A136141 -id:A136141 - cp's OEIS frontend

A354929 Numbers of the form 2*p^e, with p an odd prime and e >= 2.

18, 50, 54, 98, 162, 242, 250, 338, 486, 578, 686, 722, 1058, 1250, 1458, 1682, 1922, 2662, 2738, 3362, 3698, 4374, 4394, 4418, 4802, 5618, 6250, 6962, 7442, 8978, 9826, 10082, 10658, 12482, 13122, 13718, 13778, 15842, 18818, 20402, 21218, 22898, 23762, 24334, 25538, 29282, 31250, 32258

Offset: 1

Antti Karttunen, Jun 13 2022

nonn

Conjecturally numbers k > 1 such that A047994(k) = A344005(k) (see A354928), and k is in A265128. See comments in A346608.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A278568.
Intersection of A265128 and A354928 (conjectured).
Cf. A047994, A136141, A344005, A346608, A354924.

Select[Range[33000], IntegerExponent[#, 2] == 1 && CompositeQ[#/2] && PrimePowerQ[#/2] &] (* Amiram Eldar, Jun 18 2022 *)

isA354929(n) = ((2==(n%4)) && (isprimepower(n/2)>1));

Sum_{n>=1} 1/a(n) = (A136141 - 1/2)/2 = 0.1365783345... - Amiram Eldar, Jun 18 2022

A382294 Decimal expansion of the asymptotic mean of the excess of the number of Fermi-Dirac factors of k over the number of distinct prime factors of k when k runs over the positive integers.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Mar 21 2025

Analogous to Sum_{p prime} 1/(p*(p-1)) (A136141), which is the asymptotic mean of the excess of the number of prime factors over the number of distinct prime factors (A046660).

			0.13605447049622836522998926383768997616582469083783...

Cf. A001221, A046660, A064547, A088705, A136141, A382290.

s[n_] := Module[{c = CoefficientList[Series[-x + Sum[x^(2^k)/(1+x^(2^k)), {k, 0, n}],{x, 0, 2^n}], x]},Sum[c[[i]] * PrimeZetaP[i-1], {i, 3, Length[c]-2}]]; RealDigits[s[10], 10, 120][[1]]

default(realprecision, 120); default(parisize, 10000000);
f(x, n) = -x + sum(k = 0, n, x^(2^k)/(1+x^(2^k)));
sumeulerrat(f(1/p, 8))

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A382290(k).
Equal Sum_{k>=3} A088705(k) * P(k), where P(s) is the prime zeta function.
Equals Sum_{p prime} f(1/p), where f(x) = -x + Sum_{k>=0} x^(2^k)/(1+x^(2^k)).

A137148 a(n) = k*phi(k), where k is the n-th nonprime number.

Original entry on oeis.org

1, 8, 12, 32, 54, 40, 48, 84, 120, 128, 108, 160, 252, 220, 192, 500, 312, 486, 336, 240, 512, 660, 544, 840, 432, 684, 936, 640, 504, 880, 1080, 1012, 768, 2058, 1000, 1632, 1248, 972, 2200, 1344, 2052, 1624, 960, 1860, 2268, 2048, 3120, 1320, 2176, 3036

Offset: 1

Artur Jasinski, Jan 23 2008

Numbers that occur in A002618 but not in A036689.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002618, A018252, A036689, A065484, A136141.

a = {}; Do[If[!PrimeQ[n], AppendTo[a, n EulerPhi[n]]], {n, 1, 100}]; a

a(n) = A002618(A018252(n)). - R. J. Mathar, Jan 18 2021

Sum_{n>=1} 1/a(n) = A065484 - A136141 = 1.430699927388... . - Amiram Eldar, Oct 26 2024

A190117 a(n) = Sum_{k=1..n} k*k', where n' is the arithmetic derivative of n.

Original entry on oeis.org

0, 2, 5, 21, 26, 56, 63, 159, 213, 283, 294, 486, 499, 625, 745, 1257, 1274, 1652, 1671, 2151, 2361, 2647, 2670, 3726, 3976, 4366, 5095, 5991, 6020, 6950, 6981, 9541, 10003, 10649, 11069, 13229, 13266, 14064, 14688, 17408, 17449, 19171, 19214, 21326, 23081, 24231, 24278, 29654, 30340, 32590

Offset: 1

Giorgio Balzarotti, May 04 2011

nonn

			1*1' + 2*2' + 3*3' = 0 + 2 + 3 = 5 -> a(3) = 5.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Partial sums of A190116.

Cf. A003415, A136141.

der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2]):
seq(add(der(i)*i,i=1..n),n=1..50);

A003415[n_]:= If[Abs@n < 2, 0, n Total[#2/#1 & @@@FactorInteger[Abs@n]]];
Table[Sum[k*A003415[k], {k, 1, n}], {n, 1, 50}] (* G. C. Greubel, Dec 29 2017 *)

a(n) ~ c * n^3 / 3, where c = Sum_{p prime} 1/(p*(p-1)) = A136141. - Amiram Eldar, Jun 22 2025

A216419 Odd powers that are not prime powers.

Original entry on oeis.org

225, 441, 1089, 1225, 1521, 2025, 2601, 3025, 3249, 3375, 3969, 4225, 4761, 5625, 5929, 7225, 7569, 8281, 8649, 9025, 9261, 9801, 11025, 12321, 13225, 13689, 14161, 15129, 16641, 17689, 18225, 19881, 20449, 21025, 21609, 23409, 24025, 25281, 25921, 27225

Offset: 1

Arkadiusz Wesolowski, Sep 06 2012

nonn

Numbers in A075109 but not in A000961.

Also odd perfect powers having no primitive root (intersection of A075109 and A175594).

			81 = 9^2 as well as 81 = 3^4, therefore 81 is not a term.
225 can be expressed so in one way as (3*5)^2, therefore 225 is a term.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Primitive Root.

Cf. A008683, A075109, A136141, A175594.

[n : n in [3..27225 by 2] | IsPower(n) and EulerPhi(n) ne CarmichaelLambda(n)]; // Arkadiusz Wesolowski, Nov 09 2013

nn = 27500; lst = Union[Flatten[Table[n^i, {i, Prime[Range[PrimePi[Log[2, nn]]]]}, {n, 2, nn^(1/i)}]]]; Select[lst, OddQ[#] && ! IntegerQ@PrimitiveRoot[#] &] (* Most of the code is from T. D. Noe *)

Sum_{n>=1} 1/a(n) = 1/2 + Sum_{k>=2} mu(k)*(1-zeta(k)*(2^k-1)/2^k) - Sum_{p prime} 1/(p*(p-1)) = 0.0158808884... - Amiram Eldar, Dec 21 2020

A272531 Decimal expansion of C_2 (so named by S. Finch), a constant which is an analog of Niven's constant when mean of exponents is considered instead of maximum.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 02 2016

			0.1187309349576408430176668841155338623125786669962548878395960878789...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.2 Meissel-Mertens constants (pp. 94-95) and Section 2.6 Niven's constant p.112.

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 15.
Eric Weisstein's MathWorld, Niven's Constant

Cf. A033150, A077761, A085548, A136141.

digits = 100;
C1 = NSum[PrimeZetaP[n], {n, 2, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 2*digits] ;
M = EulerGamma - NSum[PrimeZetaP[n]/n, {n, 2, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 3*digits] ;
N0 = PrimeZetaP[2];
C2 = C1 (1 - M) - N0;
RealDigits[C2, 10, digits][[1]]

C_2 = C_1 (1 - M) - N, using Finch's notation, where C_1 is A136141, M A077761 and N A085548.

A355462 Powerful numbers divisible by exactly 2 distinct primes.

Original entry on oeis.org

36, 72, 100, 108, 144, 196, 200, 216, 225, 288, 324, 392, 400, 432, 441, 484, 500, 576, 648, 675, 676, 784, 800, 864, 968, 972, 1000, 1089, 1125, 1152, 1156, 1225, 1296, 1323, 1352, 1372, 1444, 1521, 1568, 1600, 1728, 1936, 1944, 2000, 2025, 2116, 2304, 2312, 2500

Offset: 1

Amiram Eldar, Jul 03 2022

nonn

First differs from A286708 at n = 25.
Number of the form p^i * q^j, where p != q are primes and i,j > 1.
Numbers k such that A001221(k) = 2 and A051904(k) >= 2.
The possible values of the number of the divisors (A000005) of terms in this sequence is any composite number that is not 8 or twice a prime (A264828 \ {1, 8}).
675 = 3^3*5^2 and 676 = 2^2*13^2 are 2 consecutive integers in this sequence. There are no other such pairs below 10^22 (the lesser members of such pairs are terms of A060355).

			36 is a term since 36 = 2^2 * 3^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index to sequences related to prime signature.

Intersection of A001694 and A007774.
Subsequence of A286708.
Subsequences: A085986, A143610, A162142, A179646, A179666, A179671, A179689, A179694, A179699, A179702, A179705, A189988, A189990, A189991, A190464, A190465, A303661.
Cf. A000005, A001221, A051904, A060355, A136141, A264828.

Select[Range[2500], Length[(e = FactorInteger[#][[;; , 2]])] == 2 && Min[e] > 1 &]

is(n) = {my(f=factor(n)); #f~ == 2 && vecmin(f[,2]) > 1};

Sum_{n>=1} 1/a(n) = ((Sum_{p prime} (1/(p*(p-1))))^2 - Sum_{p prime} (1/(p^2*(p-1)^2)))/2 = 0.1583860791... .

A382551 Decimal expansion of Sum_{p prime} 1/((p - 1)*p^3).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.14614660970928609293471078035798776047009787091714433668591512030391...

Index to constants which are prime zeta sums {3,1,0}.

Cf. A085541, A085548, A136141.

sumeulerrat(1/((p-1)*p^3)) \\ Amiram Eldar, Apr 01 2025

Equals -A085548 - A085541 + A136141.

Equals Sum_{k>=4} P(k), where P is the prime zeta function. - Amiram Eldar, Apr 01 2025

A382552 Decimal expansion of Sum_{p prime} 1/((p - 1)^2*p).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.60190832569884016005288567066649730356085862...

Index to constants which are prime zeta sums {1,2,0}.

Cf. A086242, A136141.

sumeulerrat(1/((p-1)^2*p)) \\ Amiram Eldar, Apr 01 2025

Equals -A136141 + A086242.

Equals Sum_{k>=3} (k-2) * P(k), where P is the prime zeta function. - Amiram Eldar, Apr 01 2025

A382584 Decimal expansion of Sum_{p prime} 1/((p - 1)^2p(p + 1)).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Mar 31 2025

			0.1902224771530221083141246173909492430368088328937868071588972676186916269020795654200305...

Index to constants which are prime zeta sums {1,2,1}.

Cf. A086242, A136141, A179119.

sumeulerrat(1/((p-1)^2*p*(p+1))) \\ Amiram Eldar, Apr 02 2025

Equals -3*A136141/4 + A086242/2 + A179119/4.

Equals Sum_{k>=2} (k-1) * (P(2*k) + P(2*k+1)), where P is the prime zeta function. - Amiram Eldar, Apr 02 2025

cp's OEIS Frontend

A354929 Numbers of the form 2*p^e, with p an odd prime and e >= 2.

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A382294 Decimal expansion of the asymptotic mean of the excess of the number of Fermi-Dirac factors of k over the number of distinct prime factors of k when k runs over the positive integers.

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A137148 a(n) = k*phi(k), where k is the n-th nonprime number.

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A190117 a(n) = Sum_{k=1..n} k*k', where n' is the arithmetic derivative of n.

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A216419 Odd powers that are not prime powers.

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Magma

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A272531 Decimal expansion of C_2 (so named by S. Finch), a constant which is an analog of Niven's constant when mean of exponents is considered instead of maximum.

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A355462 Powerful numbers divisible by exactly 2 distinct primes.

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A382551 Decimal expansion of Sum_{p prime} 1/((p - 1)*p^3).

A382584 Decimal expansion of Sum_{p prime} 1/((p - 1)^2p(p + 1)).