10, 4, 1, 2, -2, -4, -11, 3, -12, -5, 2, 6, 1, 2, -12, 0, -2, 3, -13, 5, -14, -5, 4, -1

Offset: 2

Gus Wiseman, May 07 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

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 sum instead of maximum we have A372428, zeros A372427.
Positions of zeros are A372436.
For minimum instead of maximum we have A372437, zeros {}.
For length instead of maximum we have A372441, zeros A071814.
Positions of odd terms are A372588, even A372589.
A019565 gives Heinz number of binary indices, adjoint A048675.
A029837 gives greatest binary index, least A001511.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A061395 gives greatest prime index, least A055396.
A070939 gives length of binary expansion.
A112798 lists prime indices, length A001222, reverse A296150, sum A056239.
Cf. A000720, A003963, A014499, A174090, A243055, A355536, A359495, A372429-A372432, A372589-A372591.

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Max[bix[n]]-Max[prix[n]],{n,2,100}]

a(n) = A070939(n) - A061395(n) = A029837(n) - A061395(n) for n > 1.

cp's OEIS Frontend

A048675 If n = p_i^e_i * ... * p_k^e_k, p_i < ... < p_k primes (with p_i = prime(i)), then a(n) = (1/2) * (e_i * 2^i + ... + e_k * 2^k).

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A358137 Heinz number of the partial sums of the prime indices of n.

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A358136 Irregular triangle read by rows whose n-th row lists the partial sums of the prime indices of n (row n of A112798).

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A359361 Irregular triangle read by rows whose n-th row lists the partial sums of the integer partition with Heinz number n.

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A242628 Irregular table enumerating partitions; n-th row has partitions in previous row with each part incremented, followed by partitions in previous row with an additional part of size 1.

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A372427 Numbers whose binary indices and prime indices have the same sum.

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A372436 Numbers whose binary indices and prime indices have the same maximum.

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A355534 Irregular triangle read by rows where row n lists the augmented differences of the reversed prime indices of n.

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