A004615 -id:A004615 - cp's OEIS frontend

A068871 Duplicate of A004615.

1, 11, 31, 41, 61, 71, 101, 121, 131, 151, 181, 191, 211, 241, 251, 271, 281, 311, 331

Offset: 1

dead

A030430 Primes of the form 10*n+1.

11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, 271, 281, 311, 331, 401, 421, 431, 461, 491, 521, 541, 571, 601, 631, 641, 661, 691, 701, 751, 761, 811, 821, 881, 911, 941, 971, 991, 1021, 1031, 1051, 1061, 1091, 1151, 1171, 1181, 1201, 1231, 1291

Offset: 1

Warut Roonguthai

Also primes of form 5*n+1 or equivalently 5*n+6.
Primes p such that the arithmetic mean of divisors of p^4 is an integer: A000203(p^4)/A000005(p^4) = C. - Ctibor O. Zizka, Sep 15 2008
Being a subset of A141158, this is also a subset of the primes of form x^2-5*y^2. - Tito Piezas III, Dec 28 2008
5 is quadratic residue of primes of this form. - Vincenzo Librandi, Jun 25 2014
Primes p such that 5 divides sigma(p^4), cf. A274397. - M. F. Hasler, Jul 10 2016

Michael B. Porter, Table of n, a(n) for n = 1..100000
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 519.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.

Cf. A024912, A045453, A049511, A081759, A017281, A010051, A004615 (multiplicative closure).
Cf. A001583 (subsequence).
Union of A132230 and A132232. - Ray Chandler, Apr 07 2009

a030430 n = a030430_list !! (n-1)
a030430_list = filter ((== 1) . a010051) a017281_list
-- Reinhard Zumkeller, Apr 16 2012

Select[Prime@Range[210], Mod[ #, 10] == 1 &] (* Ray Chandler, Dec 06 2006 *)
Select[Range[11,1291,10],PrimeQ] (*Zak Seidov, Aug 14 2011*)

is(n)=n%10==1 && isprime(n) \\ Charles R Greathouse IV, Sep 06 2012

lista(nn) = forprime(p=11, nn, if(p%10==1, print1(p, ", "))) \\ Iain Fox, Dec 30 2017

a(n) = 10*A024912(n)+1 = 5*A081759(n)+6.
A104146(floor(a(n)/10)) = 1.
Union of A132230 and A132232. - Ray Chandler, Apr 07 2009
a(n) ~ 4n log n. - Charles R Greathouse IV, Sep 06 2012
Intersection of A000040 and A017281. - Iain Fox, Dec 30 2017

A340004 Decimal expansion of Product_{primes p == 1 (mod 5)} p^2/(p^2-1).

Original entry on oeis.org

1, 0, 1, 0, 9, 1, 5, 1, 6, 0, 6, 0, 1, 0, 1, 9, 5, 2, 2, 6, 0, 4, 9, 5, 6, 5, 8, 4, 2, 8, 9, 5, 1, 4, 9, 2, 0, 9, 8, 4, 5, 3, 8, 6, 2, 7, 5, 8, 1, 7, 3, 8, 5, 2, 3, 7, 3, 2, 0, 2, 4, 2, 0, 0, 8, 9, 2, 5, 1, 6, 1, 3, 7, 4, 2, 4, 5, 6, 7, 2, 6, 3, 7, 0, 9, 3, 9, 6, 1, 9, 7, 6, 9, 4, 5, 5, 8, 9, 2, 1, 8

Offset: 1

Artur Jasinski, Jan 15 2021

This constant is called Euler product 2==1 modulo 5 (see Mathar's Definition 5 formula (38)) or equivalently zeta 2==1 modulo 5.

			1.01091516060101952260495658428951492...

Vaclav Kotesovec, Table of n, a(n) for n = 1..501
Salma Ettahri, Olivier Ramaré, and Léon Surel, Fast multi-precision computation of some Euler products, arXiv:1908.06808 [math.NT], 2019 p.20 (100 digits precision data).
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2014-2015, Section 3.3. zeta_{5,1}(2).

Cf. A340127, A340628, A340629, A340665, A340794, A340839.
Cf. A175646, A301429, A333240.
Cf. A175647, A248930, A248938, A335963.
Cf. A004615, A340576, A340577, A340578, A334826.

S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
Z[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = P[m, n, s*w]/w; sumz = sumz + difz; w++]; Exp[sumz]);
$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Z[5, 1, 2], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021, took 20 minutes *)

Equals Sum_{k>=1} 1/A004615(k)^2. - Amiram Eldar, Jan 24 2021

Equals exp(-gamma/2)*Pi/(A340839^2*sqrt(5*log((1 + sqrt (5))/2))). - Artur Jasinski, Jan 30 2021

A340629 Decimal expansion of Product_{primes p == 1 (mod 5)} (p^2+1)/(p^2-1).

Original entry on oeis.org

1, 0, 2, 1, 8, 7, 8, 0, 6, 0, 4, 1, 8, 7, 5, 6, 6, 7, 5, 7, 4, 4, 4, 4, 8, 9, 1, 4, 6, 0, 0, 2, 7, 0, 8, 2, 6, 1, 7, 0, 4, 6, 0, 7, 3, 7, 7, 3, 2, 5, 1, 6, 4, 0, 6, 6, 6, 0, 1, 1, 9, 4, 4, 3, 7, 7, 0, 9, 0, 4, 7, 6, 7, 0, 5, 6, 6, 0, 0, 8, 6, 0, 6, 4, 5, 5, 1, 4, 9, 9, 9, 5, 0, 0, 5, 9, 8, 4, 1, 4, 9, 9, 9, 0, 6, 2, 3, 7, 6, 0, 1, 0, 5, 2, 3, 3, 3, 2, 0, 3, 5

Offset: 1

Artur Jasinski, Jan 13 2021

			1.0218780604187566757444489146002708261704607377325...

Vaclav Kotesovec, Table of n, a(n) for n = 1..500
Salma Ettahri, Olivier Ramaré, and Léon Surel, Fast multi-precision computation of some Euler products, arXiv:1908.06808 [math.NT], 2019, p. 20 (70-digit-precision data).
Steven Finch, Greg Martin, and Pascal Sebah, Roots of unity and nullity modulo n, Proc. Amer. Math. Soc. Volume 138, Number 8, August 2010, pp. 2729-2743.
Steven Finch and Pascal Sebah, Residue of a Mod 5 Euler Product, arXiv:0912.3677 [math.NT], 2009 (formulas).
Alessandro Languasco and Alessandro Zaccagnini, On the constant in the Mertens product for arithmetic progressions. I. Identities, Funct. Approx. Comment. Math. Volume 42, Number 1 (2010), 17-27, (preliminary version).
Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Cf. A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340577, A340578, A340628.

evalf(Re(15*sqrt((1/13)*(5*((I*Pi^2*(1/150)-I*polylog(2, (-1)^(2/5)))^2+((1/150)*(11*I)*Pi^2+I*polylog(2, (-1)^(4/5)))^2)))/Pi^2), 120) # Vaclav Kotesovec, Jan 20 2021, after formula by Pascal Sebah.

S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
Z[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = P[m, n, s*w]/w; sumz = sumz + difz; PrintTemporary["iteration = ", w, ", difference = ", N[difz, digits]]; w++]; Exp[sumz]);
$MaxExtraPrecision = 1000; digits = 121; Chop[N[1/(Z[5,1,4]/Z[5,1,2]^2), digits]] (* Vaclav Kotesovec, Jan 15 2021, took over 20 minutes *)
digits = 121; digitize[c_] := RealDigits[Chop[N[c, digits]], 10, digits][[1]];
cl[x_] := I (PolyLog[2, (-1)^x] - PolyLog[2, -(-1)^(1 - x)]);
A340629 := (15 Sqrt[65]/(26 Pi^2)) Sqrt[cl[2/5]^2 + cl[4/5]^2];
digitize[A340629] (* Peter Luschny, Jan 23 2021 *)

Equals 6*sqrt(5)/(13*A340628).
Equals A340004^2/A340808. - R. J. Mathar, Jan 15 2021
Equals 15*sqrt(65)*g/(13*Pi^2) where g = sqrt(Cl2(2*Pi/5)^2 + Cl2(4*Pi/5)^2) = 1.0841621352693895..., and Cl2 is the Clausen function of order 2. Formula by Pascal Sebah (personal communication). - Artur Jasinski, Jan 20 2021
Equals Sum_{q in A004615} 2^A001221(q)/q^2. - R. J. Mathar, Jan 27 2021

Corrected and more terms from Vaclav Kotesovec, Jan 15 2021

A330348 a(n) is the number of divisors of n whose last digit equals the last digit of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 1, 1, 3, 2, 2, 1, 2, 3, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 4, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 3, 1, 1, 4, 1, 2, 1, 1, 4, 2, 2, 1, 3, 2, 1, 1, 2, 1

Offset: 1

Bernard Schott, Jun 04 2020

Inspired by Project Euler, Problem 474 (see link).
a(n) >= 1.
When n > 10 ends with 0, 1, 2 or 5, then a(n) >= 2.
The first 19 terms are the same as A038769, but a(20) = 2 and A038769(20) = 1.
From Robert Israel, Jun 04 2020: (Start)
a(10*n) = A000005(n).
If n is odd, then a(2*n) = a(n) and a(5*n) = A000005(n). (End)
Integers all of whose divisors end with the same last digit (which is necessarily 1) are in A004615. - Bernard Schott, May 07 2021

			The divisors of 12 that end in 2 are 2 and 12, so a(12) = 2.

Robert Israel, Table of n, a(n) for n = 1..10000
Project Euler, Problem 474: Last digits of divisors

Cf. A000005 (number of divisors), A004615, A010879 (last digit of n), A038769.

f:= proc(n) local t;
t:= n mod 10;
nops(select(k -> k mod 10 = t, numtheory:-divisors(n)))
end proc:
map(f, [$1..100]); # Robert Israel, Jun 04 2020

a[n_] := DivisorSum[n, 1 &, Mod[# - n, 10] == 0 &]; Array[a, 100] (* Amiram Eldar, Jun 04 2020 *)

a(n) = my(u=n%10); sumdiv(n, d, d%10 == u); \\ Michel Marcus, Jun 04 2020

from sympy import divisors
def a(n): return sum((n-d)%10 == 0 for d in divisors(n, generator=True))
print([a(n) for n in range(1, 90)]) # Michael S. Branicky, Aug 15 2022

A340808 Decimal expansion of Product_{primes p == 1 (mod 5)} 1/(1-p^(-4)).

Original entry on oeis.org

1, 0, 0, 0, 0, 6, 9, 8, 7, 2, 8, 3, 2, 1, 8, 4, 2, 6, 1, 4, 1, 4, 1, 9, 6, 3, 5, 2, 6, 4, 6, 0, 0, 6, 2, 5, 1, 5, 3, 2, 3, 6, 8, 1, 4, 6, 7, 9, 6, 1, 5, 3, 4, 0, 6, 2, 7, 2, 4, 3, 4, 4, 3, 2, 6, 2, 7, 1, 4, 9, 4, 0, 1, 4, 0, 6, 7, 9, 6, 8, 5, 8, 7, 0, 9, 5, 2, 1, 5, 1, 1, 7, 9, 4, 1, 7, 3, 0, 2, 0, 1, 8, 9, 4, 0

Offset: 1

R. J. Mathar, Jan 22 2021

Equals also the same product over the primes p == 1 (mod 10).

			1.0000698728321842614141963526460062515  = (14641/14640) * (923521/923520) * (2825761/2825760) *...

Vaclav Kotesovec, Table of n, a(n) for n = 1..500
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions... arXiv:1008.2547, Zeta_{m=5,n=1}(s=4).

Cf. A004615, A030430, A340004 (s=2), A340629, A340809, A340926, A340927.

A340629 = A340004 ^2 / this.

Equals Sum_{k>=1} 1/A004615(k)^4. - Amiram Eldar, Jan 24 2021

More digits from Vaclav Kotesovec, Jan 22 2021

A206287 Numbers with all divisors starting with digit 1.

Original entry on oeis.org

1, 11, 13, 17, 19, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1111

Offset: 1

Jaroslav Krizek, Feb 12 2012

Equivalently, integers m with all divisors starting with the same first digit of m; in fact, as 1 divides all the integers, this digit is necessarily 1; also, for these terms m: A357299(m) = A000005(m). - Bernard Schott, Sep 25 2022

			All divisors of 187 (1, 11, 17, 187) start with digit 1.

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

Disjoint union of A045707 and A206288.
Cf. A000005, A355592, A357299, A357300.
Cf. A004615 (with last digit)

filter:= proc(n) andmap(t -> floor(t/10^ilog10(t)) = 1, numtheory:-divisors(n)) end proc:
select(filter, [seq($10^d .. 2*10^d-1, d=0..3)]); # Robert Israel, Dec 25 2024

fQ[n_] := Module[{d = Divisors[n]}, Union[IntegerDigits[#][[1]] & /@ d] == {1}]; Select[Range[1111], fQ] (* T. D. Noe, Feb 13 2012 *)

A338784 a(n) is the smallest number with exactly n divisors such that all its divisors end with the same digit (which is necessarily 1).

Original entry on oeis.org

1, 11, 121, 341, 14641, 3751, 1771561, 13981, 116281, 453871, 25937424601, 153791, 3138428376721, 54918391, 14070001, 852841, 45949729863572161, 4767521, 5559917313492231481, 18608711, 1702470121, 804060162631, 81402749386839761113321, 9381251, 13521270961, 97291279678351, 195468361

Offset: 1

Bernard Schott, Nov 09 2020

As 1 is a divisor for each number, all the divisors must end with 1.

			121 is the smallest number whose 3 divisors (1, 11, 121) end with 1, hence a(3) = 121.
3751 is the smallest number whose 6 divisors (1, 11, 31, 121, 341, 3751) end with 1, hence a(6) = 121.
a(18) = 4767521 = 11^2 * 31^2 * 41 as it has 18 divisors all of which end in 1. - _David A. Corneth_, Nov 09 2020

David A. Corneth, Table of n, a(n) for n = 1..966
Project Euler, Problem 474: Last digits of divisors.

Cf. A001020, A030430, A092609, A330348.

Subsequence of A004615.

a(n) = {my(pr); if(n==1, return(1)); if(isprime(n), return(11^(n-1))); forstep(i = 1, oo, 10, f = factor(i); if(numdiv(f) == n, pr = 1; for(j = 1, #f~, if(f[j, 1]%10 != 1, pr = 0; next(2) ) ) ); if(pr, return(i)); ) } \\ David A. Corneth, Nov 09 2020

If n is prime p, then a(p) = 11^(p-1) = A001020(p-1).

For k>=1, a(2^k) = {Product_m=1..k} A030430(m) = A092609(k).

Data corrected by David A. Corneth, Nov 09 2020

A380758 Numbers which are not prime powers and their prime factors share a last digit in base 10.

Original entry on oeis.org

39, 69, 117, 119, 129, 159, 207, 219, 249, 259, 299, 309, 329, 339, 341, 351, 387, 451, 469, 477, 489, 507, 519, 551, 559, 579, 621, 629, 657, 669, 671, 679, 689, 699, 747, 749, 781, 789, 799, 833, 849, 879, 889, 897, 927, 939, 949, 959, 989, 1017, 1053, 1059

Offset: 1

Yaroslav Deryavko, Feb 01 2025

Also called the one-sided numbers.

They can end only in either 1, 3, 7 or 9.

			39 = 3*13.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
VOS 2025 Math Region stage, Problem 10.9 (in Russian).

Cf. A004615.

q:= n-> (l-> nops(l)>1 and nops({map(i-> irem(i[1], 10), l)[]})=1)(ifactors(n)[2]):
select(q, [$1..2000])[];  # Alois P. Heinz, Feb 18 2025

Sort[Times@@@Cases[Subsets[Prime[Range[100]],{2}],?(Mod[#[[1]]-#[[2]],10]==0&)]][[;;100]] (* _Shenghui Yang, Feb 18 2025 *)

isok(k) = my(f=factor(k)); (#f~ != 1) && (#Set(vector(#f~, i, f[i,1] % 10)) == 1); \\ Michel Marcus, Feb 18 2025

from sympy import factorint
def ok(n): return len(f:=factorint(n)) > 1 and len(set(p%10 for p in f)) == 1
print([k for k in range(1, 1060) if ok(k)]) # Michael S. Branicky, Feb 18 2025

A212843 Carmichael numbers that have only prime divisors of the form 10k+1.

Original entry on oeis.org

252601, 399001, 512461, 852841, 1193221, 1857241, 1909001, 2100901, 3828001, 5049001, 5148001, 5481451, 6189121, 7519441, 8341201, 9439201, 10024561, 10837321, 14676481, 15247621, 17236801, 27062101, 29111881, 31405501, 33302401, 34657141, 40430401, 42490801

Offset: 1

Marius Coman, May 28 2012

nonn

Conjecture: only Carmichael numbers of the form 10n+1 can have prime divisors of the form 10k+1 (but not all Carmichael numbers of the form 10n+1 have prime divisors of the form 10k+1).
Checked up to Carmichael number 4954039956700380001.
Conjecture: all Carmichael numbers C (not only with three prime divisors) of the form 10n+1 that have only prime divisors of the form 10k+1 can be written as C = (30a+1)*(30b+1)*(30c+1), C = (30a+11)*(30b+11)*(30c+11), or C = (30a+1)*(30b+11)*(30c+11). In other words, there are no numbers of the form C = (30a+1)*(30b+1)*(30c+11).
Checked for all Carmichael numbers from the sequence above.
The first conjecture is a consequence of Korselt's criterion. - Charles R Greathouse IV, Oct 02 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
E. W. Weisstein, Carmichael Number

Subsequence of A004615.

Terms corrected by Charles R Greathouse IV, Oct 02 2012

cp's OEIS Frontend

A068871 Duplicate of A004615.

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A030430 Primes of the form 10*n+1.

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A340004 Decimal expansion of Product_{primes p == 1 (mod 5)} p^2/(p^2-1).

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A340629 Decimal expansion of Product_{primes p == 1 (mod 5)} (p^2+1)/(p^2-1).

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A330348 a(n) is the number of divisors of n whose last digit equals the last digit of n.

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A340808 Decimal expansion of Product_{primes p == 1 (mod 5)} 1/(1-p^(-4)).

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A206287 Numbers with all divisors starting with digit 1.

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A338784 a(n) is the smallest number with exactly n divisors such that all its divisors end with the same digit (which is necessarily 1).

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