A070776 -id:A070776 - cp's OEIS frontend

A208507 Reordering of A070776 such that the cyclotomic polynomial Phi(A070776, m) is in sorted order for any integer m >= 2.

1, 2, 6, 4, 3, 10, 12, 8, 5, 14, 18, 9, 7, 20, 24, 16, 22, 11, 26, 28, 36, 13, 34, 40, 48, 32, 17, 38, 54, 27, 19, 44, 50, 25, 46, 23, 52, 56, 72, 58, 29, 62, 31, 68, 80, 96, 64, 74, 76, 108, 37, 82, 88, 100, 41, 86, 98, 49, 43, 92, 94, 47, 104, 112, 144

Offset: 1

Lei Zhou, Feb 27 2012

When p is an odd prime number and i >= 1, j >= 1, the cyclotomic polynomial
Phi(2^i*p^j, k)
  = Phi(2p,k^(2^(i-1)*p^(j-1)))
  = Phi(p, -(k^(2^(i-1)*p^(j-1))))
  = (111.....1) (p ones) base -(k^(2^(i-1)*p^(j-1)))
Phi(p^j, k)
  = Phi(p, k^(p^(j-1)))
  = (111.....1) (p ones) base k^(p^(j-1)).
For odd prime p >= 3, the above numbers can be called "Very Generic Repdigit Numbers".
This sequence is a subsequence of A206225.
The Mathematica program is rewritten to be able to generate this sequence to an arbitrary EulerPhi boundary.

			The first 20 elements of A206225 are 1, 2, 6, 4, 3, 10, 12, 8, 5, 14, 18, 9, 7, 15, 20, 24, 16, 30, 22, 11.
Among these, 15 = 3 * 5 and 30 = 2 * 3 * 5 cannot be written in the form 2^i*p^j and are thus rejected. So the first 18 terms of this sequence are 1, 2, 6, 4, 3, 10, 12, 8, 5, 14, 18, 9, 7, 20, 24, 16, 22, 11.

Cf. A070776, A206225.

eb = 48; phiinv[n_, pl_] := Module[{i, p, e, pe, val}, If[pl == {}, Return[If[n == 1, {1}, {}]]]; val = {}; p = Last[pl]; For[e = 0; pe = 1, e == 0 || Mod[n, (p - 1) pe/p] == 0, e++; pe *= p, val = Join[val, pe*phiinv[If[e == 0, n, n*p/pe/(p - 1)], Drop[pl, -1]]]]; Sort[val]]; phiinv[n_] := phiinv[n, Select[1 + Divisors[n], PrimeQ]]; elim =  Max[Table[Max[phiinv[n]], {n, 2, eb, 2}]]; t = Select[Range[elim], (a = FactorInteger[#]; b = Length[a]; ((b == 1) || ((b == 2) && (a[[1]][[1]] == 2))) && (EulerPhi[#] <= eb)) &]; SortBy[t, Cyclotomic[#, 2] &]

A002541 a(n) = Sum_{k=1..n-1} floor((n-k)/k).

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 12, 14, 17, 18, 23, 24, 27, 30, 34, 35, 40, 41, 46, 49, 52, 53, 60, 62, 65, 68, 73, 74, 81, 82, 87, 90, 93, 96, 104, 105, 108, 111, 118, 119, 126, 127, 132, 137, 140, 141, 150, 152, 157, 160, 165, 166, 173, 176, 183, 186, 189, 190, 201, 202, 205

Offset: 1

N. J. A. Sloane

Number of pairs (a, b) with 1 <= a < b <= n, a | b.
The sequence shows how many digit "skips" there have been from 2 to n, a skip being either a prime factor or product thereof. Every time you have a place where you have X skips and the next skip value is X+1, you will have a prime number since a prime number will only add exactly one more skip to get to it. a(n) = Sum_{x=2..n} floor(n/x) - Sum_{x=2..n-1} floor( (n-1)/x) = 1 when prime. - Marius-Paul Dumitrean (marius(AT)neldor.com), Feb 19 2007
A027749(a(n)+1) = n; A027749(a(n)+2) = A020639(n+1). - Reinhard Zumkeller, Nov 22 2003
Number of partitions of n into exactly 2 types of part, where one part is 1, e.g., n=7 gives 1111111, 111112, 11122, 1222, 11113, 133, 1114, 115 and 16, so a(7)=9. - Jon Perry, May 26 2004
The sequence of partial sums of A032741. Idea of proof: floor((n-k)/k) - floor((n-k-1)/k) only increases by 1 when k | n. - George Beck, Feb 12 2012
Also the number of integer partitions of n whose non-1 parts are all equal and with at least one non-1 part. - Gus Wiseman, Oct 07 2018

			From _Gus Wiseman_, Oct 07 2018: (Start)
The integer partitions whose non-1 parts are all equal and with at least one non-1 part:
  (2)  (3)   (4)    (5)     (6)      (7)       (8)        (9)
       (21)  (22)   (41)    (33)     (61)      (44)       (81)
             (31)   (221)   (51)     (331)     (71)       (333)
             (211)  (311)   (222)    (511)     (611)      (441)
                    (2111)  (411)    (2221)    (2222)     (711)
                            (2211)   (4111)    (3311)     (6111)
                            (3111)   (22111)   (5111)     (22221)
                            (21111)  (31111)   (22211)    (33111)
                                     (211111)  (41111)    (51111)
                                               (221111)   (222111)
                                               (311111)   (411111)
                                               (2111111)  (2211111)
                                                          (3111111)
                                                          (21111111)
(End)

J. P. Gram, Undersoegelser angaaende maengden af primtal under en given graense, Det Kongelige Danskevidenskabernes Selskabs Skrifter, series 6, vol. 2 (1884), 183-288; see Tab. VII: Vaerdier af Funktionen psi(n) og andre numeriske Funktioner, pp. 281-288, especially p. 281.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Antidiagonal sums of array A003988. Antidiagonal sums of A004199.
Cf. A000005, A006218, A020639, A027749, A032741.
Cf. A003238, A070776, A126656, A320224, A320225, A320226.

a002541 n = sum $ zipWith div [n - 1, n - 2 ..] [1 .. n - 1]
-- Reinhard Zumkeller, Jul 05 2013

a:= proc(n) option remember; `if`(n<2, 0,
      numtheory[tau](n)-1+a(n-1))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 12 2021

Table[Sum[Floor[(n-k)/k],{k,n-1}],{n,100}] (* Harvey P. Dale, May 02 2011 *)

a(n)=sum(k=1,n-1, n\k-1) \\ Charles R Greathouse IV, Feb 07 2017

first(n)=my(v=vector(n),s); for(k=1,n, v[k]=-k+s+=numdiv(k)); v \\ Charles R Greathouse IV, Feb 07 2017

from math import isqrt
def A002541(n): return (sum(n//k for k in range(1,isqrt(n)+1))<<1)-isqrt(n)**2-n # Chai Wah Wu, Oct 20 2023

a(n) = -n + Sum_{k=1..n} tau(k). - Vladeta Jovovic, Oct 17 2002
G.f.: 1/(1-x) * Sum_{k>=2} x^k/(1-x^k). - Benoit Cloitre, Apr 23 2003
a(n) = Sum_{i=2..n} floor(n/i). - Jon Perry, Feb 02 2004
a(n) = (Sum_{i=2..n} ceiling((n+1)/i)) - n + 1. - Jon Perry, May 26 2004 [corrected by Jason Yuen, Jul 31 2024]
a(n) = A006218(n) - n. Proof: floor((n-k)/k)+1 = floor(n/k). Then Sum_{k=1..n-1} floor((n-k)/k)+(n-1)+1 = Sum_{k=1..n-1} floor(n/k) + floor(n/n) = Sum_{k=1..n} floor(n/k); i.e., a(n) + n = A006218(n). - Philippe LALLOUET (philip.lallouet(AT)wanadoo.fr), Jun 23 2007
a(n) = A161886(n) - (2n-1). - Eric Desbiaux, Jul 10 2013
a(n+1) = Sum_{k=1..n} A004199(n-k+1,k). - L. Edson Jeffery, Aug 31 2014
a(n) = -Sum_{i=1..n} floor((n-2i+1)/(n-i+1)). - Wesley Ivan Hurt, May 08 2016
a(n) = Sum_{i=1..floor(n/2)} floor((n-i)/i). - Wesley Ivan Hurt, Nov 16 2017
a(n) = Sum_{k=1..n-1} (A000005(n-k) - 1). - Gus Wiseman, Oct 07 2018
a(n) ~ n * (log(n) + 2*EulerGamma - 2). - Rok Cestnik, Dec 19 2020

More terms from David W. Wilson

A075592 Numbers n such that number of distinct prime divisors of n is a divisor of n.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 71, 72, 73, 74, 76, 78, 79, 80, 81, 82, 83, 84, 86, 88, 89, 90

Offset: 1

Amarnath Murthy, Sep 26 2002

Numbers n such that omega(n) divides n.

The asymptotic density of this sequence is 0 (Cooper and Kennedy, 1989). - Amiram Eldar, Jul 10 2020

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
Curtis N. Cooper, Robert E. Kennedy, Chebyshev's inequality and natural density, Amer. Math. Monthly 96 (1989), no. 2, 118-124.

Different from A070776: 78 belongs to this sequence but not to A070776.

Cf. A185307 (complement), A001221, A001222, A074946 (BigOmega(n) divides n).

Select[Range[2,100],Divisible[#,Length[Select[Divisors[#], PrimeQ]]]&]  (* Harvey P. Dale, Mar 17 2011 *)

isok(n) = iferr(!(n % omega(n)), E, 0); \\ Michel Marcus, Oct 06 2017

library(gmp); omega<-function(x) length(unique(as.numeric(factorize(x))))
which(c(F,vapply(2:100,function(n) isint(n/omega(n)),T))) # Christian N. K. Anderson, Apr 25 2013

More terms from David Wasserman, Jan 20 2005

"Distinct" added to name by Christian N. K. Anderson, Apr 23 2013

A331936 Matula-Goebel numbers of semi-lone-child-avoiding rooted trees with at most one distinct non-leaf branch directly under any vertex (semi-achirality).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 14, 16, 18, 24, 26, 27, 28, 32, 36, 38, 46, 48, 49, 52, 54, 56, 64, 72, 74, 76, 81, 86, 92, 96, 98, 104, 106, 108, 112, 122, 128, 144, 148, 152, 162, 169, 172, 178, 184, 192, 196, 202, 206, 208, 212, 214, 216, 224, 243, 244, 256, 262, 288

Offset: 1

Gus Wiseman, Feb 03 2020

nonn

First differs from A331873 in lacking 69, the Matula-Goebel number of the tree ((o)((o)(o))).
A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless that child is an endpoint/leaf.
The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.
Consists of 1, 2, and all numbers equal to a power of 2 (other than 1) times a power of prime(j) for some j > 1 already in the sequence.

			The sequence of rooted trees ranked by this sequence together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   8: (ooo)
   9: ((o)(o))
  12: (oo(o))
  14: (o(oo))
  16: (oooo)
  18: (o(o)(o))
  24: (ooo(o))
  26: (o(o(o)))
  27: ((o)(o)(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  38: (o(ooo))
  46: (o((o)(o)))
  48: (oooo(o))
  49: ((oo)(oo))
The sequence of terms together with their prime indices begins:
    1: {}              52: {1,1,6}            152: {1,1,1,8}
    2: {1}             54: {1,2,2,2}          162: {1,2,2,2,2}
    4: {1,1}           56: {1,1,1,4}          169: {6,6}
    6: {1,2}           64: {1,1,1,1,1,1}      172: {1,1,14}
    8: {1,1,1}         72: {1,1,1,2,2}        178: {1,24}
    9: {2,2}           74: {1,12}             184: {1,1,1,9}
   12: {1,1,2}         76: {1,1,8}            192: {1,1,1,1,1,1,2}
   14: {1,4}           81: {2,2,2,2}          196: {1,1,4,4}
   16: {1,1,1,1}       86: {1,14}             202: {1,26}
   18: {1,2,2}         92: {1,1,9}            206: {1,27}
   24: {1,1,1,2}       96: {1,1,1,1,1,2}      208: {1,1,1,1,6}
   26: {1,6}           98: {1,4,4}            212: {1,1,16}
   27: {2,2,2}        104: {1,1,1,6}          214: {1,28}
   28: {1,1,4}        106: {1,16}             216: {1,1,1,2,2,2}
   32: {1,1,1,1,1}    108: {1,1,2,2,2}        224: {1,1,1,1,1,4}
   36: {1,1,2,2}      112: {1,1,1,1,4}        243: {2,2,2,2,2}
   38: {1,8}          122: {1,18}             244: {1,1,18}
   46: {1,9}          128: {1,1,1,1,1,1,1}    256: {1,1,1,1,1,1,1,1}
   48: {1,1,1,1,2}    144: {1,1,1,1,2,2}      262: {1,32}
   49: {4,4}          148: {1,1,12}           288: {1,1,1,1,1,2,2}

David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

A superset of A000079.
The non-lone-child-avoiding version is A320230.
The non-semi version is A320269.
These trees are counted by A331933.
Not requiring semi-achirality gives A331935.
The fully-achiral case is A331992.
Achiral trees are counted by A003238.
Numbers with at most one distinct odd prime factor are A070776.
Matula-Goebel numbers of achiral rooted trees are A214577.
Matula-Goebel numbers of semi-identity trees are A306202.
Numbers S with at most one distinct prime index in S are A331912.
Cf. A001678, A007097, A050381, A061775, A196050, A291636, A331784, A331873, A331914, A331934, A331965, A331967, A331991.

msQ[n_]:=n<=2||!PrimeQ[n]&&Length[DeleteCases[FactorInteger[n],{2,_}]]<=1&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[100],msQ]

Intersection of A320230 and A331935.

A324106 Multiplicative with a(p^e) = A005940(p^e).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 15, 12, 25, 18, 15, 16, 11, 14, 21, 20, 27, 30, 45, 24, 49, 50, 75, 36, 125, 30, 81, 32, 45, 22, 45, 28, 55, 42, 75, 40, 77, 54, 105, 60, 35, 90, 135, 48, 121, 98, 33, 100, 245, 150, 75, 72, 63, 250, 375, 60, 625, 162, 63, 64, 125, 90, 39, 44, 135, 90, 99, 56, 91, 110, 147, 84, 135, 150, 189, 80, 143, 154, 231, 108, 55

Offset: 1

Antti Karttunen, Feb 15 2019

Question: are there any other numbers n besides 1 and those in A070776, for which a(n) = A005940(n)? At least not below 2^25. This is probably easy to prove.

			For n = 85 = 5*17, a(85) = A005940(5) * A005940(17) = 5*11 = 55. Note that A005940(5) is obtained from the binary expansion of 5-1 = 4, which is "100", and A005940(17) is obtained from the binary expansion of 17-1 = 16, which is "1000".

Antti Karttunen, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Fan style binary tree of a(n), n = 1..2^12, color coded to show the smallest values in the range r = (2^r - 1)..2^(r+1) in blue and highlighting the largest with red.
Index entries for sequences related to binary expansion of n

Cf. A005940, A070776, A324107 (fixed points), A324108, A324109.

nn = 128; Array[Set[a[#], #] &, 2]; Do[If[EvenQ[n], Set[a[n], 2 a[n/2]], Set[a[n], Times @@ Power @@@ Map[{Prime[PrimePi[#1] + 1], #2} & @@ # &, FactorInteger[a[(n + 1)/2]]]]], {n, 3, nn}]; Array[Times @@ Map[a, Power @@@ FactorInteger[#]] &, nn] (* Michael De Vlieger, Sep 18 2022 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ From A005940
A324106(n) = { my(f=factor(n)); prod(i=1, #f~, A005940(f[i,1]^f[i,2])); };

A320230 Matula-Goebel numbers of rooted trees in which the non-leaf branches directly under any given node are all equal.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 36, 37, 38, 40, 41, 43, 44, 46, 48, 49, 50, 52, 53, 54, 56, 58, 59, 61, 62, 64, 67, 68, 71, 72, 74, 76, 79, 80, 81, 82, 83, 86, 88, 89, 92, 96, 97

Offset: 1

Gus Wiseman, Oct 08 2018

nonn

A number is in the sequence iff it belongs to A070776 and its prime indices already belong to the sequence. A prime index of n is a number m such that prime(m) divides n.

Cf. A002541, A070776, A214577, A294336, A317710, A320222, A320226.

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

is(n) = while((n>>=valuation(n,2)) > 1, isprimepower(n,&n) || return(0); n=primepi(n)); 1; \\ Kevin Ryde, Apr 04 2021

A320269 Matula-Goebel numbers of lone-child-avoiding rooted trees in which the non-leaf branches directly under any given node are all equal (semi-achirality).

Original entry on oeis.org

1, 4, 8, 14, 16, 28, 32, 38, 49, 56, 64, 76, 86, 98, 106, 112, 128, 152, 172, 196, 212, 214, 224, 256, 262, 304, 326, 343, 344, 361, 392, 424, 428, 448, 454, 512, 524, 526, 608, 622, 652, 686, 688, 722, 766, 784, 848, 856, 886, 896, 908, 1024, 1042, 1048, 1052

Offset: 1

Gus Wiseman, Oct 08 2018

nonn

First differs from A331871 in lacking 1589.
Lone-child-avoiding means there are no unary branchings.
The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

			The sequence of rooted trees together with their Matula-Goebel numbers begins:
    1: o
    4: (oo)
    8: (ooo)
   14: (o(oo))
   16: (oooo)
   28: (oo(oo))
   32: (ooooo)
   38: (o(ooo))
   49: ((oo)(oo))
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   86: (o(o(oo)))
   98: (o(oo)(oo))
  106: (o(oooo))
  112: (oooo(oo))
  128: (ooooooo)
  152: (ooo(ooo))
  172: (oo(o(oo)))
  196: (oo(oo)(oo))

Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Cf. A002541, A070776, A167865, A214577, A317710, A320222, A320226.
The same-tree version is A291441.
Not requiring lone-child-avoidance gives A320230.
The enumeration of these trees by vertices is A320268.
The semi-lone-child-avoiding version is A331936.
If the non-leaf branches are all different instead of equal we get A331965.
The fully-achiral case is A331967.
Achiral rooted trees are counted by A003238.
MG-numbers of lone-child-avoiding rooted trees are A291636.
Cf. A061775, A196050, A276625, A280996, A306202, A331912, A331916, A331966.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
hmakQ[n_]:=And[!PrimeQ[n],SameQ@@DeleteCases[primeMS[n],1],And@@hmakQ/@primeMS[n]];Select[Range[1000],hmakQ[#]&]

Updated with corrected terminology by Gus Wiseman, Feb 06 2020

A320268 Number of unlabeled series-reduced rooted trees with n nodes where the non-leaf branches directly under any given node are all equal.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 6, 9, 16, 26, 44, 70, 119, 189, 314, 506, 830, 1336, 2186, 3522, 5737, 9266, 15047, 24313, 39444, 63759, 103322, 167098, 270616, 437714, 708676, 1146390, 1855582, 3002017, 4858429, 7860454, 12720310, 20580764, 33303260, 53884144, 87190964

Offset: 1

Gus Wiseman, Oct 08 2018

nonn

This is a weaker condition than achirality (cf. A167865).

A rooted tree is series-reduced if every non-leaf node has at least two branches.

			The a(3) = 1 through a(8) = 9 rooted trees:
  (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)    (ooooooo)
               (o(oo))  (o(ooo))  (o(oooo))   (o(ooooo))
                        (oo(oo))  (oo(ooo))   (oo(oooo))
                                  (ooo(oo))   (ooo(ooo))
                                  ((oo)(oo))  (oooo(oo))
                                  (o(o(oo)))  (o(o(ooo)))
                                              (o(oo)(oo))
                                              (o(oo(oo)))
                                              (oo(o(oo)))

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A001678, A002541, A003238, A010766, A070776, A014668, A126656, A167865, A317099, A317100, A317712, A320222, A320226, A320269.

saum[n_]:=Sum[If[DeleteCases[ptn,1]=={},1,saum[DeleteCases[ptn,1][[1]]]],{ptn,Select[IntegerPartitions[n-1],And[Length[#]!=1,SameQ@@DeleteCases[#,1]]&]}];
Array[saum,20]

seq(n)={my(v=vector(n)); v[1]=1; for(n=3, n, v[n] = 1 + sum(k=2, n-2, (n-1)\k*v[k])); v} \\ Andrew Howroyd, Oct 26 2018

a(1) = 1; a(2) = 0; a(n > 2) = 1 + Sum_{k = 2..n-2} floor((n-1)/k) * a(k).

A324109 Numbers n such that A324108(n) = A324054(n-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 36, 37, 38, 40, 41, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 56, 58, 59, 61, 62, 64, 67, 68, 71, 72, 73, 74, 76, 79, 80, 81, 82, 83, 86, 87, 88, 89, 92, 94, 96, 97, 98, 100, 101, 103, 104, 106, 107, 108, 109, 112, 113, 116, 118, 121

Offset: 1

Antti Karttunen and David A. Corneth, Feb 15 2019

nonn

Numbers n such that A324054(n-1) is equal to A324108(n), which is a multiplicative function with A324108(p^e) = A324054((p^e)-1).
Prime powers (A000961) is a subsequence by definition.
Also A070776 is a subsequence. This follows because for every n of the form 2^i * p^j (where p is an odd prime, and i >= 0, j >= 0), we have A324108(2^i * p^j) = A324054(2^i - 1)*A324054(p^j - 1) = sigma(A005940(2^i)) * sigma(A005940(p^j)). Because A005940(1) = 1, and A005940(2n) = 2*A005940(n), the powers of two are among the fixed points of A005940 (cf. A029747), thus the left half of product is sigma(2^i), while on the other hand, we know that A005940(p^j) is odd (because A005940 also preserves parity), and thus the whole product is equal to sigma(2^i * A005940(p^j)) = sigma(A005940(2^i * p^j)) = A324054((2^i * p^j)-1).
See subsequence A324111 for less regular solutions.

Antti Karttunen, Table of n, a(n) for n = 1..10001

Union of A070776 and A324111.

Cf. A000961 (a subsequence), A029747, A324054, A324107, A324108, A324110 (complement).

A324054(n) = { my(p=2,mp=p*p,m=1); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, if(3==(n%4),mp *= p,m *= (mp-1)/(p-1))); n>>=1); (m); };
A324108(n) = { my(f=factor(n)); prod(i=1, #f~, A324054((f[i,1]^f[i,2])-1)); };
isA324109(n) = (A324054(n-1)==A324108(n));
for(n=1,121,if(isA324109(n), print1(n,", ")));

A324111 Numbers n for which A324108(n) = A324054(n-1) and which are neither prime powers nor of the form 2^i * p^j, where p is an odd prime, with either exponent i or j > 0.

Original entry on oeis.org

1, 87, 174, 348, 696, 1392, 2091, 2784, 4182, 5568, 8364, 11136, 16683, 16728, 22272, 33215, 33366, 33456, 44544, 66430, 66732, 66912, 89088, 132860, 133464, 133824, 178176, 265720, 266928, 267179, 267648, 356352, 531440, 533856, 534358, 535296, 712704, 1062880, 1066877, 1067712, 1068716, 1070592, 1319235, 1425408

Offset: 1

Antti Karttunen, Feb 15 2019

nonn

Setwise difference of A324109 and A070776.
Setwise difference of A070537 and A324110.
If an odd number n > 1 is present, then all 2^k * n are present also. Odd terms > 1 are given in A324112.

			87 is a term, as 87 = 3*29, A324054(3-1) = 4, A324054(29-1) = 156, and A324108(87) = 4*156 = 624 = A324054(87-1).

Antti Karttunen, Table of n, a(n) for n = 1..173

Cf. A070537, A070776, A324109, A324110, A324112.

A000265(n) = (n/2^valuation(n, 2));
A324054(n) = { my(p=2,mp=p*p,m=1); while(n, if(!(n%2), p=nextprime(1+p); mp = p*p, if(3==(n%4),mp *= p,m *= (mp-1)/(p-1))); n>>=1); (m); };
A324108(n) = { my(f=factor(n)); prod(i=1, #f~, A324054((f[i,1]^f[i,2])-1)); };
isA324111(n) = ((1!=omega(n))&&(1!=omega(A000265(n)))&&(A324054(n-1)==A324108(n)));
for(n=1,2^20,if(isA324111(n), print1(n,", ")))

cp's OEIS Frontend

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