A029940 -id:A029940 - cp's OEIS frontend

A280131 Partial sums of A029940 (Product_{d|n} phi(d)).

1, 2, 4, 6, 10, 14, 20, 28, 40, 56, 66, 98, 110, 146, 210, 274, 290, 434, 452, 708, 852, 952, 974, 1998, 2078, 2222, 2438, 3302, 3330, 7426, 7456, 8480, 8880, 9136, 9712, 23536, 23572, 23896, 24472, 40856, 40896, 61632, 61674, 65674, 74890, 75374, 75420

Offset: 1

Jaroslav Krizek, Dec 27 2016

nonn

phi(n) is the number of totatives of n (A000010).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A029940, A232533 (partial sums of A057661), A280132 (partial products of A029940).

[&+[&*[EulerPhi(d): d in Divisors(k)]: k in [1..n]]: n in [1..100]]

A029940:= [seq(mul(numtheory:-phi(d),d=numtheory:-divisors(n)),n=1..100)]:
ListTools:-PartialSums(A029940); # Robert Israel, Jan 11 2017

Accumulate@ Array[Product[EulerPhi@ d, {d, Divisors@ #}] &, 47] (* Michael De Vlieger, Dec 27 2016 *)

a(n) = Sum_{i=1..n} A029940(i).

A280132 Partial products of A029940 (Product_{d|n} phi(d)).

Original entry on oeis.org

1, 1, 2, 4, 16, 64, 384, 3072, 36864, 589824, 5898240, 188743680, 2264924160, 81537269760, 5218385264640, 333976656936960, 5343626510991360, 769482217582755840, 13850679916489605120, 3545774058621338910720, 510591464441472803143680, 51059146444147280314368000

Offset: 1

Jaroslav Krizek, Dec 27 2016

nonn

phi(n) is the number of totatives of n (A000010).

Cf. A000010, A029940, A280131 (partial sums of A029940), A280133 (partial products of A057661).

[&*[&*[EulerPhi(d): d in Divisors(k)]: k in [1..n]]: n in [1..100]];

FoldList[Times[#1, #2] &, Array[Product[EulerPhi@ d, {d, Divisors@ #}] &, 22]] (* Michael De Vlieger, Dec 27 2016 *)

a(n) = Product_{i=1..n} A029940(i).

A248795 Numbers k such that Product_{d|k} phi(d) = Product_{d|(k+1)} phi(d) where phi(x) = Euler totient function (A000010).

Original entry on oeis.org

1, 3, 5, 15, 255, 65535, 2200694, 2619705, 6372794, 40588485, 76466985, 81591194, 118018094, 206569605, 470542485, 525644385, 726638834, 791937614, 971122514, 991172805, 1268457014, 1384822005, 1613055045, 1709460753, 1861556654, 1872619665, 2507927414, 2659263945

Offset: 1

Jaroslav Krizek, Nov 19 2014

nonn

Numbers k such that A029940(k) = A029940(k+1).

4294967295 is in this sequence.

			15 is in the sequence because A029940(15) = A029940(16) = 64.

Cf. A000010, A019434, A029940, A248796.

[n: n in [1..100000] | (&*[EulerPhi(d): d in Divisors(n)]) eq (&*[EulerPhi(d): d in Divisors(n+1)])];

a248795[n_Integer] := Select[Range[n],
Product[EulerPhi[i], {i, Divisors[#]}] ==
Product[EulerPhi[j], {j, Divisors[# + 1]}] &]; a248795[10^5] (* Michael De Vlieger, Nov 30 2014 *)

lista(nn) = {d = divisors(1); vcur = prod(k=1, #d, eulerphi(d[k])); for (n=2, nn, d = divisors(n); vnext = prod(k=1, #d, eulerphi(d[k])); if (vnext == vcur, print1(n-1, ", ")); vcur = vnext;);} \\ Michel Marcus, Nov 23 2014

a(n) = A248796(n) - 2.

a(7)-a(9) from Michel Marcus, Nov 21 2014
a(10)-a(20) from Michel Marcus, Nov 23 2014
More terms from Jinyuan Wang, Jul 27 2025

A248796 Numbers k such that Product_{d|(k-2)} phi(d) = Product_{d|(k-1)} phi(d) where phi(x) = Euler totient function (A000010).

Original entry on oeis.org

3, 5, 7, 17, 257, 65537, 2200696, 2619707, 6372796, 40588487, 76466987, 81591196, 118018096, 206569607, 470542487, 525644387, 726638836, 791937616, 971122516, 991172807, 1268457016, 1384822007, 1613055047, 1709460755, 1861556656, 1872619667, 2507927416, 2659263947

Offset: 1

Jaroslav Krizek, Nov 19 2014

nonn

Numbers k such that A029940(k-2) = A029940(k-1).

The first 5 known Fermat primes (A019434) are terms of this sequence.

			17 is in the sequence because A029940(15) = A029940(16) = 64.

Supersequence of A247164 and A247203.

Cf. A000010, A019434, A029940, A248795.

[n: n in [3..100000] | (&*[EulerPhi(d): d in Divisors(n-2)]) eq (&*[EulerPhi(d): d in Divisors(n-1)])];

a(n) = A248795(n) + 2.

A029940(a(n)) = a(n) - 1 if a(n) = prime.

a(7)-a(9) using A248795 by Jaroslav Krizek, Nov 19 2014
a(10)-a(20) using A248795 by Jaroslav Krizek, Nov 25 2014
More terms from Jinyuan Wang, Jul 27 2025

A247164 Primes p such that Product_{d|(p-2)} phi(d) = Product_{d|(p-1)} phi(d) where phi(x) = Euler totient function (A000010).

Original entry on oeis.org

3, 5, 7, 17, 257, 65537, 991172807, 1872619667, 4081364447

Offset: 1

Jaroslav Krizek, Nov 21 2014

Primes p such that A029940(p-2) = A029940(p-1).

First 5 known terms of Fermat primes (A019434) are terms of this sequence.

			Prime 17 is in the sequence because A029940(15) = A029940(16) = 64.

Subsequence of A248796. Supersequence of A247203.

Cf. A000010, A019434, A029940, A248795.

[n: n in [3..100000] |  IsPrime(n) and (&*[EulerPhi(d): d in Divisors(n-2)]) eq (&*[EulerPhi(d): d in Divisors(n-1)])];

A029940(a(n)) = a(n) - 1.

a(8)-a(9) from Jinyuan Wang, Jul 27 2025

A247203 Primes p such that phi(p-2) = phi(p-1) and simultaneously Product_{d|(p-2)} phi(d) = Product_{d|(p-1)} phi(d) where phi(x) = Euler totient function (A000010).

Original entry on oeis.org

3, 5, 17, 257, 65537, 991172807, 1872619667, 4081364447

Offset: 1

Jaroslav Krizek, Nov 25 2014

Primes p such that A000010(p-2) = A000010(p-1) and simultaneously A029940(p-2) = A029940(p-1).

The first 5 known Fermat primes (A019434) are terms of this sequence.

			17 is in the sequence because phi(15) = phi(16) = 8 and simultaneously Product_{d|15} phi(d) = Product_{d|16} phi(d) = 64.

Subsequence of A247164 and A248796.

Cf. A000010, A029940.

[p: p in PrimesInInterval(3, 10^7) | (&*[EulerPhi(d): d in Divisors(p-2)]) eq (&*[EulerPhi(d): d in Divisors(p-1)]) and EulerPhi(p-2) eq EulerPhi(p-1)];

[n: n in [A248796(n)] | IsPrime(n) and EulerPhi(n-2) eq EulerPhi(n-1)];

[n: n in [A247164(n)] | IsPrime(n) and EulerPhi(n-2) eq EulerPhi(n-1)];

a(7)-a(8) from Jinyuan Wang, Jul 27 2025

A120065 Number of permutations on 1..n where gcd(s_i,n) = gcd(i,n). Also Product_{d divides n} phi(d)!.

Original entry on oeis.org

1, 1, 2, 2, 24, 4, 720, 48, 1440, 576, 3628800, 192, 479001600, 518400, 1935360, 1935360, 20922789888000, 2073600, 6402373705728000, 46448640, 689762304000, 13168189440000, 1124000727777607680000, 185794560, 58389648196239360000

Offset: 1

Martin Fuller, Jun 06 2006

nonn

The values of this sequence also represents the size of the search space for pandigital polydivisible numbers, PPN, in some even base n. PPN in some base b are defined as numbers that contain all the nonzero digits 1..b without repetition, arranged such that the first k digits are divisible by k for the entire length of the number, e.g., in base 10: 381654729 or in base 14: 9C3A5476B812D. It can be shown that for a base, b, the i-th digit, d, is limited to values such that gcd(i,b)=gcd(d,b). Thus the search space for some base is the factorial applied to the counts of numbers that share a gcd in that base. - Nicholas Stefan Georgescu, Mar 06 2023

			a(8) = 48 = 4! * 2! * 1! * 1! because we can permute [1,3,5,7] in 4! ways, [2,6] in 2! ways and 4 and 8 are fixed.

Skirtle's Den, Pandigital Polydivisible Numbers

Cf. A029940 Product phi(d); d divides n.

Cf. A000010 Euler totient function phi(n).

a(n) = prod(i=1, n, if(n%i==0, eulerphi(i)!, 1))

from sympy import factorial, gcd
from numpy import product
from collections import Counter
[int(product(list(map(factorial,Counter([gcd(i,n) for i in range(1,n)]).values())))) for n in range(1,20)] # Nicholas Stefan Georgescu, Mar 06 2023

A247174 Numbers k such that phi(k) = phi(k+1) and simultaneously Product_{d|k} phi(d) = Product_{d|(k+1)} phi(d) where phi(x) = Euler totient function (A000010).

Original entry on oeis.org

1, 3, 15, 255, 65535, 2200694, 2619705, 6372794, 40588485, 76466985, 81591194, 118018094, 206569605, 470542485, 525644385, 726638834, 791937614, 971122514, 991172805

Offset: 1

Jaroslav Krizek, Nov 22 2014

nonn

Numbers n such that A000010(n) = A000010(n+1) and simultaneously A029940(n) = A029940(n+1).
4294967295 is also a term of this sequence.
Intersection of A001274 and A248795.

			15 is in the sequence because phi(15) = phi(16) = 8 and simultaneously Product_{d|15} phi(d) = Product_{d|(15+1)} phi(d) = 64.

Amiram Eldar, Table of n, a(n) for n = 1..213 (terms below 10^13, calculated using the b-file at A001274)

Cf. A000010, A001274, A029940, A248795.

[n: n in [1..100000] |  (&*[EulerPhi(d): d in Divisors(n)]) eq (&*[EulerPhi(d): d in Divisors(n+1)]) and EulerPhi(n) eq EulerPhi(n+1)]

[n: n in [A248795(n)] | EulerPhi(n) eq EulerPhi(n+1)]

a247174[n_Integer] := Module[{a001274, a248795},
  a001274[m_] := Select[Range[m], EulerPhi[#] == EulerPhi[# + 1] &];
  a248795[m_] :=
   Select[Range[m],
    Product[EulerPhi[i], {i, Divisors[#]}] ==
      Product[EulerPhi[j], {j, Divisors[# + 1]}] &];
Intersection[a001274[n], a248795[n]]] (* Michael De Vlieger, Dec 01 2014 *)

a(6)-a(19) using A248795 by Jaroslav Krizek, Nov 25 2014

A280133 Partial products of A057661 (Sum_{d|n} psi(d)).

Original entry on oeis.org

1, 2, 8, 48, 528, 5808, 127776, 2811072, 87143232, 2788583424, 156160671744, 6090266198016, 481131029643264, 31273516926812160, 2314240252584099840, 199024661722232586240, 27266378655945864314880, 2508506836347019516968960, 431463175851687356918661120

Offset: 1

Jaroslav Krizek, Dec 27 2016

nonn

psi(n) is the sum of the totatives of n (A023896).

Cf. A000010, A057661, A232533 (partial sums of A057661), A280132 (partial products of A029940).

[&*[&+[&+[h: h in [1..d] | GCD(h,d) eq 1]: d in Divisors(k)]: k in [1..n]]: n in [1..1000]]

FoldList[Times[#1, #2] &, Array[Sum[Total@ Select[Range@ d, CoprimeQ[#, d] &], {d, Divisors@ #}] &, 19]] (* Michael De Vlieger, Dec 27 2016 *)

a(n) = Product_{i=1..n} A057661(i).

A349741 a(n) = Product_{k=1..n-1} phi(gcd(n,k)).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 4, 4, 1, 32, 1, 6, 256, 16, 1, 96, 1, 1024, 2304, 10, 1, 16384, 256, 12, 2304, 13824, 1, 524288, 1, 2048, 102400, 16, 5308416, 14155776, 1, 18, 589824, 134217728, 1, 63700992, 1, 1024000, 86973087744, 22, 1, 8589934592, 46656, 1310720

Offset: 1

Ilya Gutkovskiy, Nov 28 2021

nonn

Cf. A000010, A001088, A029935, A029940, A046022 (positions of 1's), A051190.

Table[Product[EulerPhi[GCD[n, k]], {k, 1, n - 1}], {n, 1, 50}]

a(n) = prod(k=1, n-1, eulerphi(gcd(n, k))); \\ Michel Marcus, Nov 28 2021

a(n) = Product_{d|n, d < n} phi(d)^phi(n/d).

cp's OEIS Frontend

A280131 Partial sums of A029940 (Product_{d|n} phi(d)).

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A280132 Partial products of A029940 (Product_{d|n} phi(d)).

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A248795 Numbers k such that Product_{d|k} phi(d) = Product_{d|(k+1)} phi(d) where phi(x) = Euler totient function (A000010).

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A248796 Numbers k such that Product_{d|(k-2)} phi(d) = Product_{d|(k-1)} phi(d) where phi(x) = Euler totient function (A000010).

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A247164 Primes p such that Product_{d|(p-2)} phi(d) = Product_{d|(p-1)} phi(d) where phi(x) = Euler totient function (A000010).

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A247203 Primes p such that phi(p-2) = phi(p-1) and simultaneously Product_{d|(p-2)} phi(d) = Product_{d|(p-1)} phi(d) where phi(x) = Euler totient function (A000010).

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A120065 Number of permutations on 1..n where gcd(s_i,n) = gcd(i,n). Also Product_{d divides n} phi(d)!.

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